This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table.
1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).xq=0⟹(xr,yr)=(xp,yp).
+
1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:xq=0⟹(xr,yr)=(xp,yp).(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).
A 255-bit scalar from Fq. We decompose a full-width scalar α into 853-bit windows:
α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23).
-
The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255).
-If k0..84 is witnessed directly then no issue of canonicity arises. If the scalar is given as a base field element, then
-care must be taken to ensure a canonical representation, since 2255>p. This occurs, for example, in the scalar
-multiplication for the nullifier computation of the Action circuit.
At the point of witnessing the scalar, we range-constrain each 3-bit word of its decomposition.
-Degree9Constraintqwitness-scalar-fixed⋅range_check(ki,8)=0
-where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).
Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..85)[0..8) such that:
-
-
for the first 84 rows M[0..84)[0..8): M[w][k]=[(k+2)⋅(23)w]B
-
in the last row M[84][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B
-
-
The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑8323j+1.
-
-
Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0.
-In this case, the window table entries evaluate to the same point:
-
-
M[0][k0]=[(7+1)∗(23)0]B=[8]B,
-
M[1][k1]=[(0+1)∗(23)1]B=[8]B.
-
-
In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition.
-Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2).
-
-
For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..84):
-
-
Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e.
-Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..84);for w=84; and
-
Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square.
-
-
Repeating this for all 85 windows, we end up with:
-
-
an 85×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form
-Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7,
-where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and
-
a length-85 array Z of zw's.
-
-
We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.
Given a decomposed scalar α and a fixed base B, we compute [α]B as follows:
-
-
For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw].
-
Check that (xw,yw) is on the curve: yw2=xw3+b.
-
Witness uw such that yw+zw=uw2.
-
For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B.
-
For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result.
-
-
-
Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling.
This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where
-−(264−1)≤vold−vnew≤264−1
-
vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard).
-
Witness the sign s and magnitude m such that
-s∈{−1,1}m∈[0,264)vold−vnew=s⋅m
-
Then compute P=[m]V, and conditionally negate P using (x,y)↦(x,s⋅y).
-
We compute the window table in a similar way to full-width fixed-base scalar multiplication,
-but with only ceil(64/3)=22 windows:
-
αshort=k0+k1⋅(23)1+⋯+k21⋅(23)21,ki∈[0..23).
-
In addition to the 21 full-width 3-bit constraints, we have two additional constraints:
-
Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.
We support using a base field element as the scalar in fixed-base multiplication.
-This is used in
-DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition.
-
Decompose the base field element α into three-bit windows using a running sum z, where zi+1=(zi−ai)/23 for α=a0+23a1+...+23⋅84a84. (This running sum z is not to be confused with the Z array used to check the y-coordinate of a fixed-base window.)
-
We set z0=α, which means:
-z1z2z84z85=(α−k0)/23, (subtract the lowest 3 bits)=k1+23k2+...+23⋅83k84,=(z1−k1)/23=k2+23k3+...+23⋅82k84,⋯,=k84=(z84−k84)/23=0.
-
We must range-constrain the difference at each step of the running sum. We also constrain the final output of the running sum to be 0.
-Degree92Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅23,23)=0qdecompose-base-field⋅z85=0
-
We also enforce canonicity of this decomposition. That is, we want to check that 0≤α<p, where p the is Pallas base field modulus p=2254+tp=2254+45560315531419706090280762371685220353. Note that tp<2130.
+
The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255).
+
Degree9Constraintqscalar-fixed⋅(i=0∑7w−i)=0
+
We range-constrain each 3-bit word of the scalar decomposition using a polynomial range-check constraint:
+Degree9Constraintqdecompose-base-field⋅range_check(word,23)=0
+where range_check(word,range)=word⋅(1−word)⋯(range−1−word).
We support using a base field element as the scalar in fixed-base multiplication. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition.
+
Decompose the base field element α into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=85,K=3.
+
If k0..84 is witnessed directly then no issue of canonicity arises. However, because the scalar is given as a base field element here, care must be taken to ensure a canonical representation, since 2255>p. That is, we must check that 0≤α<p, where p the is Pallas base field modulus p=2254+tp=2254+45560315531419706090280762371685220353. Note that tp<2130.
To do this, we decompose α into three pieces: α=α0 (252 bits) ∣∣α1 (2 bits) ∣∣α2 (1 bit) .
We check the correctness of this decomposition by:
Degree532Constraintqcanon-base-field⋅range_check(α1,22)=0qcanon-base-field⋅range_check(α2,21)=0qcanon-base-field⋅(z84−(α1+α2⋅22))=0
@@ -271,8 +206,67 @@ If the MSB z44⟹alpha_0_hi_120=k44+23k45+⋯+23⋅(84−44)k84=z44−23⋅(84−44)k84=z44−23⋅(40)z84.
Then, we constrain bits 130..=131 of α0 to be zeroes; in other words, we constrain the three-bit word k43=α[129..=131]=α0[129..=131]∈{0,1}. We make use of the running sum decomposition to obtain k43=z43−z44⋅23.
-
Define α0′=α0+2130−tp. To check that 0≤α0′<2130, we use 13 ten-bit lookups, where we constrain the z13 running sum output of the lookup to be 0 if α2=1.
+
Define α0′=α0+2130−tp. To check that 0≤α0′<2130, we use 13 ten-bit lookups, where we constrain the z13 running sum output of the lookup to be 0 if α2=1.Degree3343Constraintqcanon-base-field⋅α2⋅α1=0qcanon-base-field⋅α2⋅alpha_0_hi_120=0qcanon-base-field⋅α2⋅k43⋅(1−k43)=0qcanon-base-field⋅α2⋅z13(lookup(α0′,13))=0Commentα2=1⟹α1=0Constrain α0 to be a 132-bit valueConstrain α0[130..=131] to 0α2=1⟹0≤α0′<2130
A short signed scalar is witnessed as a magnitude m and sign s such that
+s∈{−1,1}m∈[0,264)vold−vnew=s⋅m.
+
This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where
+−(264−1)≤vold−vnew≤264−1
+
vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard).
+
Decompose the magnitude m into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=22,K=3.
+
We have two additional constraints:
+Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.
Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..W)[0..8) such that:
+
+
for the first (W-1) rows M[0..(W−1))[0..8): M[w][k]=[(k+2)⋅(23)w]B
+
in the last row M[W−1][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B
+
+
The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑W−223j+1.
+
+
Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0.
+In this case, the window table entries evaluate to the same point:
+
+
M[0][k0]=[(7+1)∗(23)0]B=[8]B,
+
M[1][k1]=[(0+1)∗(23)1]B=[8]B.
+
+
In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition.
+Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2).
+
+
For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..(W−1)):
+
+
Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e.
+Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..(W−1));for w=84; and
+
Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square.
+
+
Repeating this for all W windows, we end up with:
+
+
an W×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form
+Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7,
+where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and
+
a length-W array Z of zw's.
+
+
We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.
Given a decomposed scalar α and a fixed base B, we compute [α]B as follows:
+
+
For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw].
+
Check that (xw,yw) is on the curve: yw2=xw3+b.
+
Witness uw such that yw+zw=uw2.
+
For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B.
+
For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result.
+
+
+
Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling.
Recall that the signed short exponent is witnessed as a 64−bit magnitude m, and a sign s∈1,−1. Using the above algorithm, we compute P=[m]B. Then, to get the final result P′, we conditionally negate P using (x,y)↦(x,s⋅y).
Note: this doesn't include the last row that uses complete addition. In the implementation this is allocated in a different region.
diff --git a/design/circuit/gadgets/ecc/var-base-scalar-mul.html b/design/circuit/gadgets/ecc/var-base-scalar-mul.html
index 9d25de51..917d8465 100644
--- a/design/circuit/gadgets/ecc/var-base-scalar-mul.html
+++ b/design/circuit/gadgets/ecc/var-base-scalar-mul.html
@@ -95,7 +95,7 @@
@@ -306,7 +306,7 @@ where
Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0
where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero.
We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.
+
We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.
This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Firstly, express α=k0+2K⋅k1+22K⋅k2+⋯, where each ki a K-bit value.
-
We initialize the running sum z0=s, and compute subsequent terms zi=2Kzi−1−ki−1. This gives us:
1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).xq=0⟹(xr,yr)=(xp,yp).
+
1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:xq=0⟹(xr,yr)=(xp,yp).(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).
A 255-bit scalar from Fq. We decompose a full-width scalar α into 853-bit windows:
α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23).
-
The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255).
-If k0..84 is witnessed directly then no issue of canonicity arises. If the scalar is given as a base field element, then
-care must be taken to ensure a canonical representation, since 2255>p. This occurs, for example, in the scalar
-multiplication for the nullifier computation of the Action circuit.
At the point of witnessing the scalar, we range-constrain each 3-bit word of its decomposition.
-Degree9Constraintqwitness-scalar-fixed⋅range_check(ki,8)=0
-where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).
Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..85)[0..8) such that:
-
-
for the first 84 rows M[0..84)[0..8): M[w][k]=[(k+2)⋅(23)w]B
-
in the last row M[84][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B
-
-
The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑8323j+1.
-
-
Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0.
-In this case, the window table entries evaluate to the same point:
-
-
M[0][k0]=[(7+1)∗(23)0]B=[8]B,
-
M[1][k1]=[(0+1)∗(23)1]B=[8]B.
-
-
In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition.
-Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2).
-
-
For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..84):
-
-
Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e.
-Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..84);for w=84; and
-
Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square.
-
-
Repeating this for all 85 windows, we end up with:
-
-
an 85×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form
-Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7,
-where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and
-
a length-85 array Z of zw's.
-
-
We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.
Given a decomposed scalar α and a fixed base B, we compute [α]B as follows:
-
-
For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw].
-
Check that (xw,yw) is on the curve: yw2=xw3+b.
-
Witness uw such that yw+zw=uw2.
-
For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B.
-
For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result.
-
-
-
Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling.
This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where
-−(264−1)≤vold−vnew≤264−1
-
vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard).
-
Witness the sign s and magnitude m such that
-s∈{−1,1}m∈[0,264)vold−vnew=s⋅m
-
Then compute P=[m]V, and conditionally negate P using (x,y)↦(x,s⋅y).
-
We compute the window table in a similar way to full-width fixed-base scalar multiplication,
-but with only ceil(64/3)=22 windows:
-
αshort=k0+k1⋅(23)1+⋯+k21⋅(23)21,ki∈[0..23).
-
In addition to the 21 full-width 3-bit constraints, we have two additional constraints:
-
Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.
We support using a base field element as the scalar in fixed-base multiplication.
-This is used in
-DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition.
-
Decompose the base field element α into three-bit windows using a running sum z, where zi+1=(zi−ai)/23 for α=a0+23a1+...+23⋅84a84. (This running sum z is not to be confused with the Z array used to check the y-coordinate of a fixed-base window.)
-
We set z0=α, which means:
-z1z2z84z85=(α−k0)/23, (subtract the lowest 3 bits)=k1+23k2+...+23⋅83k84,=(z1−k1)/23=k2+23k3+...+23⋅82k84,⋯,=k84=(z84−k84)/23=0.
-
We must range-constrain the difference at each step of the running sum. We also constrain the final output of the running sum to be 0.
-Degree92Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅23,23)=0qdecompose-base-field⋅z85=0
-
We also enforce canonicity of this decomposition. That is, we want to check that 0≤α<p, where p the is Pallas base field modulus p=2254+tp=2254+45560315531419706090280762371685220353. Note that tp<2130.
+
The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255).
+
Degree9Constraintqscalar-fixed⋅(i=0∑7w−i)=0
+
We range-constrain each 3-bit word of the scalar decomposition using a polynomial range-check constraint:
+Degree9Constraintqdecompose-base-field⋅range_check(word,23)=0
+where range_check(word,range)=word⋅(1−word)⋯(range−1−word).
We support using a base field element as the scalar in fixed-base multiplication. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition.
+
Decompose the base field element α into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=85,K=3.
+
If k0..84 is witnessed directly then no issue of canonicity arises. However, because the scalar is given as a base field element here, care must be taken to ensure a canonical representation, since 2255>p. That is, we must check that 0≤α<p, where p the is Pallas base field modulus p=2254+tp=2254+45560315531419706090280762371685220353. Note that tp<2130.
To do this, we decompose α into three pieces: α=α0 (252 bits) ∣∣α1 (2 bits) ∣∣α2 (1 bit) .
We check the correctness of this decomposition by:
Degree532Constraintqcanon-base-field⋅range_check(α1,22)=0qcanon-base-field⋅range_check(α2,21)=0qcanon-base-field⋅(z84−(α1+α2⋅22))=0
@@ -984,8 +919,67 @@ If the MSB z44⟹alpha_0_hi_120=k44+23k45+⋯+23⋅(84−44)k84=z44−23⋅(84−44)k84=z44−23⋅(40)z84.
Then, we constrain bits 130..=131 of α0 to be zeroes; in other words, we constrain the three-bit word k43=α[129..=131]=α0[129..=131]∈{0,1}. We make use of the running sum decomposition to obtain k43=z43−z44⋅23.
-
Define α0′=α0+2130−tp. To check that 0≤α0′<2130, we use 13 ten-bit lookups, where we constrain the z13 running sum output of the lookup to be 0 if α2=1.
+
Define α0′=α0+2130−tp. To check that 0≤α0′<2130, we use 13 ten-bit lookups, where we constrain the z13 running sum output of the lookup to be 0 if α2=1.Degree3343Constraintqcanon-base-field⋅α2⋅α1=0qcanon-base-field⋅α2⋅alpha_0_hi_120=0qcanon-base-field⋅α2⋅k43⋅(1−k43)=0qcanon-base-field⋅α2⋅z13(lookup(α0′,13))=0Commentα2=1⟹α1=0Constrain α0 to be a 132-bit valueConstrain α0[130..=131] to 0α2=1⟹0≤α0′<2130
A short signed scalar is witnessed as a magnitude m and sign s such that
+s∈{−1,1}m∈[0,264)vold−vnew=s⋅m.
+
This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where
+−(264−1)≤vold−vnew≤264−1
+
vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard).
+
Decompose the magnitude m into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=22,K=3.
+
We have two additional constraints:
+Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.
Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..W)[0..8) such that:
+
+
for the first (W-1) rows M[0..(W−1))[0..8): M[w][k]=[(k+2)⋅(23)w]B
+
in the last row M[W−1][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B
+
+
The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑W−223j+1.
+
+
Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0.
+In this case, the window table entries evaluate to the same point:
+
+
M[0][k0]=[(7+1)∗(23)0]B=[8]B,
+
M[1][k1]=[(0+1)∗(23)1]B=[8]B.
+
+
In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition.
+Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2).
+
+
For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..(W−1)):
+
+
Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e.
+Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..(W−1));for w=84; and
+
Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square.
+
+
Repeating this for all W windows, we end up with:
+
+
an W×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form
+Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7,
+where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and
+
a length-W array Z of zw's.
+
+
We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.
Given a decomposed scalar α and a fixed base B, we compute [α]B as follows:
+
+
For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw].
+
Check that (xw,yw) is on the curve: yw2=xw3+b.
+
Witness uw such that yw+zw=uw2.
+
For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B.
+
For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result.
+
+
+
Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling.
Recall that the signed short exponent is witnessed as a 64−bit magnitude m, and a sign s∈1,−1. Using the above algorithm, we compute P=[m]B. Then, to get the final result P′, we conditionally negate P using (x,y)↦(x,s⋅y).
We're trying to compute [α]T for α∈[0,q). Set k=α+tq and n=254. Then we can compute
[2254+(α+tq)]T=[2254+(α+tq)−(2254+tq)]T=[α]T
provided that α+tq∈[0,2n+1), i.e. α<2n+1−tq which covers the whole range we need because in fact 2255−tq>q.
@@ -1132,7 +1126,7 @@ where
Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0
where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero.
We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.
+
We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.
This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Firstly, express α=k0+2K⋅k1+22K⋅k2+⋯, where each ki a K-bit value.
-
We initialize the running sum z0=s, and compute subsequent terms zi=2Kzi−1−ki−1. This gives us:
This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table.
diff --git a/searchindex.js b/searchindex.js
index 5d486371..5e4a5b4b 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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: This library is being actively developed and should not be used in production software. Requires Rust 1.51+.","breadcrumbs":"orchard Crates.io","id":"0","title":"orchard Crates.io"},"1":{"body":"The Orchard Book Crate documentation","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"","breadcrumbs":"Design","id":"10","title":"Design"},"11":{"body":"","breadcrumbs":"General design notes","id":"11","title":"General design notes"},"12":{"body":"Keep the design close to Sapling, while eliminating aspects we don't like.","breadcrumbs":"Requirements","id":"12","title":"Requirements"},"13":{"body":"Delegated proving with privacy from the prover. We know how to do this, but it would require a discrete log equality proof, and the most efficient way to do this would be to do RedDSA and this at the same time, which means more work for e.g. hardware wallets.","breadcrumbs":"Non-requirements","id":"13","title":"Non-requirements"},"14":{"body":"Should we have one memo per output, or one memo per transaction, or 0..n memos? Variable, or (1 or n), is a potential privacy leak. Need to consider the privacy issue related to light clients requesting individual memos vs being able to fetch all memos.","breadcrumbs":"Open issues","id":"14","title":"Open issues"},"15":{"body":"TODO: UDAs: arbitrary vs whitelisted","breadcrumbs":"Note structure","id":"15","title":"Note structure"},"16":{"body":"For Sapling, we have encountered multiple places where the specification uses typed variables to define the consensus rules, but the C++ implementation in zcashd relied on byte encodings to implement them. This resulted in subtly-different consensus rules being deployed than were intended, for example where a particular type was not round-trip encodable. In Orchard, we avoid this by defining the consensus rules in terms of the byte encodings of all variables, and being explicit about any types that are not round-trip encodable. This makes consensus compatibility between strongly-typed implementations (such as this crate) and byte-oriented implementations easier to achieve.","breadcrumbs":"Typed variables vs byte encodings","id":"16","title":"Typed variables vs byte encodings"},"17":{"body":"Orchard keys and payment addresses are structurally similar to Sapling. The main change is that Orchard keys use the Pallas curve instead of Jubjub, in order to enable the future use of the Pallas-Vesta curve cycle in the Orchard protocol. (We already use Vesta as the curve on which Halo 2 proofs are computed, but this doesn't yet require a cycle.) Using the Pallas curve and making the most efficient use of the Halo 2 proof system involves corresponding changes to the key derivation process, such as using Sinsemilla for Pallas-efficient commitments. We also take the opportunity to remove all uses of expensive general-purpose hashes (such as BLAKE2s) from the circuit. We make several structural changes, building on the lessons learned from Sapling: The nullifier private key nsk is removed. Its purpose in Sapling was as defense-in-depth, in case RedDSA was found to have weaknesses; an adversary who could recover ask would not be able to spend funds. In practice it has not been feasible to manage nsk much more securely than a full viewing key, as the computational power required to generate Sapling proofs has made it necessary to perform this step on the same device that is creating the overall transaction (rather than on a more constrained device like a hardware wallet). We are also more confident in RedDSA now. nk is now a field element instead of a curve point, making it more efficient to generate nullifiers. ovk is now derived from fvk, instead of being derived in parallel. This places it in a similar position within the key structure to ivk, and also removes an issue where two full viewing keys could be constructed that have the same ivk but different ovks. Users still have control over whether ovk is used when constructing a transaction. All diversifiers now result in valid payment addresses, due to group hashing into Pallas being specified to be infallible. This removes significant complexity from the use cases for diversified addresses. The fact that Pallas is a prime-order curve simplifies the protocol and removes the need for cofactor multiplication in key agreement. Unlike Sapling, we define public (including ephemeral) and private keys used for note encryption to exclude the zero point and the zero scalar. Without this change, the implementation of the Orchard Action circuit would need special cases for the zero point, since Pallas is a short Weierstrass rather than an Edwards curve. This also has the advantage of ensuring that the key agreement has \"contributory behaviour\" — that is, if either party contributes a random scalar, then the shared secret will be random to an observer who does not know that scalar and cannot break Diffie–Hellman. Other than the above, Orchard retains the same design rationale for its keys and addresses as Sapling. For example, diversifiers remain at 11 bytes, so that a raw Orchard address is the same length as a raw Sapling address. Orchard payment addresses do not have a stand-alone string encoding. Instead, we define \"unified addresses\" that can bundle together addresses of different types, including Orchard. Unified addresses have a Human-Readable Part of \"u\" on Mainnet, i.e. they will have the prefix \"u1\". For specifications of this and other formats (e.g. for Orchard viewing and spending keys), see section 5.6.4 of the NU5 protocol specification [#NU5-orchardencodings].","breadcrumbs":"Design » Keys and addresses","id":"17","title":"Keys and addresses"},"18":{"body":"When designing Sapling, we defined a BIP 32 -like mechanism for generating hierarchical deterministic wallets in ZIP 32 . We decided at the time to stick closely to the design of BIP 32, on the assumption that there were Bitcoin use cases that used both hardened and non-hardened derivation that we might not be aware of. This decision created significant complexity for Sapling: we needed to handle derivation separately for each component of the expanded spending key and full viewing key (whereas for transparent addresses there is only a single component in the spending key). Non-hardened derivation enables creating a multi-level path of child addresses below some parent address, without involving the parent spending key. The primary use case for this is HD wallets for transparent addresses, which use the following structure defined in BIP 44 : (H) BIP 44 (H) Coin type: Zcash (H) Account 0 (N) Normal addresses (N) Address 0 (N) Address 1... (N) Change addresses (N) Change address 0 (N) Change address 1... (H) Account 1... Shielded accounts do not require separating change addresses from normal addresses, because addresses are not revealed in transactions. Similarly, there is also no need to generate a fresh spending key for every transaction, and in fact this would cause a linear slow-down in wallet scanning. But for users who do want to generate multiple addresses per account, they can generate the following structure, which does not use non-hardened derivation: (H) ZIP 32 (H) Coin type: Zcash (H) Account 0 Diversified address 0 Diversified address 1... (H) Account 1... Non-hardened derivation is therefore only required for use-cases that require the ability to derive more than one child layer of addresses. However, in the years since Sapling was deployed, we have not seen any such use cases appear. Therefore, for Orchard we only define hardened derivation, and do so with a much simpler design than ZIP 32. All derivations produce an opaque binary spending key, from which the keys and addresses are then derived. As a side benefit, this makes key formats shorter. (The formats that will actually be used in practice for Orchard will correspond to the simpler Sapling formats in the protocol specification, rather than the longer and more complicated \"extended\" ones defined by ZIP 32.)","breadcrumbs":"Design » Hierarchical deterministic wallets","id":"18","title":"Hierarchical deterministic wallets"},"19":{"body":"In Sprout, we had a single proof that represented two spent notes and two new notes. This was necessary in order to faciliate spending multiple notes in a single transaction (to balance value, an output of one JoinSplit could be spent in the next one), but also provided a minimal level of arity-hiding: single-JoinSplit transactions all looked like 2-in 2-out transactions, and in multi-JoinSplit transactions each JoinSplit looked like a 1-in 1-out. In Sapling, we switched to using value commitments to balance the transaction, removing the min-2 arity requirement. We opted for one proof per spent note and one (much simpler) proof per output note, which greatly improved the performance of generating outputs, but removed any arity-hiding from the proofs (instead having the transaction builder pad transactions to 1-in, 2-out). For Orchard, we take a combined approach: we define an Orchard transaction as containing a bundle of actions, where each action is both a spend and an output. This provides the same inherent arity-hiding as multi-JoinSplit Sprout, but using Sapling value commitments to balance the transaction without doubling its size. TODO: Depending on the circuit cost, we may switch to having an action internally represent either a spend or an output. Externally spends and outputs would still be indistinguishable, but the transaction would be larger.","breadcrumbs":"Design » Actions","id":"19","title":"Actions"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Bootstrap Open Source Licence, version 1.0, or at your option, any later version. See the file LICENSE-BOSL for the terms of the Bootstrap Open Source Licence, version 1.0. The purpose of the BOSL is to allow commercial improvements to the package while ensuring that all improvements are open source. See here for why the BOSL exists.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"TODO: One memo per tx vs one memo per output","breadcrumbs":"Design » Memo fields","id":"20","title":"Memo fields"},"21":{"body":"As in Sapling, we require two kinds of commitment schemes in Orchard: HomomorphicCommit is a linearly homomorphic commitment scheme with perfect hiding, and strong binding reducible to DL. Commit and ShortCommit are commitment schemes with perfect hiding, and strong binding reducible to DL. By \"strong binding\" we mean that the scheme is collision resistant on the input and randomness. We instantiate HomomorphicCommit with a Pedersen commitment, and use it for value commitments: cv=HomomorphicCommitrcvcv(v) We instantiate Commit and ShortCommit with Sinsemilla, and use them for all other commitments: ivk=ShortCommitrivkivk(ak,nk) cm=Commitrcmcm(rest of note) This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient Sinsemilla instead of Bowe--Hopwood Pedersen hashes. Note that for ivk, we also deviate from Sapling in two ways: We use ShortCommit to derive ivk instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit, at the cost of requiring rivk to be part of the full viewing key. We define ivk as an integer in [1,qP); that is, we exclude ivk=0. For Sapling, we relied on BLAKE2s to make ivk=0 infeasible to produce, but it was still technically possible. For Orchard, we get this by construction: 0 is not a valid x-coordinate for any Pallas point. SinsemillaShortCommit internally maps points to field elements by replacing the identity (which has no affine coordinates) with 0. But SinsemillaCommit is defined using incomplete addition, and thus will never produce the identity.","breadcrumbs":"Design » Commitments","id":"21","title":"Commitments"},"22":{"body":"The commitment tree structure for Orchard is identical to Sapling: A single global commitment tree of fixed depth 32. Note commitments are appended to the tree in-order from the block. Valid Orchard anchors correspond to the global tree state at block boundaries (after all commitments from a block have been appended, and before any commitments from the next block have been appended). The only difference is that we instantiate MerkleCRHOrchard with Sinsemilla (whereas MerkleCRHSapling used a Bowe--Hopwood Pedersen hash).","breadcrumbs":"Design » Commitment tree","id":"22","title":"Commitment tree"},"23":{"body":"The fixed-depth incremental Merkle trees that we use (in Sprout and Sapling, and again in Orchard) require specifying an \"empty\" or \"uncommitted\" leaf - a value that will never be appended to the tree as a regular leaf. For Sprout (and trees composed of the outputs of bit-twiddling hash functions), we use the all-zeroes array; the probability of a real note having a colliding note commitment is cryptographically negligible. For Sapling, where leaves are u-coordinates of Jubjub points, we use the value 1 which is not the u-coordinate of any Jubjub point. Orchard note commitments are the x-coordinates of Pallas points; thus we take the same approach as Sapling, using a value that is not the x-coordinate of any Pallas point as the uncommitted leaf value. We use the value 2 for both Pallas and Vesta, because 23+5 is not a square in either Fp or Fq: sage: p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001\nsage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: EllipticCurve(GF(p), [0, 5]).count_points() == q\nTrue\nsage: EllipticCurve(GF(q), [0, 5]).count_points() == p\nTrue\nsage: Mod(13, p).is_square()\nFalse\nsage: Mod(13, q).is_square()\nFalse Note: There are also no Pallas points with x-coordinate 0, but we map the identity to (0,0) within the circuit. Although SinsemillaCommit cannot return the identity (the incomplete addition would return ⊥ instead), it would arguably be confusing to rely on that.","breadcrumbs":"Design » Uncommitted leaves","id":"23","title":"Uncommitted leaves"},"24":{"body":"We considered splitting the commitment tree into several sub-trees: Bundle tree, that accumulates the commitments within a single bundle (and thus a single transaction). Block tree, that accumulates the bundle tree roots within a single block. Global tree, that accumulates the block tree roots. Each of these trees would have had a fixed depth (necessary for being able to create proofs). Chains that integrated Orchard could have decoupled the limits on commitments-per-subtree from higher-layer constraints like block size, by enabling their blocks and transactions to be structured internally as a series of Orchard blocks or txs (e.g. a Zcash block would have contained a Vec, that each were appended in-order). The motivation for considering this change was to improve the lives of light client wallet developers. When a new note is received, the wallet derives its incremental witness from the state of the global tree at the point when the note's commitment is appended; this incremental state then needs to be updated with every subsequent commitment in the block in-order. Wallets can't get help from the server to create these for new notes without leaking the specific note that was received. We decided that this was too large a change from Sapling, and that it should be possible to improve the Incremental Merkle Tree implementation to work around the efficiency issues without domain-separating the tree.","breadcrumbs":"Design » Considered alternatives","id":"24","title":"Considered alternatives"},"25":{"body":"The nullifier design we use for Orchard is nf=ExtractP([(Fnk(ρ)+ψ)modp]G+cm), where: F is a keyed circuit-efficient PRF (such as Rescue or Poseidon). ρ is unique to this output. As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. Given that an action consists of a single spend and a single output, we set ρ to be the nullifier of the spent note. ψ is sender-controlled randomness. It is not required to be unique, and in practice is derived from both ρ and a sender-selected random value rseed: ψ=KDFψ(ρ,rseed). G is a fixed independent base. ExtractP extracts the x-coordinate of a Pallas curve point. This gives a note structure of (addr,v,ρ,ψ,rcm). The note plaintext includes rseed in place of ψ and rcm, and omits ρ (which is a public part of the action).","breadcrumbs":"Design » Nullifiers","id":"25","title":"Nullifiers"},"26":{"body":"We care about several security properties for our nullifiers: Balance: can I forge money? Note Privacy: can I gain information about notes only from the public block chain? This describes notes sent in-band. Note Privacy (OOB): can I gain information about notes sent out-of-band, only from the public block chain? In this case, we assume privacy of the channel over which the note is sent, and that the adversary does not have access to any notes sent to the same address which are then spent (so that the nullifier is on the block chain somewhere). Spend Unlinkability: given the incoming viewing key for an address, and not the full viewing key, can I (possibly the sender) detect spends of any notes sent to that address? We're giving ivk to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address. Faerie Resistance: can I perform a Faerie Gold attack (i.e. cause notes to be accepted that are unspendable)? We're giving the full viewing key to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address, and be able to derive every nullifier. We assume (and instantiate elsewhere) the following primitives: GH is a cryptographic hash into the group (such as BLAKE2s with simplified SWU), used to derive all fixed independent bases. E is an elliptic curve (such as Pallas). KDF is the note encryption key derivation function. For our chosen design, our desired security properties rely on the following assumptions: BalanceNote PrivacyNote Privacy (OOB)Spend UnlinkabilityFaerie ResistanceDLEHashDHEKDFNear perfect‡DDHE†∨PRFFDLE HashDHEKDF is computational Diffie-Hellman using KDF for the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker than DDHE but stronger than DLE. We omit ROGH as a security assumption because we only rely on the random oracle applied to fixed inputs defined by the protocol, i.e. to generate the fixed base G, not to attacker-specified inputs. † We additionally assume that for any input x, {Fnk(x):nk∈E} gives a scalar in an adequate range for DDHE. (Otherwise, F could be trivial, e.g. independent of nk.) ‡ Statistical distance <2−167.8 from perfect.","breadcrumbs":"Design » Security properties","id":"26","title":"Security properties"},"27":{"body":"⚠ Caution: be skeptical of the claims in this table about what problem(s) each security property depends on. They may not be accurate and are definitely not fully rigorous. The entries in this table omit the application of ExtractP, which is an optimization to halve the nullifier length. That optimization requires its own security analysis, but because it is a deterministic mapping, only Faerie Resistance could be affected by it. nf[nk][θ]H[nk]H+[rnf]IHash([nk][θ]H)Hash([nk]H+[rnf]I)[Fnk(ψ)][θ]H[Fnk(ψ)]H+[rnf]I[Fnk(ψ)]G+[θ]H[Fnk(ψ)]H+cm[Fnk(ρ,ψ)]G+cm[Fnk(ρ)]G+cm[Fnk(ρ,ψ)]Gv+[rnf]I[Fnk(ρ)]Gv+[rnf]I[(Fnk(ρ)+ψ)modp]Gv[Fnk(ρ,ψ)]G+Commitrnfnf(v,ρ)[Fnk(ρ)]G+Commitrnfnf(v,ρ)Note(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,rnf,ψ,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,ψ,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)BalanceDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLENote PrivacyHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFNote Priv OOBPerfectPerfectPerfectPerfectPerfectPerfectPerfectDDHE†DDHE†DDHE†PerfectPerfectNear perfect‡PerfectPerfectSpend UnlinkabilityDDHEDDHEDDHE∨PreHashDDHE∨PreHashDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFFaerie ResistanceROGH∧DLEROGH∧DLECollHash∧ROGH∧DLECollHash∧ROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEDLEDLECollF∧DLECollF∧DLEbrokenDLEDLEReason not to useNo SU for DL-breakingNo SU for DL-breakingCollHash for FRCollHash for FRPerf. (2 var-base)Perf. (1 var+1 fix-base)Perf. (1 var+1 fix-base)NP(OOB) not perfectNP(OOB) not perfectNP(OOB) not perfectCollF for FRCollF for FRbroken for FRPerf. (2 fix-base)Perf. (2 fix-base) In the above alternatives: Hash is a keyed circuit-efficient hash (such as Rescue). I is an fixed independent base, independent of G and any others returned by GH. Gv is a pair of fixed independent bases (independent of all others), where the specific choice of base depends on whether the note has zero value. H is a base unique to this output. For non-zero-valued notes, H=GH(ρ). As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. For zero-valued notes, H is constrained by the circuit to a fixed base independent of I and any others returned by GH.","breadcrumbs":"Design » Considered alternatives","id":"27","title":"Considered alternatives"},"28":{"body":"In order to satisfy the Balance security property, we require that the circuit must be able to enforce that only one nullifier is accepted for a given note. As in Sprout and Sapling, we achieve this by ensuring that the nullifier deterministically depends only on values committed to (directly or indirectly) by the note commitment. As in Sapling, this involves arguing that: There can be only one ivk for a given addr. This is true because the circuit checks that pkd=[ivk]gd, and the mapping ivk↦[ivk]gd is an injection for any gd. (ivk is in the base field of E, which must be smaller than its scalar field, as is the case for Pallas.) There can be only one nk for a given ivk. This is true because the circuit checks that ivk=ShortCommitrivkivk(ak,nk) where ShortCommit is binding (see Commitments ).","breadcrumbs":"Design » Rationale","id":"28","title":"Rationale"},"29":{"body":"Faerie Resistance requires that nullifiers be unique. This is primarily achieved by taking a unique value (checked for uniqueness by the public consensus rules) as an input to the nullifier. However, it is also necessary to ensure that the transformations applied to this value preserve its uniqueness. Meanwhile, to achieve Spend Unlinkability , we require that the nullifier does not reveal any information about the unique value it is derived from. The design alternatives fall into two categories in terms of how they balance these requirements: Publish a unique value ρ at note creation time, and blind that value within the nullifier computation. This is similar to the approach taken in Sprout and Sapling, which both implemented nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while Sapling uses BLAKE2s. Derive a unique base H from some unique value, publish that unique base at note creation time, and then blind the base (either additively or multiplicatively) during nullifier computation. For Spend Unlinkability , the only value unknown to the adversary is nk, and the cryptographic assumptions only involve the first term (other terms like cm or [rnf]I cannot be extracted directly from the observed nullifiers, but can be subtracted from them). We therefore ensure that the first term does not commit directly to the note (to avoid a DL-breaking adversary from immediately breaking SU ). We were considering using a design involving H with the goal of eliminating all usages of a PRF inside the circuit, for two reasons: Instantiating PRFF with a traditional hash function is expensive in the circuit. We didn't want to solely rely on an algebraic hash function satisfying PRFF to achieve Spend Unlinkability . However, those designs rely on both ROGH and DLE for Faerie Resistance , while still requiring DDHE for Spend Unlinkability . (There are two designs for which this is not the case, but they rely on DDHE† for Note Privacy (OOB) which was not acceptable). By contrast, several designs involving ρ (including the chosen design) have weaker assumptions for Faerie Resistance (only relying on DLE), and Spend Unlinkability does not require PRFF to hold: they can fall back on the same DDHE assumption as the H designs (along with an additional assumption about the output of F which is easily satisfied).","breadcrumbs":"Design » Use of ρ","id":"29","title":"Use of ρ"},"3":{"body":"","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"Most of the designs include either a multiplicative blinding term [θ]H, or an additive blinding term [rnf]I, in order to achieve perfect Note Privacy (OOB) (to an adversary who does not know the note). The chosen design is effectively using [ψ]G for this purpose; a DL-breaking adversary only learns Fnk(ρ)+ψ(modp). This reduces Note Privacy (OOB) from perfect to statistical, but given that ψ is from a distribution statistically close to uniform on [0,q), this is statistically close to better than 2−128. The benefit is that it does not require an additional scalar multiplication, making it more efficient inside the circuit. ψ's derivation has two motivations: Deriving from a random value rseed enables multiple derived values to be conveyed to the recipient within an action (such as the ephemeral secret esk, per ZIP 212 ), while keeping the note plaintext short. Mixing ρ into the derivation ensures that the sender can't repeat ψ across two notes, which could have enabled spend linkability attacks in some designs. The note that is committed to, and which the circuit takes as input, only includes ψ (i.e. the circuit does not check the derivation from rseed). However, an adversarial sender is still constrained by this derivation, because the recipient recomputes ψ during note decryption. If an action were created using an arbitrary ψ (for which the adversary did not have a corresponding rseed), the recipient would derive a note commitment that did not match the action's commitment field, and reject it (as in Sapling).","breadcrumbs":"Design » Use of ψ","id":"30","title":"Use of ψ"},"31":{"body":"The nullifier commits to the note value via cm for two reasons: It domain-separates nullifiers for zero-valued notes from other notes. This is necessary because we do not require zero-valued notes to exist in the commitment tree. Designs that bind the nullifier to Fnk(ρ) require CollF to achieve Faerie Resistance (and similarly where Hash is applied to a value derived from H). Adding cm to the nullifier avoids this assumption: all of the bases used to derive cm are fixed and independent of G, and so the nullifier can be viewed as a Pedersen hash where the input includes ρ directly. The Commitnf variants were considered to avoid directly depending on cm (which in its native type is a base field element, not a group element). We decided instead to follow Sapling by defining an intermediate representation of cm as a group element, that is only used in nullifier computation. The circuit already needs to compute cm, so this improves performance by removing We also considered variants that used a choice of fixed bases Gv to provide domain separation for zero-valued notes. The most performant design (similar to the chosen design) does not achieve Faerie Resistance for an adversary that knows the recipient's full viewing key (ψ could be brute-forced to cancel out Fnk(ρ), causing a collision), and the other variants require assuming CollF as mentioned above.","breadcrumbs":"Design » Use of cm","id":"31","title":"Use of cm"},"32":{"body":"Orchard signatures are an instantiation of RedDSA with a cofactor of 1. TODO: Should it be possible to sign partial transactions? If we're going to merge down all the signatures into a single one, and also want this, we need to ensure there's a feasible MPC.","breadcrumbs":"Design » Signatures","id":"32","title":"Signatures"},"33":{"body":"","breadcrumbs":"Design » Circuit","id":"33","title":"Circuit"},"34":{"body":"","breadcrumbs":"Design » Circuit » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"We will use formulae for curve arithmetic using affine coordinates on short Weierstrass curves, derived from section 4.1 of Hüseyin Hışıl's thesis . Inputs: P=(xp,yp),Q=(xq,yq) Output: R=P⸭Q=(xr,yr) The formulae from Hışıl's thesis are: x3=(x1−x2y1−y2)2−x1−x2 y3=x1−x2y1−y2⋅(x1−x3)−y1 Rename: (x1,y1) to (xq,yq) (x2,y2) to (xp,yp) (x3,y3) to (xr,yr). Let λ=xq−xpyq−yp=xp−xqyp−yq, which we implement as λ⋅(xp−xq)=yp−yq Also, xr=λ2−xq−xp yr=λ⋅(xq−xr)−yq which is equivalent to xr+xq+xp=λ2 Assuming xp=xq, ⟹⟹⟹(xr+xq+xp)⋅(xp−xq)2(xr+xq+xp)⋅(xp−xq)2yryr+yq(yr+yq)⋅(xp−xq)=====λ2⋅(xp−xq)2(λ⋅(xp−xq))2λ⋅(xq−xr)−yqλ⋅(xq−xr)λ⋅(xp−xq)⋅(xq−xr) Substituting for λ⋅(xp−xq), we get the constraints: (xr+xq+xp)⋅(xp−xq)2−(yp−yq)2=0 Note that this constraint is unsatisfiable for P⸭(−P) (when P=O), and so cannot be used with arbitrary inputs. (yr+yq)(xp−xq)−(yp−yq)(xq−xr)=0","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Incomplete addition","id":"35","title":"Incomplete addition"},"36":{"body":"OO(xp,yp)(x,y)(x,y)(xp,yp)++++++O(xq,yq)O(x,y)(x,−y)(xq,yq)=O=(xq,yq)=(xp,yp)=[2](x,y)=O=(xp,yp)⸭(xq,yq),if xp=xq Suppose that we represent O as (0,0). (0 is not an x-coordinate of a valid point because we would need y2=x3+5, and 5 is not square in Fq. Also 0 is not a y-coordinate of a valid point because −5 is not a cube in Fq.) P+Q(xp,yp)+(xq,yq)λxryr=R=(xr,yr)=xq−xpyq−yp=λ2−xp−xq=λ(xp−xr)−yp For the doubling case, Hışıl's thesis tells us that λ has to instead be computed as 2y3x2. Define inv0(x)={0,1/x,if x=0otherwise. Witness α,β,γ,δ,λ where: α=inv0(xq−xp) β=inv0(xp) γ=inv0(xq) δ={inv0(yq+yp),0,if xq=xpotherwise λ=⎩⎨⎧xq−xpyq−yp,2yp3xp20,if xq=xpif xq=xp∧yp=0otherwise.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Complete addition","id":"36","title":"Complete addition"},"37":{"body":"Degree456666444444Constraintqadd⋅(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))qadd⋅(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅(1−xp⋅β)⋅(xr−xq)qadd⋅(1−xp⋅β)⋅(yr−yq)qadd⋅(1−xq⋅γ)⋅(xr−xp)qadd⋅(1−xq⋅γ)⋅(yr−yp)qadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xrqadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr============000000000000Meaningxq=xp⟹λ=xq−xpyq−yp{xq=xp∧yp=0⟹λ=2yp3xp2xq=xp∧yp=0⟹xp=0xp=0∧xq=0∧xq=xp⟹xr=λ2−xp−xqxp=0∧xq=0∧xq=xp⟹yr=λ⋅(xp−xr)−ypxp=0∧xq=0∧yq=−yp⟹xr=λ2−xp−xqxp=0∧xq=0∧yq=−yp⟹yr=λ⋅(xp−xr)−ypxp=0⟹xr=xqxp=0⟹yr=yqxq=0⟹xr=xpxq=0⟹yr=ypxq=xp∧yq=−yp⟹xr=0xq=xp∧yq=−yp⟹yr=0 Max degree: 6","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraints","id":"37","title":"Constraints"},"38":{"body":"1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).xq=0⟹(xr,yr)=(xp,yp). Propositions: (1)(2)(3)(4)(5)(6)xq=xp⟹λ=(yq−yp)/(xq−xp).(xq=xp)∧yp=0⟹λ=3xp2/2yp(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))⟹(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp)xp=0⟹(xr,yr)=(xq,yq)xq=0⟹(xr,yr)=(xp,yp)xq=xp∧yq=−yp⟹(xr,yr)=(0,0) Test cases: (xp,yp)+(xq,yq)=(xr,yr) (0,0)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because yp=0 holds because xp=0 holds because xp=0 only when xr=0,yr=0 holds because xq=0 only when xr=0,yr=0 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (x,y)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xq=0 holds because xp=0 because 0 is not a valid x-coordinate only when β=xp−1 holds because xq=0 only when (xr,yr)=(xp,yp) holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xp,yp) is the only solution (0,0)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xp=0 holds because xp=0 only when (xr,yr)=(xq,yq) holds because xq=0 because 0 is not a valid x-coordinate only when γ=xq−1 holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xq,yq) is the only solution (x,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate) only when λ=3xp2/2yp holds because xp=0∧xq=0andyq=−yp only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xp=0 only when γ=xq−1 holds because yq=−yp only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution (x,y)+(x,−y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate)only when λ=3xp2/2yp holds because xp=0∧xq=0 but yq=−yp∧xq=xp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (ζx,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xp=xp only when α=(xq−xp)−1 which is defined because xq=xp holds because (xp=0)∧(xq=0)∧(xq=xp) only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp only when δ=0 Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution All remaining cases (x,y)+(x′,y′) are identical to the case (ζx,y)+(x,y) when λαβγδ=(yq−yp)/(xq−xp)=(xq−xp)−1=xp−1=xq−1=0 because in all remaining cases, xq=xp,xp=0,xq=0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Analysis of constraints","id":"38","title":"Analysis of constraints"},"39":{"body":"There are 6 fixed bases in the Orchard protocol: KOrchard, used in deriving the nullifier; GOrchard, used in spend authorization; R base for NoteCommitOrchard; V and R bases for ValueCommitOrchard; and R base for Commitivk.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"39","title":"Fixed-base scalar multiplication"},"4":{"body":"","breadcrumbs":"Concepts » Preliminaries","id":"4","title":"Preliminaries"},"40":{"body":"In most cases, we multiply the fixed bases by 255−bit scalars from Fq. We decompose a full-width scalar α into 85 3-bit windows: α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23). The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255). If k0..84 is witnessed directly then no issue of canonicity arises. If the scalar is given as a base field element, then care must be taken to ensure a canonical representation, since 2255>p. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit. Degree9Constraintqscalar-fixed⋅1⋅(i=0∑7w−i)=0 At the point of witnessing the scalar, we range-constrain each 3-bit word of its decomposition. Degree9Constraintqwitness-scalar-fixed⋅range_check(ki,8)=0 where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"40","title":"Witness scalar"},"41":{"body":"Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..85)[0..8) such that: for the first 84 rows M[0..84)[0..8): M[w][k]=[(k+2)⋅(23)w]B in the last row M[84][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑8323j+1. Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0. In this case, the window table entries evaluate to the same point: M[0][k0]=[(7+1)∗(23)0]B=[8]B, M[1][k1]=[(0+1)∗(23)1]B=[8]B. In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition. Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2). For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..84): Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e. Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..84);for w=84; and Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square. Repeating this for all 85 windows, we end up with: an 85×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7, where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and a length-85 array Z of zw's. We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Load fixed base","id":"41","title":"Load fixed base"},"42":{"body":"Given a decomposed scalar α and a fixed base B, we compute [α]B as follows: For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw]. Check that (xw,yw) is on the curve: yw2=xw3+b. Witness uw such that yw+zw=uw2. For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B. For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result. Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling. Constraints: Degree843Constraintqmul-fixed⋅(Lx[w](kw)−xw)=0qmul-fixed⋅(yw2−xw3−b)=0qmul-fixed⋅(uw2−yw−Z[w])=0 where b=5 (from the Pallas curve equation).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"42","title":"Fixed-base scalar multiplication"},"43":{"body":"This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where −(264−1)≤vold−vnew≤264−1 vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard). Witness the sign s and magnitude m such that s∈{−1,1}m∈[0,264)vold−vnew=s⋅m Then compute P=[m]V, and conditionally negate P using (x,y)↦(x,s⋅y). We compute the window table in a similar way to full-width fixed-base scalar multiplication, but with only ceil(64/3)=22 windows: αshort=k0+k1⋅(23)1+⋯+k21⋅(23)21,ki∈[0..23). In addition to the 21 full-width 3-bit constraints, we have two additional constraints: Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication with signed short exponent","id":"43","title":"Fixed-base scalar multiplication with signed short exponent"},"44":{"body":"We support using a base field element as the scalar in fixed-base multiplication. This is used in DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition. Decompose the base field element α into three-bit windows using a running sum z, where zi+1=(zi−ai)/23 for α=a0+23a1+...+23⋅84a84. (This running sum z is not to be confused with the Z array used to check the y-coordinate of a fixed-base window.) We set z0=α, which means: z1z2z84z85=(α−k0)/23, (subtract the lowest 3 bits)=k1+23k2+...+23⋅83k84,=(z1−k1)/23=k2+23k3+...+23⋅82k84,⋯,=k84=(z84−k84)/23=0. We must range-constrain the difference at each step of the running sum. We also constrain the final output of the running sum to be 0. Degree92Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅23,23)=0qdecompose-base-field⋅z85=0 We also enforce canonicity of this decomposition. That is, we want to check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq. (We use big-endian bit order for convenient input into the variable-base scalar multiplication algorithm.) k=k254⋅2254+k253⋅2253+⋯+k0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"47","title":"Witness scalar"},"48":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication","id":"48","title":"Variable-base scalar multiplication"},"49":{"body":"Define a running sum zj=∑i=jn(ki⋅2i−j), where n=254 and: zn+1=0,zn=kn,(most significant bit)z0=k. Initialize A254=[2]T. for i from 254 down to 4: (ki)(ki−1)=0zi=2zi+1+kixP,i=xTyP,i=(2ki−1)⋅yT(conditionally negate)λ1,i⋅(xA,i−xP,i)=yA,i−yP,iλ1,i2=xR,i+xA,i+xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)=2yA,iλ2,i2=xA,i−1+xR,i+xA,iλ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1, where xR,i=(λ1,i2−xA,i−xT). After substitution of xP,i,yP,i,xR,i,yA,i, and yA,i−1, this becomes: Initialize A254=[2]T. for i from 254 down to 4: // let ki=zi−2zi+1// let yA,i=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT))(ki)(ki−1)=0λ1,i⋅(xA,i−xT)=yA,i−(2ki−1)⋅yTλ2,i2=xA,i−1+λ1,i2−xT {λ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1,λ2,4⋅(xA,4−xA,3)=yA,4+yA,3witnessed,if i>4if i=4. Here, yA,3witnessed is assigned to a cell. This is unlike previous yA,i's, which were implicitly derived from λ1,i,λ2,i,xA,i,xT, but never actually assigned. The bits k3…1 are used in three further steps, using complete addition : for i from 3 down to 1: // let ki=zi−2zi+1(ki)(ki−1)=0(xA,i−1,yA,i−1)=((xA,i,yA,i)+(xT,yT))+(xA,i,yA,i) If the least significant bit is set k0=1, we return the accumulator A. Else, if k0=0, we return A−T (also using complete addition). Let B={(0,0),(xT,−yT), if k0=1, otherwise. Output (xA,0,yA,0)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraint program for optimized double-and-add (incomplete addition)","id":"49","title":"Constraint program for optimized double-and-add (incomplete addition)"},"5":{"body":"","breadcrumbs":"User Documentation","id":"5","title":"User Documentation"},"50":{"body":"We need six advice columns to witness (xT,yT,λ1,λ2,xA,i,zi). However, since (xT,yT) are the same, we can perform two incomplete additions in a single row, reusing the same (xT,yT). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xTxTxT⋮xTyTyTyT⋮yTzhiz255=0z254z253⋮z130xAhixA,254=2[T]xxA,253⋮xA,130xA,129λ1hiyA,254=2[T]yλ1,254λ1,253⋮λ1,130yA,129λ2hiλ2,254λ2,253⋮λ2,130qmulhi122⋮3zloz130z129z128⋮z5z4xAloxA,129xA,128⋮xA,5xA,4xA,3λ1loyA,129λ1,129λ1,128⋮λ1,5λ1,4yA,3λ2loλ2,129λ2,128⋮λ2,5λ2,4qmullo122⋮23 For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. qmul=1 This gate is only used on the first row (before the for loop). We check that λ1,λ2 are initialized to values consistent with the initial yA. Degree5Constraintqmulone⋅(yA,nwitnessed−yA,n)=0 where qmuloneyA,nyA,nwitnessed=qmul⋅(2−qmul)⋅(3−qmul),=2(λ1,n+λ2,n)⋅(xA,n−(λ1,n2−xA,n−xT)), is witnessed. qmul=2 This gate is used on all rows corresponding to the for loop except the last. Degree445655Constraintqmultwo⋅(xT,cur−xT,next)=0qmultwo⋅(yT,cur−yT,next)=0qmultwo⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmultwo⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmultwo⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmultwo⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1)=0 where qmultwoyA,iyA,i−1=qmul⋅(1−qmul)⋅(3−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)),=2(λ1,i−1+λ2,i−1)⋅(xA,i−1−(λ1,i−12−xA,i−1−xT)), qmul=3 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA has been witnessed correctly. Degree5655Constraintqmulthree⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmulthree⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmulthree⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmulthree⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1witnessed)=0 where qmulthreeyA,iyA,i−1witnessed=qmul⋅(1−qmul)⋅(2−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)), is witnessed.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Circuit design","id":"50","title":"Circuit design"},"51":{"body":"zi cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0=α+tq can overflow [0,p). Since overflow can only occur in the final step that constrains z0=2⋅z1+k0, we have z0=k(modp). It is then sufficient to also check that z0=α+tq(modp) (so that k=α+tq(modp)) and that k∈[tq,p+tq). These conditions together imply that k=α+tq as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0 do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check","id":"51","title":"Overflow check"},"52":{"body":"Since tp+tq<2130, we have [tq,p+tq)=[tq,tq+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq−2130,p+tq)). We may assume that k=α+tq(modp). Therefore, k∈[tq,p+tq)⇔⇔⇔(k∈[tq,tq+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(k∈[tq,tq+2130)∨k∈[2130,2254)))∧(k254=1⟹(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq−2130,p+tq) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq to give k+2130−p−tq∈[0,2130); observe that k+2130−p−tq is guaranteed to be in [2130−tp−tq,2131−tp−tq) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq(modp), observe that (k+2130−p−tq)modp=(α+tq+2130−p−tq)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq,p+tq)⇔⇔(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254=0⟹(α∈[0,2130)∨k253..130 are not all 0))∧(k254=1⟹(k253..130 are all 0∧(α+2130)modp∈[0,2130))) Constraining k253..130 to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi=∑h=in(kh⋅2h−i), so we have: z130z130z130−k254⋅2124===∑h=130254(kh⋅2h−130)k254⋅2254−130+∑h=130253(kh⋅2h−130)∑h=130253(kh⋅2h−130) So k253..130 are all 0 exactly when z130=k254⋅2124. Finally, we can merge the 130-bit decompositions for the k254=0 and k254=1 cases by checking that (α+k254⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Optimized check for k∈[tq,p+tq)","id":"52","title":"Optimized check for k∈[tq,p+tq)"},"53":{"body":"Let s=α+k254⋅2130. The constraints for the overflow check are: z0k254=1⟹(z130k254=0⟹(z130=α+tq(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130)) Define inv0(x)={0,1/x,if x=0otherwise. Witness η=inv0(z130), and decompose smodp as s129..0. Then the needed gates are: Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0 where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check constraints","id":"53","title":"Overflow check constraints"},"54":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » Sinsemilla","id":"54","title":"Sinsemilla"},"55":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Circuit » Gadgets » Overview","id":"55","title":"Overview"},"56":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi. let Acc0:=Q for i from 0 up to n−1: let Acci+1:=(Acci⸭P[mi+1])⸭Acci return Accn Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Circuit » Gadgets » Description","id":"56","title":"Description"},"57":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr(M)=Hash(M)⸭[r]H. Let ShortCommitr(M) be the x-coordinate of Commitr(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Circuit » Gadgets » Use as a commitment scheme","id":"57","title":"Use as a commitment scheme"},"58":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Circuit » Gadgets » Efficient implementation","id":"58","title":"Efficient implementation"},"59":{"body":"Let P={(j,xP[j],yP[j]) for j∈{0..2k−1}}. Input: m1..=n. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i,yA,i) corresponds to Acci in the protocol spec.) Output: (xA,n,yA,n). (xA,0,yA,0)=Q for i from 0 up to n−1: yP,i=yA,i−λ1,i⋅(xA,i−xP,i) xR,i=λ1,i2−xA,i−xP,i 2⋅yA,i=(λ1,i+λ2,i)⋅(xA,i−xR,i) (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i λ2,i⋅(xA,i−xA,i+1)=yA,i+yA,i+1","breadcrumbs":"Design » Circuit » Gadgets » Constraint program","id":"59","title":"Constraint program"},"6":{"body":"","breadcrumbs":"User Documentation » Creating keys and addresses","id":"6","title":"Creating keys and addresses"},"60":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » PLONK / Halo 2 constraints","id":"60","title":"PLONK / Halo 2 constraints"},"61":{"body":"We have an n-bit message m=m1+2km2+...+2k⋅(n−1)mn. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0=α and define zi+1:=2Kzi−mi+1. We will end up with zn=0. Rearranging gives us an expression for each word of the original message mi+1=zi−2k⋅zi+1, which we can look up in the table. In other words, zn−i=h=0∑i−12kh⋅mh+1. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2 selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1 that effectively forces zn to zero for the last step of each element, as indicated by qS2. This allows the cell that would have been zn to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Circuit » Gadgets » Message decomposition","id":"61","title":"Message decomposition"},"62":{"body":"Note: qS3 is synthesized from qS1 and qS2; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′xAxQxA,1xA,2⋮xA,n−1xA,0′xA,1′xA,2′⋮xA,n−1′xA,n′xPxP[m1]xP[m2]xP[m3]⋮xP[mn]xP[m1′]xP[m2′]xP[m3′]⋮xP[mn′]bitsz0z1z2⋮zn−1z0′z1′z2′⋮zn−1′λ1λ1,0λ1,1λ1,2⋮λ1,n−1λ1,0′λ1,1′λ1,2′⋮λ1,n−1′yA,nλ2λ2,0λ2,1λ2,2⋮λ2,n−1λ2,0′λ2,1′λ2,2′⋮λ2,n−1′qS111111111110qS211110111120qS300000000020fixed_yQyQ0000000000tableidx012⋮⋮⋮⋮⋮⋮⋮⋮tablexxP[0]xP[1]xP[2]⋮⋮⋮⋮⋮⋮⋮⋮tableyyP[0]yP[1]yP[2]⋮⋮⋮⋮⋮⋮⋮⋮ xQ, z0, z0′, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Circuit » Gadgets » Layout","id":"62","title":"Layout"},"63":{"body":"For i∈[0,n), letxR,iYA,iyP,imi+1qS3=====λ1,i2−xA,i−xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)YA,i/2−λ1,i⋅(xA,i−xP,i)zi−2k⋅(qS2,i−qS3,i)⋅zi+1qS2⋅(qS2−1) The Halo 2 circuit API can automatically substitute yP,i, xR,i, yA,i, and yA,i+1, so we don't need to do that manually. xA,0=xQ 2⋅yQ=YA,0 for i from 0 up to n−1: (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i 4⋅λ2,i⋅(xA,i−xA,i+1)=2⋅YA,i+(2−qS3)⋅YA,i+1+2qS3⋅yA,n Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. Degree4536ConstraintfixedyQ⋅(2⋅fixedyQ−YA,0)=0qS1,i⇒(mi+1,xP,i,yP,i)∈PqS1,i⋅(λ2,i2−(xA,i+1+xR,i+xA,i))qS1,i⋅(4⋅λ2,i⋅(xA,i−xA,i+1)−(2⋅YA,i+(2−qS3,i)⋅YA,i+1+2⋅qS3,i⋅yA,n))=0 By gating the lookup expression on qS1, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (qS1⋅mi+1,qS1⋅xP,i+(1−qS1)⋅xP,0,qS1⋅yP,i+(1−qS1)⋅yP,0) This increases the degree of the lookup argument to 6.","breadcrumbs":"Design » Circuit » Gadgets » Optimized Sinsemilla gate","id":"63","title":"Optimized Sinsemilla gate"},"64":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Firstly, express α=k0+2K⋅k1+22K⋅k2+⋯, where each ki a K-bit value. We initialize the running sum z0=s, and compute subsequent terms zi=2Kzi−1−ki−1. This gives us: z0z1z2=α=k0+2K⋅k1+22K⋅k2+23K⋅k3+⋯,=(z0−k0)/2K=k1+2K⋅k2+22K⋅k3+⋯,=(z1−k1)/2K=k2+2K⋅k3+⋯,⋮ Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table. This gadget takes in α,n and returns (z0,z1,⋯,zn).","breadcrumbs":"Design » Circuit » Gadgets » Lookup decomposition","id":"64","title":"Lookup decomposition"},"65":{"body":"Strict mode constrains the running sum output zn to be zero, thus range-constraining the field element to be within n⋅K bits.","breadcrumbs":"Design » Circuit » Gadgets » Strict mode","id":"65","title":"Strict mode"},"66":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. 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: This library is being actively developed and should not be used in production software. Requires Rust 1.51+.","breadcrumbs":"orchard Crates.io","id":"0","title":"orchard Crates.io"},"1":{"body":"The Orchard Book Crate documentation","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"","breadcrumbs":"Design","id":"10","title":"Design"},"11":{"body":"","breadcrumbs":"General design notes","id":"11","title":"General design notes"},"12":{"body":"Keep the design close to Sapling, while eliminating aspects we don't like.","breadcrumbs":"Requirements","id":"12","title":"Requirements"},"13":{"body":"Delegated proving with privacy from the prover. We know how to do this, but it would require a discrete log equality proof, and the most efficient way to do this would be to do RedDSA and this at the same time, which means more work for e.g. hardware wallets.","breadcrumbs":"Non-requirements","id":"13","title":"Non-requirements"},"14":{"body":"Should we have one memo per output, or one memo per transaction, or 0..n memos? Variable, or (1 or n), is a potential privacy leak. Need to consider the privacy issue related to light clients requesting individual memos vs being able to fetch all memos.","breadcrumbs":"Open issues","id":"14","title":"Open issues"},"15":{"body":"TODO: UDAs: arbitrary vs whitelisted","breadcrumbs":"Note structure","id":"15","title":"Note structure"},"16":{"body":"For Sapling, we have encountered multiple places where the specification uses typed variables to define the consensus rules, but the C++ implementation in zcashd relied on byte encodings to implement them. This resulted in subtly-different consensus rules being deployed than were intended, for example where a particular type was not round-trip encodable. In Orchard, we avoid this by defining the consensus rules in terms of the byte encodings of all variables, and being explicit about any types that are not round-trip encodable. This makes consensus compatibility between strongly-typed implementations (such as this crate) and byte-oriented implementations easier to achieve.","breadcrumbs":"Typed variables vs byte encodings","id":"16","title":"Typed variables vs byte encodings"},"17":{"body":"Orchard keys and payment addresses are structurally similar to Sapling. The main change is that Orchard keys use the Pallas curve instead of Jubjub, in order to enable the future use of the Pallas-Vesta curve cycle in the Orchard protocol. (We already use Vesta as the curve on which Halo 2 proofs are computed, but this doesn't yet require a cycle.) Using the Pallas curve and making the most efficient use of the Halo 2 proof system involves corresponding changes to the key derivation process, such as using Sinsemilla for Pallas-efficient commitments. We also take the opportunity to remove all uses of expensive general-purpose hashes (such as BLAKE2s) from the circuit. We make several structural changes, building on the lessons learned from Sapling: The nullifier private key nsk is removed. Its purpose in Sapling was as defense-in-depth, in case RedDSA was found to have weaknesses; an adversary who could recover ask would not be able to spend funds. In practice it has not been feasible to manage nsk much more securely than a full viewing key, as the computational power required to generate Sapling proofs has made it necessary to perform this step on the same device that is creating the overall transaction (rather than on a more constrained device like a hardware wallet). We are also more confident in RedDSA now. nk is now a field element instead of a curve point, making it more efficient to generate nullifiers. ovk is now derived from fvk, instead of being derived in parallel. This places it in a similar position within the key structure to ivk, and also removes an issue where two full viewing keys could be constructed that have the same ivk but different ovks. Users still have control over whether ovk is used when constructing a transaction. All diversifiers now result in valid payment addresses, due to group hashing into Pallas being specified to be infallible. This removes significant complexity from the use cases for diversified addresses. The fact that Pallas is a prime-order curve simplifies the protocol and removes the need for cofactor multiplication in key agreement. Unlike Sapling, we define public (including ephemeral) and private keys used for note encryption to exclude the zero point and the zero scalar. Without this change, the implementation of the Orchard Action circuit would need special cases for the zero point, since Pallas is a short Weierstrass rather than an Edwards curve. This also has the advantage of ensuring that the key agreement has \"contributory behaviour\" — that is, if either party contributes a random scalar, then the shared secret will be random to an observer who does not know that scalar and cannot break Diffie–Hellman. Other than the above, Orchard retains the same design rationale for its keys and addresses as Sapling. For example, diversifiers remain at 11 bytes, so that a raw Orchard address is the same length as a raw Sapling address. Orchard payment addresses do not have a stand-alone string encoding. Instead, we define \"unified addresses\" that can bundle together addresses of different types, including Orchard. Unified addresses have a Human-Readable Part of \"u\" on Mainnet, i.e. they will have the prefix \"u1\". For specifications of this and other formats (e.g. for Orchard viewing and spending keys), see section 5.6.4 of the NU5 protocol specification [#NU5-orchardencodings].","breadcrumbs":"Design » Keys and addresses","id":"17","title":"Keys and addresses"},"18":{"body":"When designing Sapling, we defined a BIP 32 -like mechanism for generating hierarchical deterministic wallets in ZIP 32 . We decided at the time to stick closely to the design of BIP 32, on the assumption that there were Bitcoin use cases that used both hardened and non-hardened derivation that we might not be aware of. This decision created significant complexity for Sapling: we needed to handle derivation separately for each component of the expanded spending key and full viewing key (whereas for transparent addresses there is only a single component in the spending key). Non-hardened derivation enables creating a multi-level path of child addresses below some parent address, without involving the parent spending key. The primary use case for this is HD wallets for transparent addresses, which use the following structure defined in BIP 44 : (H) BIP 44 (H) Coin type: Zcash (H) Account 0 (N) Normal addresses (N) Address 0 (N) Address 1... (N) Change addresses (N) Change address 0 (N) Change address 1... (H) Account 1... Shielded accounts do not require separating change addresses from normal addresses, because addresses are not revealed in transactions. Similarly, there is also no need to generate a fresh spending key for every transaction, and in fact this would cause a linear slow-down in wallet scanning. But for users who do want to generate multiple addresses per account, they can generate the following structure, which does not use non-hardened derivation: (H) ZIP 32 (H) Coin type: Zcash (H) Account 0 Diversified address 0 Diversified address 1... (H) Account 1... Non-hardened derivation is therefore only required for use-cases that require the ability to derive more than one child layer of addresses. However, in the years since Sapling was deployed, we have not seen any such use cases appear. Therefore, for Orchard we only define hardened derivation, and do so with a much simpler design than ZIP 32. All derivations produce an opaque binary spending key, from which the keys and addresses are then derived. As a side benefit, this makes key formats shorter. (The formats that will actually be used in practice for Orchard will correspond to the simpler Sapling formats in the protocol specification, rather than the longer and more complicated \"extended\" ones defined by ZIP 32.)","breadcrumbs":"Design » Hierarchical deterministic wallets","id":"18","title":"Hierarchical deterministic wallets"},"19":{"body":"In Sprout, we had a single proof that represented two spent notes and two new notes. This was necessary in order to faciliate spending multiple notes in a single transaction (to balance value, an output of one JoinSplit could be spent in the next one), but also provided a minimal level of arity-hiding: single-JoinSplit transactions all looked like 2-in 2-out transactions, and in multi-JoinSplit transactions each JoinSplit looked like a 1-in 1-out. In Sapling, we switched to using value commitments to balance the transaction, removing the min-2 arity requirement. We opted for one proof per spent note and one (much simpler) proof per output note, which greatly improved the performance of generating outputs, but removed any arity-hiding from the proofs (instead having the transaction builder pad transactions to 1-in, 2-out). For Orchard, we take a combined approach: we define an Orchard transaction as containing a bundle of actions, where each action is both a spend and an output. This provides the same inherent arity-hiding as multi-JoinSplit Sprout, but using Sapling value commitments to balance the transaction without doubling its size. TODO: Depending on the circuit cost, we may switch to having an action internally represent either a spend or an output. Externally spends and outputs would still be indistinguishable, but the transaction would be larger.","breadcrumbs":"Design » Actions","id":"19","title":"Actions"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Bootstrap Open Source Licence, version 1.0, or at your option, any later version. See the file LICENSE-BOSL for the terms of the Bootstrap Open Source Licence, version 1.0. The purpose of the BOSL is to allow commercial improvements to the package while ensuring that all improvements are open source. See here for why the BOSL exists.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"TODO: One memo per tx vs one memo per output","breadcrumbs":"Design » Memo fields","id":"20","title":"Memo fields"},"21":{"body":"As in Sapling, we require two kinds of commitment schemes in Orchard: HomomorphicCommit is a linearly homomorphic commitment scheme with perfect hiding, and strong binding reducible to DL. Commit and ShortCommit are commitment schemes with perfect hiding, and strong binding reducible to DL. By \"strong binding\" we mean that the scheme is collision resistant on the input and randomness. We instantiate HomomorphicCommit with a Pedersen commitment, and use it for value commitments: cv=HomomorphicCommitrcvcv(v) We instantiate Commit and ShortCommit with Sinsemilla, and use them for all other commitments: ivk=ShortCommitrivkivk(ak,nk) cm=Commitrcmcm(rest of note) This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient Sinsemilla instead of Bowe--Hopwood Pedersen hashes. Note that for ivk, we also deviate from Sapling in two ways: We use ShortCommit to derive ivk instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit, at the cost of requiring rivk to be part of the full viewing key. We define ivk as an integer in [1,qP); that is, we exclude ivk=0. For Sapling, we relied on BLAKE2s to make ivk=0 infeasible to produce, but it was still technically possible. For Orchard, we get this by construction: 0 is not a valid x-coordinate for any Pallas point. SinsemillaShortCommit internally maps points to field elements by replacing the identity (which has no affine coordinates) with 0. But SinsemillaCommit is defined using incomplete addition, and thus will never produce the identity.","breadcrumbs":"Design » Commitments","id":"21","title":"Commitments"},"22":{"body":"The commitment tree structure for Orchard is identical to Sapling: A single global commitment tree of fixed depth 32. Note commitments are appended to the tree in-order from the block. Valid Orchard anchors correspond to the global tree state at block boundaries (after all commitments from a block have been appended, and before any commitments from the next block have been appended). The only difference is that we instantiate MerkleCRHOrchard with Sinsemilla (whereas MerkleCRHSapling used a Bowe--Hopwood Pedersen hash).","breadcrumbs":"Design » Commitment tree","id":"22","title":"Commitment tree"},"23":{"body":"The fixed-depth incremental Merkle trees that we use (in Sprout and Sapling, and again in Orchard) require specifying an \"empty\" or \"uncommitted\" leaf - a value that will never be appended to the tree as a regular leaf. For Sprout (and trees composed of the outputs of bit-twiddling hash functions), we use the all-zeroes array; the probability of a real note having a colliding note commitment is cryptographically negligible. For Sapling, where leaves are u-coordinates of Jubjub points, we use the value 1 which is not the u-coordinate of any Jubjub point. Orchard note commitments are the x-coordinates of Pallas points; thus we take the same approach as Sapling, using a value that is not the x-coordinate of any Pallas point as the uncommitted leaf value. We use the value 2 for both Pallas and Vesta, because 23+5 is not a square in either Fp or Fq: sage: p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001\nsage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: EllipticCurve(GF(p), [0, 5]).count_points() == q\nTrue\nsage: EllipticCurve(GF(q), [0, 5]).count_points() == p\nTrue\nsage: Mod(13, p).is_square()\nFalse\nsage: Mod(13, q).is_square()\nFalse Note: There are also no Pallas points with x-coordinate 0, but we map the identity to (0,0) within the circuit. Although SinsemillaCommit cannot return the identity (the incomplete addition would return ⊥ instead), it would arguably be confusing to rely on that.","breadcrumbs":"Design » Uncommitted leaves","id":"23","title":"Uncommitted leaves"},"24":{"body":"We considered splitting the commitment tree into several sub-trees: Bundle tree, that accumulates the commitments within a single bundle (and thus a single transaction). Block tree, that accumulates the bundle tree roots within a single block. Global tree, that accumulates the block tree roots. Each of these trees would have had a fixed depth (necessary for being able to create proofs). Chains that integrated Orchard could have decoupled the limits on commitments-per-subtree from higher-layer constraints like block size, by enabling their blocks and transactions to be structured internally as a series of Orchard blocks or txs (e.g. a Zcash block would have contained a Vec, that each were appended in-order). The motivation for considering this change was to improve the lives of light client wallet developers. When a new note is received, the wallet derives its incremental witness from the state of the global tree at the point when the note's commitment is appended; this incremental state then needs to be updated with every subsequent commitment in the block in-order. Wallets can't get help from the server to create these for new notes without leaking the specific note that was received. We decided that this was too large a change from Sapling, and that it should be possible to improve the Incremental Merkle Tree implementation to work around the efficiency issues without domain-separating the tree.","breadcrumbs":"Design » Considered alternatives","id":"24","title":"Considered alternatives"},"25":{"body":"The nullifier design we use for Orchard is nf=ExtractP([(Fnk(ρ)+ψ)modp]G+cm), where: F is a keyed circuit-efficient PRF (such as Rescue or Poseidon). ρ is unique to this output. As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. Given that an action consists of a single spend and a single output, we set ρ to be the nullifier of the spent note. ψ is sender-controlled randomness. It is not required to be unique, and in practice is derived from both ρ and a sender-selected random value rseed: ψ=KDFψ(ρ,rseed). G is a fixed independent base. ExtractP extracts the x-coordinate of a Pallas curve point. This gives a note structure of (addr,v,ρ,ψ,rcm). The note plaintext includes rseed in place of ψ and rcm, and omits ρ (which is a public part of the action).","breadcrumbs":"Design » Nullifiers","id":"25","title":"Nullifiers"},"26":{"body":"We care about several security properties for our nullifiers: Balance: can I forge money? Note Privacy: can I gain information about notes only from the public block chain? This describes notes sent in-band. Note Privacy (OOB): can I gain information about notes sent out-of-band, only from the public block chain? In this case, we assume privacy of the channel over which the note is sent, and that the adversary does not have access to any notes sent to the same address which are then spent (so that the nullifier is on the block chain somewhere). Spend Unlinkability: given the incoming viewing key for an address, and not the full viewing key, can I (possibly the sender) detect spends of any notes sent to that address? We're giving ivk to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address. Faerie Resistance: can I perform a Faerie Gold attack (i.e. cause notes to be accepted that are unspendable)? We're giving the full viewing key to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address, and be able to derive every nullifier. We assume (and instantiate elsewhere) the following primitives: GH is a cryptographic hash into the group (such as BLAKE2s with simplified SWU), used to derive all fixed independent bases. E is an elliptic curve (such as Pallas). KDF is the note encryption key derivation function. For our chosen design, our desired security properties rely on the following assumptions: BalanceNote PrivacyNote Privacy (OOB)Spend UnlinkabilityFaerie ResistanceDLEHashDHEKDFNear perfect‡DDHE†∨PRFFDLE HashDHEKDF is computational Diffie-Hellman using KDF for the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker than DDHE but stronger than DLE. We omit ROGH as a security assumption because we only rely on the random oracle applied to fixed inputs defined by the protocol, i.e. to generate the fixed base G, not to attacker-specified inputs. † We additionally assume that for any input x, {Fnk(x):nk∈E} gives a scalar in an adequate range for DDHE. (Otherwise, F could be trivial, e.g. independent of nk.) ‡ Statistical distance <2−167.8 from perfect.","breadcrumbs":"Design » Security properties","id":"26","title":"Security properties"},"27":{"body":"⚠ Caution: be skeptical of the claims in this table about what problem(s) each security property depends on. They may not be accurate and are definitely not fully rigorous. The entries in this table omit the application of ExtractP, which is an optimization to halve the nullifier length. That optimization requires its own security analysis, but because it is a deterministic mapping, only Faerie Resistance could be affected by it. nf[nk][θ]H[nk]H+[rnf]IHash([nk][θ]H)Hash([nk]H+[rnf]I)[Fnk(ψ)][θ]H[Fnk(ψ)]H+[rnf]I[Fnk(ψ)]G+[θ]H[Fnk(ψ)]H+cm[Fnk(ρ,ψ)]G+cm[Fnk(ρ)]G+cm[Fnk(ρ,ψ)]Gv+[rnf]I[Fnk(ρ)]Gv+[rnf]I[(Fnk(ρ)+ψ)modp]Gv[Fnk(ρ,ψ)]G+Commitrnfnf(v,ρ)[Fnk(ρ)]G+Commitrnfnf(v,ρ)Note(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,rnf,ψ,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,ψ,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)BalanceDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLENote PrivacyHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFNote Priv OOBPerfectPerfectPerfectPerfectPerfectPerfectPerfectDDHE†DDHE†DDHE†PerfectPerfectNear perfect‡PerfectPerfectSpend UnlinkabilityDDHEDDHEDDHE∨PreHashDDHE∨PreHashDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFFaerie ResistanceROGH∧DLEROGH∧DLECollHash∧ROGH∧DLECollHash∧ROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEDLEDLECollF∧DLECollF∧DLEbrokenDLEDLEReason not to useNo SU for DL-breakingNo SU for DL-breakingCollHash for FRCollHash for FRPerf. (2 var-base)Perf. (1 var+1 fix-base)Perf. (1 var+1 fix-base)NP(OOB) not perfectNP(OOB) not perfectNP(OOB) not perfectCollF for FRCollF for FRbroken for FRPerf. (2 fix-base)Perf. (2 fix-base) In the above alternatives: Hash is a keyed circuit-efficient hash (such as Rescue). I is an fixed independent base, independent of G and any others returned by GH. Gv is a pair of fixed independent bases (independent of all others), where the specific choice of base depends on whether the note has zero value. H is a base unique to this output. For non-zero-valued notes, H=GH(ρ). As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. For zero-valued notes, H is constrained by the circuit to a fixed base independent of I and any others returned by GH.","breadcrumbs":"Design » Considered alternatives","id":"27","title":"Considered alternatives"},"28":{"body":"In order to satisfy the Balance security property, we require that the circuit must be able to enforce that only one nullifier is accepted for a given note. As in Sprout and Sapling, we achieve this by ensuring that the nullifier deterministically depends only on values committed to (directly or indirectly) by the note commitment. As in Sapling, this involves arguing that: There can be only one ivk for a given addr. This is true because the circuit checks that pkd=[ivk]gd, and the mapping ivk↦[ivk]gd is an injection for any gd. (ivk is in the base field of E, which must be smaller than its scalar field, as is the case for Pallas.) There can be only one nk for a given ivk. This is true because the circuit checks that ivk=ShortCommitrivkivk(ak,nk) where ShortCommit is binding (see Commitments ).","breadcrumbs":"Design » Rationale","id":"28","title":"Rationale"},"29":{"body":"Faerie Resistance requires that nullifiers be unique. This is primarily achieved by taking a unique value (checked for uniqueness by the public consensus rules) as an input to the nullifier. However, it is also necessary to ensure that the transformations applied to this value preserve its uniqueness. Meanwhile, to achieve Spend Unlinkability , we require that the nullifier does not reveal any information about the unique value it is derived from. The design alternatives fall into two categories in terms of how they balance these requirements: Publish a unique value ρ at note creation time, and blind that value within the nullifier computation. This is similar to the approach taken in Sprout and Sapling, which both implemented nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while Sapling uses BLAKE2s. Derive a unique base H from some unique value, publish that unique base at note creation time, and then blind the base (either additively or multiplicatively) during nullifier computation. For Spend Unlinkability , the only value unknown to the adversary is nk, and the cryptographic assumptions only involve the first term (other terms like cm or [rnf]I cannot be extracted directly from the observed nullifiers, but can be subtracted from them). We therefore ensure that the first term does not commit directly to the note (to avoid a DL-breaking adversary from immediately breaking SU ). We were considering using a design involving H with the goal of eliminating all usages of a PRF inside the circuit, for two reasons: Instantiating PRFF with a traditional hash function is expensive in the circuit. We didn't want to solely rely on an algebraic hash function satisfying PRFF to achieve Spend Unlinkability . However, those designs rely on both ROGH and DLE for Faerie Resistance , while still requiring DDHE for Spend Unlinkability . (There are two designs for which this is not the case, but they rely on DDHE† for Note Privacy (OOB) which was not acceptable). By contrast, several designs involving ρ (including the chosen design) have weaker assumptions for Faerie Resistance (only relying on DLE), and Spend Unlinkability does not require PRFF to hold: they can fall back on the same DDHE assumption as the H designs (along with an additional assumption about the output of F which is easily satisfied).","breadcrumbs":"Design » Use of ρ","id":"29","title":"Use of ρ"},"3":{"body":"","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"Most of the designs include either a multiplicative blinding term [θ]H, or an additive blinding term [rnf]I, in order to achieve perfect Note Privacy (OOB) (to an adversary who does not know the note). The chosen design is effectively using [ψ]G for this purpose; a DL-breaking adversary only learns Fnk(ρ)+ψ(modp). This reduces Note Privacy (OOB) from perfect to statistical, but given that ψ is from a distribution statistically close to uniform on [0,q), this is statistically close to better than 2−128. The benefit is that it does not require an additional scalar multiplication, making it more efficient inside the circuit. ψ's derivation has two motivations: Deriving from a random value rseed enables multiple derived values to be conveyed to the recipient within an action (such as the ephemeral secret esk, per ZIP 212 ), while keeping the note plaintext short. Mixing ρ into the derivation ensures that the sender can't repeat ψ across two notes, which could have enabled spend linkability attacks in some designs. The note that is committed to, and which the circuit takes as input, only includes ψ (i.e. the circuit does not check the derivation from rseed). However, an adversarial sender is still constrained by this derivation, because the recipient recomputes ψ during note decryption. If an action were created using an arbitrary ψ (for which the adversary did not have a corresponding rseed), the recipient would derive a note commitment that did not match the action's commitment field, and reject it (as in Sapling).","breadcrumbs":"Design » Use of ψ","id":"30","title":"Use of ψ"},"31":{"body":"The nullifier commits to the note value via cm for two reasons: It domain-separates nullifiers for zero-valued notes from other notes. This is necessary because we do not require zero-valued notes to exist in the commitment tree. Designs that bind the nullifier to Fnk(ρ) require CollF to achieve Faerie Resistance (and similarly where Hash is applied to a value derived from H). Adding cm to the nullifier avoids this assumption: all of the bases used to derive cm are fixed and independent of G, and so the nullifier can be viewed as a Pedersen hash where the input includes ρ directly. The Commitnf variants were considered to avoid directly depending on cm (which in its native type is a base field element, not a group element). We decided instead to follow Sapling by defining an intermediate representation of cm as a group element, that is only used in nullifier computation. The circuit already needs to compute cm, so this improves performance by removing We also considered variants that used a choice of fixed bases Gv to provide domain separation for zero-valued notes. The most performant design (similar to the chosen design) does not achieve Faerie Resistance for an adversary that knows the recipient's full viewing key (ψ could be brute-forced to cancel out Fnk(ρ), causing a collision), and the other variants require assuming CollF as mentioned above.","breadcrumbs":"Design » Use of cm","id":"31","title":"Use of cm"},"32":{"body":"Orchard signatures are an instantiation of RedDSA with a cofactor of 1. TODO: Should it be possible to sign partial transactions? If we're going to merge down all the signatures into a single one, and also want this, we need to ensure there's a feasible MPC.","breadcrumbs":"Design » Signatures","id":"32","title":"Signatures"},"33":{"body":"","breadcrumbs":"Design » Circuit","id":"33","title":"Circuit"},"34":{"body":"","breadcrumbs":"Design » Circuit » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"We will use formulae for curve arithmetic using affine coordinates on short Weierstrass curves, derived from section 4.1 of Hüseyin Hışıl's thesis . Inputs: P=(xp,yp),Q=(xq,yq) Output: R=P⸭Q=(xr,yr) The formulae from Hışıl's thesis are: x3=(x1−x2y1−y2)2−x1−x2 y3=x1−x2y1−y2⋅(x1−x3)−y1 Rename: (x1,y1) to (xq,yq) (x2,y2) to (xp,yp) (x3,y3) to (xr,yr). Let λ=xq−xpyq−yp=xp−xqyp−yq, which we implement as λ⋅(xp−xq)=yp−yq Also, xr=λ2−xq−xp yr=λ⋅(xq−xr)−yq which is equivalent to xr+xq+xp=λ2 Assuming xp=xq, ⟹⟹⟹(xr+xq+xp)⋅(xp−xq)2(xr+xq+xp)⋅(xp−xq)2yryr+yq(yr+yq)⋅(xp−xq)=====λ2⋅(xp−xq)2(λ⋅(xp−xq))2λ⋅(xq−xr)−yqλ⋅(xq−xr)λ⋅(xp−xq)⋅(xq−xr) Substituting for λ⋅(xp−xq), we get the constraints: (xr+xq+xp)⋅(xp−xq)2−(yp−yq)2=0 Note that this constraint is unsatisfiable for P⸭(−P) (when P=O), and so cannot be used with arbitrary inputs. (yr+yq)(xp−xq)−(yp−yq)(xq−xr)=0","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Incomplete addition","id":"35","title":"Incomplete addition"},"36":{"body":"OO(xp,yp)(x,y)(x,y)(xp,yp)++++++O(xq,yq)O(x,y)(x,−y)(xq,yq)=O=(xq,yq)=(xp,yp)=[2](x,y)=O=(xp,yp)⸭(xq,yq),if xp=xq Suppose that we represent O as (0,0). (0 is not an x-coordinate of a valid point because we would need y2=x3+5, and 5 is not square in Fq. Also 0 is not a y-coordinate of a valid point because −5 is not a cube in Fq.) P+Q(xp,yp)+(xq,yq)λxryr=R=(xr,yr)=xq−xpyq−yp=λ2−xp−xq=λ(xp−xr)−yp For the doubling case, Hışıl's thesis tells us that λ has to instead be computed as 2y3x2. Define inv0(x)={0,1/x,if x=0otherwise. Witness α,β,γ,δ,λ where: α=inv0(xq−xp) β=inv0(xp) γ=inv0(xq) δ={inv0(yq+yp),0,if xq=xpotherwise λ=⎩⎨⎧xq−xpyq−yp,2yp3xp20,if xq=xpif xq=xp∧yp=0otherwise.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Complete addition","id":"36","title":"Complete addition"},"37":{"body":"Degree456666444444Constraintqadd⋅(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))qadd⋅(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅(1−xp⋅β)⋅(xr−xq)qadd⋅(1−xp⋅β)⋅(yr−yq)qadd⋅(1−xq⋅γ)⋅(xr−xp)qadd⋅(1−xq⋅γ)⋅(yr−yp)qadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xrqadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr============000000000000Meaningxq=xp⟹λ=xq−xpyq−yp{xq=xp∧yp=0⟹λ=2yp3xp2xq=xp∧yp=0⟹xp=0xp=0∧xq=0∧xq=xp⟹xr=λ2−xp−xqxp=0∧xq=0∧xq=xp⟹yr=λ⋅(xp−xr)−ypxp=0∧xq=0∧yq=−yp⟹xr=λ2−xp−xqxp=0∧xq=0∧yq=−yp⟹yr=λ⋅(xp−xr)−ypxp=0⟹xr=xqxp=0⟹yr=yqxq=0⟹xr=xpxq=0⟹yr=ypxq=xp∧yq=−yp⟹xr=0xq=xp∧yq=−yp⟹yr=0 Max degree: 6","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraints","id":"37","title":"Constraints"},"38":{"body":"1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:xq=0⟹(xr,yr)=(xp,yp).(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0). Propositions: (1)(2)(3)(4)(5)(6)xq=xp⟹λ=(yq−yp)/(xq−xp).(xq=xp)∧yp=0⟹λ=3xp2/2yp(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))⟹(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp)xp=0⟹(xr,yr)=(xq,yq)xq=0⟹(xr,yr)=(xp,yp)xq=xp∧yq=−yp⟹(xr,yr)=(0,0) Test cases: (xp,yp)+(xq,yq)=(xr,yr) (0,0)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because yp=0 holds because xp=0 holds because xp=0 only when xr=0,yr=0 holds because xq=0 only when xr=0,yr=0 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (x,y)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xq=0 holds because xp=0 because 0 is not a valid x-coordinate only when β=xp−1 holds because xq=0 only when (xr,yr)=(xp,yp) holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xp,yp) is the only solution (0,0)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xp=0 holds because xp=0 only when (xr,yr)=(xq,yq) holds because xq=0 because 0 is not a valid x-coordinate only when γ=xq−1 holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xq,yq) is the only solution (x,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate) only when λ=3xp2/2yp holds because xp=0∧xq=0andyq=−yp only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xp=0 only when γ=xq−1 holds because yq=−yp only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution (x,y)+(x,−y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate)only when λ=3xp2/2yp holds because xp=0∧xq=0 but yq=−yp∧xq=xp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (ζx,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xp=xp only when α=(xq−xp)−1 which is defined because xq=xp holds because (xp=0)∧(xq=0)∧(xq=xp) only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp only when δ=0 Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution All remaining cases (x,y)+(x′,y′) are identical to the case (ζx,y)+(x,y) when λαβγδ=(yq−yp)/(xq−xp)=(xq−xp)−1=xp−1=xq−1=0 because in all remaining cases, xq=xp,xp=0,xq=0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Analysis of constraints","id":"38","title":"Analysis of constraints"},"39":{"body":"There are 6 fixed bases in the Orchard protocol: KOrchard, used in deriving the nullifier; GOrchard, used in spend authorization; R base for NoteCommitOrchard; V and R bases for ValueCommitOrchard; and R base for Commitivk.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"39","title":"Fixed-base scalar multiplication"},"4":{"body":"","breadcrumbs":"Concepts » Preliminaries","id":"4","title":"Preliminaries"},"40":{"body":"We support fixed-base scalar multiplication with three types of scalars:","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Decompose scalar","id":"40","title":"Decompose scalar"},"41":{"body":"A 255-bit scalar from Fq. We decompose a full-width scalar α into 85 3-bit windows: α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23). The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255). Degree9Constraintqscalar-fixed⋅(i=0∑7w−i)=0 We range-constrain each 3-bit word of the scalar decomposition using a polynomial range-check constraint: Degree9Constraintqdecompose-base-field⋅range_check(word,23)=0 where range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Full-width scalar","id":"41","title":"Full-width scalar"},"42":{"body":"We support using a base field element as the scalar in fixed-base multiplication. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition. Decompose the base field element α into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=85,K=3. If k0..84 is witnessed directly then no issue of canonicity arises. However, because the scalar is given as a base field element here, care must be taken to ensure a canonical representation, since 2255>p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq. (We use big-endian bit order for convenient input into the variable-base scalar multiplication algorithm.) k=k254⋅2254+k253⋅2253+⋯+k0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"49","title":"Witness scalar"},"5":{"body":"","breadcrumbs":"User Documentation","id":"5","title":"User Documentation"},"50":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication","id":"50","title":"Variable-base scalar multiplication"},"51":{"body":"Define a running sum zj=∑i=jn(ki⋅2i−j), where n=254 and: zn+1=0,zn=kn,(most significant bit)z0=k. Initialize A254=[2]T. for i from 254 down to 4: (ki)(ki−1)=0zi=2zi+1+kixP,i=xTyP,i=(2ki−1)⋅yT(conditionally negate)λ1,i⋅(xA,i−xP,i)=yA,i−yP,iλ1,i2=xR,i+xA,i+xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)=2yA,iλ2,i2=xA,i−1+xR,i+xA,iλ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1, where xR,i=(λ1,i2−xA,i−xT). After substitution of xP,i,yP,i,xR,i,yA,i, and yA,i−1, this becomes: Initialize A254=[2]T. for i from 254 down to 4: // let ki=zi−2zi+1// let yA,i=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT))(ki)(ki−1)=0λ1,i⋅(xA,i−xT)=yA,i−(2ki−1)⋅yTλ2,i2=xA,i−1+λ1,i2−xT {λ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1,λ2,4⋅(xA,4−xA,3)=yA,4+yA,3witnessed,if i>4if i=4. Here, yA,3witnessed is assigned to a cell. This is unlike previous yA,i's, which were implicitly derived from λ1,i,λ2,i,xA,i,xT, but never actually assigned. The bits k3…1 are used in three further steps, using complete addition : for i from 3 down to 1: // let ki=zi−2zi+1(ki)(ki−1)=0(xA,i−1,yA,i−1)=((xA,i,yA,i)+(xT,yT))+(xA,i,yA,i) If the least significant bit is set k0=1, we return the accumulator A. Else, if k0=0, we return A−T (also using complete addition). Let B={(0,0),(xT,−yT), if k0=1, otherwise. Output (xA,0,yA,0)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraint program for optimized double-and-add (incomplete addition)","id":"51","title":"Constraint program for optimized double-and-add (incomplete addition)"},"52":{"body":"We need six advice columns to witness (xT,yT,λ1,λ2,xA,i,zi). However, since (xT,yT) are the same, we can perform two incomplete additions in a single row, reusing the same (xT,yT). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xTxTxT⋮xTyTyTyT⋮yTzhiz255=0z254z253⋮z130xAhixA,254=2[T]xxA,253⋮xA,130xA,129λ1hiyA,254=2[T]yλ1,254λ1,253⋮λ1,130yA,129λ2hiλ2,254λ2,253⋮λ2,130qmulhi122⋮3zloz130z129z128⋮z5z4xAloxA,129xA,128⋮xA,5xA,4xA,3λ1loyA,129λ1,129λ1,128⋮λ1,5λ1,4yA,3λ2loλ2,129λ2,128⋮λ2,5λ2,4qmullo122⋮23 For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. qmul=1 This gate is only used on the first row (before the for loop). We check that λ1,λ2 are initialized to values consistent with the initial yA. Degree5Constraintqmulone⋅(yA,nwitnessed−yA,n)=0 where qmuloneyA,nyA,nwitnessed=qmul⋅(2−qmul)⋅(3−qmul),=2(λ1,n+λ2,n)⋅(xA,n−(λ1,n2−xA,n−xT)), is witnessed. qmul=2 This gate is used on all rows corresponding to the for loop except the last. Degree445655Constraintqmultwo⋅(xT,cur−xT,next)=0qmultwo⋅(yT,cur−yT,next)=0qmultwo⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmultwo⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmultwo⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmultwo⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1)=0 where qmultwoyA,iyA,i−1=qmul⋅(1−qmul)⋅(3−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)),=2(λ1,i−1+λ2,i−1)⋅(xA,i−1−(λ1,i−12−xA,i−1−xT)), qmul=3 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA has been witnessed correctly. Degree5655Constraintqmulthree⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmulthree⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmulthree⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmulthree⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1witnessed)=0 where qmulthreeyA,iyA,i−1witnessed=qmul⋅(1−qmul)⋅(2−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)), is witnessed.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Circuit design","id":"52","title":"Circuit design"},"53":{"body":"zi cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0=α+tq can overflow [0,p). Since overflow can only occur in the final step that constrains z0=2⋅z1+k0, we have z0=k(modp). It is then sufficient to also check that z0=α+tq(modp) (so that k=α+tq(modp)) and that k∈[tq,p+tq). These conditions together imply that k=α+tq as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0 do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check","id":"53","title":"Overflow check"},"54":{"body":"Since tp+tq<2130, we have [tq,p+tq)=[tq,tq+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq−2130,p+tq)). We may assume that k=α+tq(modp). Therefore, k∈[tq,p+tq)⇔⇔⇔(k∈[tq,tq+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(k∈[tq,tq+2130)∨k∈[2130,2254)))∧(k254=1⟹(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq−2130,p+tq) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq to give k+2130−p−tq∈[0,2130); observe that k+2130−p−tq is guaranteed to be in [2130−tp−tq,2131−tp−tq) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq(modp), observe that (k+2130−p−tq)modp=(α+tq+2130−p−tq)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq,p+tq)⇔⇔(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254=0⟹(α∈[0,2130)∨k253..130 are not all 0))∧(k254=1⟹(k253..130 are all 0∧(α+2130)modp∈[0,2130))) Constraining k253..130 to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi=∑h=in(kh⋅2h−i), so we have: z130z130z130−k254⋅2124===∑h=130254(kh⋅2h−130)k254⋅2254−130+∑h=130253(kh⋅2h−130)∑h=130253(kh⋅2h−130) So k253..130 are all 0 exactly when z130=k254⋅2124. Finally, we can merge the 130-bit decompositions for the k254=0 and k254=1 cases by checking that (α+k254⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Optimized check for k∈[tq,p+tq)","id":"54","title":"Optimized check for k∈[tq,p+tq)"},"55":{"body":"Let s=α+k254⋅2130. The constraints for the overflow check are: z0k254=1⟹(z130k254=0⟹(z130=α+tq(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130)) Define inv0(x)={0,1/x,if x=0otherwise. Witness η=inv0(z130), and decompose smodp as s129..0. Then the needed gates are: Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0 where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check constraints","id":"55","title":"Overflow check constraints"},"56":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » Sinsemilla","id":"56","title":"Sinsemilla"},"57":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Circuit » Gadgets » Overview","id":"57","title":"Overview"},"58":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi. let Acc0:=Q for i from 0 up to n−1: let Acci+1:=(Acci⸭P[mi+1])⸭Acci return Accn Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Circuit » Gadgets » Description","id":"58","title":"Description"},"59":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr(M)=Hash(M)⸭[r]H. Let ShortCommitr(M) be the x-coordinate of Commitr(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Circuit » Gadgets » Use as a commitment scheme","id":"59","title":"Use as a commitment scheme"},"6":{"body":"","breadcrumbs":"User Documentation » Creating keys and addresses","id":"6","title":"Creating keys and addresses"},"60":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Circuit » Gadgets » Efficient implementation","id":"60","title":"Efficient implementation"},"61":{"body":"Let P={(j,xP[j],yP[j]) for j∈{0..2k−1}}. Input: m1..=n. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i,yA,i) corresponds to Acci in the protocol spec.) Output: (xA,n,yA,n). (xA,0,yA,0)=Q for i from 0 up to n−1: yP,i=yA,i−λ1,i⋅(xA,i−xP,i) xR,i=λ1,i2−xA,i−xP,i 2⋅yA,i=(λ1,i+λ2,i)⋅(xA,i−xR,i) (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i λ2,i⋅(xA,i−xA,i+1)=yA,i+yA,i+1","breadcrumbs":"Design » Circuit » Gadgets » Constraint program","id":"61","title":"Constraint program"},"62":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » PLONK / Halo 2 constraints","id":"62","title":"PLONK / Halo 2 constraints"},"63":{"body":"We have an n-bit message m=m1+2km2+...+2k⋅(n−1)mn. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0=α and define zi+1:=2Kzi−mi+1. We will end up with zn=0. Rearranging gives us an expression for each word of the original message mi+1=zi−2k⋅zi+1, which we can look up in the table. In other words, zn−i=h=0∑i−12kh⋅mh+1. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2 selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1 that effectively forces zn to zero for the last step of each element, as indicated by qS2. This allows the cell that would have been zn to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Circuit » Gadgets » Message decomposition","id":"63","title":"Message decomposition"},"64":{"body":"Note: qS3 is synthesized from qS1 and qS2; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′xAxQxA,1xA,2⋮xA,n−1xA,0′xA,1′xA,2′⋮xA,n−1′xA,n′xPxP[m1]xP[m2]xP[m3]⋮xP[mn]xP[m1′]xP[m2′]xP[m3′]⋮xP[mn′]bitsz0z1z2⋮zn−1z0′z1′z2′⋮zn−1′λ1λ1,0λ1,1λ1,2⋮λ1,n−1λ1,0′λ1,1′λ1,2′⋮λ1,n−1′yA,nλ2λ2,0λ2,1λ2,2⋮λ2,n−1λ2,0′λ2,1′λ2,2′⋮λ2,n−1′qS111111111110qS211110111120qS300000000020fixed_yQyQ0000000000tableidx012⋮⋮⋮⋮⋮⋮⋮⋮tablexxP[0]xP[1]xP[2]⋮⋮⋮⋮⋮⋮⋮⋮tableyyP[0]yP[1]yP[2]⋮⋮⋮⋮⋮⋮⋮⋮ xQ, z0, z0′, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Circuit » Gadgets » Layout","id":"64","title":"Layout"},"65":{"body":"For i∈[0,n), letxR,iYA,iyP,imi+1qS3=====λ1,i2−xA,i−xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)YA,i/2−λ1,i⋅(xA,i−xP,i)zi−2k⋅(qS2,i−qS3,i)⋅zi+1qS2⋅(qS2−1) The Halo 2 circuit API can automatically substitute yP,i, xR,i, yA,i, and yA,i+1, so we don't need to do that manually. xA,0=xQ 2⋅yQ=YA,0 for i from 0 up to n−1: (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i 4⋅λ2,i⋅(xA,i−xA,i+1)=2⋅YA,i+(2−qS3)⋅YA,i+1+2qS3⋅yA,n Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. Degree4536ConstraintfixedyQ⋅(2⋅fixedyQ−YA,0)=0qS1,i⇒(mi+1,xP,i,yP,i)∈PqS1,i⋅(λ2,i2−(xA,i+1+xR,i+xA,i))qS1,i⋅(4⋅λ2,i⋅(xA,i−xA,i+1)−(2⋅YA,i+(2−qS3,i)⋅YA,i+1+2⋅qS3,i⋅yA,n))=0 By gating the lookup expression on qS1, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (qS1⋅mi+1,qS1⋅xP,i+(1−qS1)⋅xP,0,qS1⋅yP,i+(1−qS1)⋅yP,0) This increases the degree of the lookup argument to 6.","breadcrumbs":"Design » Circuit » Gadgets » Optimized Sinsemilla gate","id":"65","title":"Optimized Sinsemilla gate"},"66":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0+2K⋅k1+22K⋅k2+⋯+2(W−1)K⋅kW−1 where each ki a K-bit value. This is done using a running sum zi,i∈[0..W). We initialize the running sum z0=α, and compute subsequent terms zi+1=2Kzi−ki. This gives us: z0z1z2↓zW=α=k0+2K⋅k1+22K⋅k2+23K⋅k3+⋯,=(z0−k0)/2K=k1+2K⋅k2+22K⋅k3+⋯,=(z1−k1)/2K=k2+2K⋅k3+⋯,⋮ (in strict mode)=(zW−1−kW−1)/2K=0 (because zW−1=kW−1)","breadcrumbs":"Design » Circuit » Gadgets » Decomposition","id":"66","title":"Decomposition"},"67":{"body":"Strict mode constrains the running sum output zW to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1=kW−1 gives us the last window in the decomposition.","breadcrumbs":"Design » Circuit » Gadgets » Strict mode","id":"67","title":"Strict mode"},"68":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table.","breadcrumbs":"Design » Circuit » Gadgets » Lookup decomposition","id":"68","title":"Lookup decomposition"},"69":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"69","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"70":{"body":"For a short range (for instance, 2K≤8), we can range-constrain each word using a polynomial constraint instead of a lookup: Degree2K+12Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅2K,2K)=0qstrict⋅zW=0 where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).","breadcrumbs":"Design » Circuit » Gadgets » Short range decomposition","id":"70","title":"Short range decomposition"},"8":{"body":"","breadcrumbs":"User Documentation » Spending notes","id":"8","title":"Spending notes"},"9":{"body":"","breadcrumbs":"User Documentation » Integration into an existing chain","id":"9","title":"Integration into an existing 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: This library is being actively developed and should not be used in production software. Requires Rust 1.51+.","breadcrumbs":"orchard Crates.io","id":"0","title":"orchard Crates.io"},"1":{"body":"The Orchard Book Crate documentation","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"","breadcrumbs":"Design","id":"10","title":"Design"},"11":{"body":"","breadcrumbs":"General design notes","id":"11","title":"General design notes"},"12":{"body":"Keep the design close to Sapling, while eliminating aspects we don't like.","breadcrumbs":"Requirements","id":"12","title":"Requirements"},"13":{"body":"Delegated proving with privacy from the prover. We know how to do this, but it would require a discrete log equality proof, and the most efficient way to do this would be to do RedDSA and this at the same time, which means more work for e.g. hardware wallets.","breadcrumbs":"Non-requirements","id":"13","title":"Non-requirements"},"14":{"body":"Should we have one memo per output, or one memo per transaction, or 0..n memos? Variable, or (1 or n), is a potential privacy leak. Need to consider the privacy issue related to light clients requesting individual memos vs being able to fetch all memos.","breadcrumbs":"Open issues","id":"14","title":"Open issues"},"15":{"body":"TODO: UDAs: arbitrary vs whitelisted","breadcrumbs":"Note structure","id":"15","title":"Note structure"},"16":{"body":"For Sapling, we have encountered multiple places where the specification uses typed variables to define the consensus rules, but the C++ implementation in zcashd relied on byte encodings to implement them. This resulted in subtly-different consensus rules being deployed than were intended, for example where a particular type was not round-trip encodable. In Orchard, we avoid this by defining the consensus rules in terms of the byte encodings of all variables, and being explicit about any types that are not round-trip encodable. This makes consensus compatibility between strongly-typed implementations (such as this crate) and byte-oriented implementations easier to achieve.","breadcrumbs":"Typed variables vs byte encodings","id":"16","title":"Typed variables vs byte encodings"},"17":{"body":"Orchard keys and payment addresses are structurally similar to Sapling. The main change is that Orchard keys use the Pallas curve instead of Jubjub, in order to enable the future use of the Pallas-Vesta curve cycle in the Orchard protocol. (We already use Vesta as the curve on which Halo 2 proofs are computed, but this doesn't yet require a cycle.) Using the Pallas curve and making the most efficient use of the Halo 2 proof system involves corresponding changes to the key derivation process, such as using Sinsemilla for Pallas-efficient commitments. We also take the opportunity to remove all uses of expensive general-purpose hashes (such as BLAKE2s) from the circuit. We make several structural changes, building on the lessons learned from Sapling: The nullifier private key nsk is removed. Its purpose in Sapling was as defense-in-depth, in case RedDSA was found to have weaknesses; an adversary who could recover ask would not be able to spend funds. In practice it has not been feasible to manage nsk much more securely than a full viewing key, as the computational power required to generate Sapling proofs has made it necessary to perform this step on the same device that is creating the overall transaction (rather than on a more constrained device like a hardware wallet). We are also more confident in RedDSA now. nk is now a field element instead of a curve point, making it more efficient to generate nullifiers. ovk is now derived from fvk, instead of being derived in parallel. This places it in a similar position within the key structure to ivk, and also removes an issue where two full viewing keys could be constructed that have the same ivk but different ovks. Users still have control over whether ovk is used when constructing a transaction. All diversifiers now result in valid payment addresses, due to group hashing into Pallas being specified to be infallible. This removes significant complexity from the use cases for diversified addresses. The fact that Pallas is a prime-order curve simplifies the protocol and removes the need for cofactor multiplication in key agreement. Unlike Sapling, we define public (including ephemeral) and private keys used for note encryption to exclude the zero point and the zero scalar. Without this change, the implementation of the Orchard Action circuit would need special cases for the zero point, since Pallas is a short Weierstrass rather than an Edwards curve. This also has the advantage of ensuring that the key agreement has \"contributory behaviour\" — that is, if either party contributes a random scalar, then the shared secret will be random to an observer who does not know that scalar and cannot break Diffie–Hellman. Other than the above, Orchard retains the same design rationale for its keys and addresses as Sapling. For example, diversifiers remain at 11 bytes, so that a raw Orchard address is the same length as a raw Sapling address. Orchard payment addresses do not have a stand-alone string encoding. Instead, we define \"unified addresses\" that can bundle together addresses of different types, including Orchard. Unified addresses have a Human-Readable Part of \"u\" on Mainnet, i.e. they will have the prefix \"u1\". For specifications of this and other formats (e.g. for Orchard viewing and spending keys), see section 5.6.4 of the NU5 protocol specification [#NU5-orchardencodings].","breadcrumbs":"Design » Keys and addresses","id":"17","title":"Keys and addresses"},"18":{"body":"When designing Sapling, we defined a BIP 32 -like mechanism for generating hierarchical deterministic wallets in ZIP 32 . We decided at the time to stick closely to the design of BIP 32, on the assumption that there were Bitcoin use cases that used both hardened and non-hardened derivation that we might not be aware of. This decision created significant complexity for Sapling: we needed to handle derivation separately for each component of the expanded spending key and full viewing key (whereas for transparent addresses there is only a single component in the spending key). Non-hardened derivation enables creating a multi-level path of child addresses below some parent address, without involving the parent spending key. The primary use case for this is HD wallets for transparent addresses, which use the following structure defined in BIP 44 : (H) BIP 44 (H) Coin type: Zcash (H) Account 0 (N) Normal addresses (N) Address 0 (N) Address 1... (N) Change addresses (N) Change address 0 (N) Change address 1... (H) Account 1... Shielded accounts do not require separating change addresses from normal addresses, because addresses are not revealed in transactions. Similarly, there is also no need to generate a fresh spending key for every transaction, and in fact this would cause a linear slow-down in wallet scanning. But for users who do want to generate multiple addresses per account, they can generate the following structure, which does not use non-hardened derivation: (H) ZIP 32 (H) Coin type: Zcash (H) Account 0 Diversified address 0 Diversified address 1... (H) Account 1... Non-hardened derivation is therefore only required for use-cases that require the ability to derive more than one child layer of addresses. However, in the years since Sapling was deployed, we have not seen any such use cases appear. Therefore, for Orchard we only define hardened derivation, and do so with a much simpler design than ZIP 32. All derivations produce an opaque binary spending key, from which the keys and addresses are then derived. As a side benefit, this makes key formats shorter. (The formats that will actually be used in practice for Orchard will correspond to the simpler Sapling formats in the protocol specification, rather than the longer and more complicated \"extended\" ones defined by ZIP 32.)","breadcrumbs":"Design » Hierarchical deterministic wallets","id":"18","title":"Hierarchical deterministic wallets"},"19":{"body":"In Sprout, we had a single proof that represented two spent notes and two new notes. This was necessary in order to faciliate spending multiple notes in a single transaction (to balance value, an output of one JoinSplit could be spent in the next one), but also provided a minimal level of arity-hiding: single-JoinSplit transactions all looked like 2-in 2-out transactions, and in multi-JoinSplit transactions each JoinSplit looked like a 1-in 1-out. In Sapling, we switched to using value commitments to balance the transaction, removing the min-2 arity requirement. We opted for one proof per spent note and one (much simpler) proof per output note, which greatly improved the performance of generating outputs, but removed any arity-hiding from the proofs (instead having the transaction builder pad transactions to 1-in, 2-out). For Orchard, we take a combined approach: we define an Orchard transaction as containing a bundle of actions, where each action is both a spend and an output. This provides the same inherent arity-hiding as multi-JoinSplit Sprout, but using Sapling value commitments to balance the transaction without doubling its size. TODO: Depending on the circuit cost, we may switch to having an action internally represent either a spend or an output. Externally spends and outputs would still be indistinguishable, but the transaction would be larger.","breadcrumbs":"Design » Actions","id":"19","title":"Actions"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Bootstrap Open Source Licence, version 1.0, or at your option, any later version. See the file LICENSE-BOSL for the terms of the Bootstrap Open Source Licence, version 1.0. The purpose of the BOSL is to allow commercial improvements to the package while ensuring that all improvements are open source. See here for why the BOSL exists.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"TODO: One memo per tx vs one memo per output","breadcrumbs":"Design » Memo fields","id":"20","title":"Memo fields"},"21":{"body":"As in Sapling, we require two kinds of commitment schemes in Orchard: HomomorphicCommit is a linearly homomorphic commitment scheme with perfect hiding, and strong binding reducible to DL. Commit and ShortCommit are commitment schemes with perfect hiding, and strong binding reducible to DL. By \"strong binding\" we mean that the scheme is collision resistant on the input and randomness. We instantiate HomomorphicCommit with a Pedersen commitment, and use it for value commitments: cv=HomomorphicCommitrcvcv(v) We instantiate Commit and ShortCommit with Sinsemilla, and use them for all other commitments: ivk=ShortCommitrivkivk(ak,nk) cm=Commitrcmcm(rest of note) This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient Sinsemilla instead of Bowe--Hopwood Pedersen hashes. Note that for ivk, we also deviate from Sapling in two ways: We use ShortCommit to derive ivk instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit, at the cost of requiring rivk to be part of the full viewing key. We define ivk as an integer in [1,qP); that is, we exclude ivk=0. For Sapling, we relied on BLAKE2s to make ivk=0 infeasible to produce, but it was still technically possible. For Orchard, we get this by construction: 0 is not a valid x-coordinate for any Pallas point. SinsemillaShortCommit internally maps points to field elements by replacing the identity (which has no affine coordinates) with 0. But SinsemillaCommit is defined using incomplete addition, and thus will never produce the identity.","breadcrumbs":"Design » Commitments","id":"21","title":"Commitments"},"22":{"body":"The commitment tree structure for Orchard is identical to Sapling: A single global commitment tree of fixed depth 32. Note commitments are appended to the tree in-order from the block. Valid Orchard anchors correspond to the global tree state at block boundaries (after all commitments from a block have been appended, and before any commitments from the next block have been appended). The only difference is that we instantiate MerkleCRHOrchard with Sinsemilla (whereas MerkleCRHSapling used a Bowe--Hopwood Pedersen hash).","breadcrumbs":"Design » Commitment tree","id":"22","title":"Commitment tree"},"23":{"body":"The fixed-depth incremental Merkle trees that we use (in Sprout and Sapling, and again in Orchard) require specifying an \"empty\" or \"uncommitted\" leaf - a value that will never be appended to the tree as a regular leaf. For Sprout (and trees composed of the outputs of bit-twiddling hash functions), we use the all-zeroes array; the probability of a real note having a colliding note commitment is cryptographically negligible. For Sapling, where leaves are u-coordinates of Jubjub points, we use the value 1 which is not the u-coordinate of any Jubjub point. Orchard note commitments are the x-coordinates of Pallas points; thus we take the same approach as Sapling, using a value that is not the x-coordinate of any Pallas point as the uncommitted leaf value. We use the value 2 for both Pallas and Vesta, because 23+5 is not a square in either Fp or Fq: sage: p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001\nsage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: EllipticCurve(GF(p), [0, 5]).count_points() == q\nTrue\nsage: EllipticCurve(GF(q), [0, 5]).count_points() == p\nTrue\nsage: Mod(13, p).is_square()\nFalse\nsage: Mod(13, q).is_square()\nFalse Note: There are also no Pallas points with x-coordinate 0, but we map the identity to (0,0) within the circuit. Although SinsemillaCommit cannot return the identity (the incomplete addition would return ⊥ instead), it would arguably be confusing to rely on that.","breadcrumbs":"Design » Uncommitted leaves","id":"23","title":"Uncommitted leaves"},"24":{"body":"We considered splitting the commitment tree into several sub-trees: Bundle tree, that accumulates the commitments within a single bundle (and thus a single transaction). Block tree, that accumulates the bundle tree roots within a single block. Global tree, that accumulates the block tree roots. Each of these trees would have had a fixed depth (necessary for being able to create proofs). Chains that integrated Orchard could have decoupled the limits on commitments-per-subtree from higher-layer constraints like block size, by enabling their blocks and transactions to be structured internally as a series of Orchard blocks or txs (e.g. a Zcash block would have contained a Vec, that each were appended in-order). The motivation for considering this change was to improve the lives of light client wallet developers. When a new note is received, the wallet derives its incremental witness from the state of the global tree at the point when the note's commitment is appended; this incremental state then needs to be updated with every subsequent commitment in the block in-order. Wallets can't get help from the server to create these for new notes without leaking the specific note that was received. We decided that this was too large a change from Sapling, and that it should be possible to improve the Incremental Merkle Tree implementation to work around the efficiency issues without domain-separating the tree.","breadcrumbs":"Design » Considered alternatives","id":"24","title":"Considered alternatives"},"25":{"body":"The nullifier design we use for Orchard is nf=ExtractP([(Fnk(ρ)+ψ)modp]G+cm), where: F is a keyed circuit-efficient PRF (such as Rescue or Poseidon). ρ is unique to this output. As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. Given that an action consists of a single spend and a single output, we set ρ to be the nullifier of the spent note. ψ is sender-controlled randomness. It is not required to be unique, and in practice is derived from both ρ and a sender-selected random value rseed: ψ=KDFψ(ρ,rseed). G is a fixed independent base. ExtractP extracts the x-coordinate of a Pallas curve point. This gives a note structure of (addr,v,ρ,ψ,rcm). The note plaintext includes rseed in place of ψ and rcm, and omits ρ (which is a public part of the action).","breadcrumbs":"Design » Nullifiers","id":"25","title":"Nullifiers"},"26":{"body":"We care about several security properties for our nullifiers: Balance: can I forge money? Note Privacy: can I gain information about notes only from the public block chain? This describes notes sent in-band. Note Privacy (OOB): can I gain information about notes sent out-of-band, only from the public block chain? In this case, we assume privacy of the channel over which the note is sent, and that the adversary does not have access to any notes sent to the same address which are then spent (so that the nullifier is on the block chain somewhere). Spend Unlinkability: given the incoming viewing key for an address, and not the full viewing key, can I (possibly the sender) detect spends of any notes sent to that address? We're giving ivk to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address. Faerie Resistance: can I perform a Faerie Gold attack (i.e. cause notes to be accepted that are unspendable)? We're giving the full viewing key to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address, and be able to derive every nullifier. We assume (and instantiate elsewhere) the following primitives: GH is a cryptographic hash into the group (such as BLAKE2s with simplified SWU), used to derive all fixed independent bases. E is an elliptic curve (such as Pallas). KDF is the note encryption key derivation function. For our chosen design, our desired security properties rely on the following assumptions: BalanceNote PrivacyNote Privacy (OOB)Spend UnlinkabilityFaerie ResistanceDLEHashDHEKDFNear perfect‡DDHE†∨PRFFDLE HashDHEKDF is computational Diffie-Hellman using KDF for the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker than DDHE but stronger than DLE. We omit ROGH as a security assumption because we only rely on the random oracle applied to fixed inputs defined by the protocol, i.e. to generate the fixed base G, not to attacker-specified inputs. † We additionally assume that for any input x, {Fnk(x):nk∈E} gives a scalar in an adequate range for DDHE. (Otherwise, F could be trivial, e.g. independent of nk.) ‡ Statistical distance <2−167.8 from perfect.","breadcrumbs":"Design » Security properties","id":"26","title":"Security properties"},"27":{"body":"⚠ Caution: be skeptical of the claims in this table about what problem(s) each security property depends on. They may not be accurate and are definitely not fully rigorous. The entries in this table omit the application of ExtractP, which is an optimization to halve the nullifier length. That optimization requires its own security analysis, but because it is a deterministic mapping, only Faerie Resistance could be affected by it. nf[nk][θ]H[nk]H+[rnf]IHash([nk][θ]H)Hash([nk]H+[rnf]I)[Fnk(ψ)][θ]H[Fnk(ψ)]H+[rnf]I[Fnk(ψ)]G+[θ]H[Fnk(ψ)]H+cm[Fnk(ρ,ψ)]G+cm[Fnk(ρ)]G+cm[Fnk(ρ,ψ)]Gv+[rnf]I[Fnk(ρ)]Gv+[rnf]I[(Fnk(ρ)+ψ)modp]Gv[Fnk(ρ,ψ)]G+Commitrnfnf(v,ρ)[Fnk(ρ)]G+Commitrnfnf(v,ρ)Note(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,rnf,ψ,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,ψ,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)BalanceDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLENote PrivacyHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFNote Priv OOBPerfectPerfectPerfectPerfectPerfectPerfectPerfectDDHE†DDHE†DDHE†PerfectPerfectNear perfect‡PerfectPerfectSpend UnlinkabilityDDHEDDHEDDHE∨PreHashDDHE∨PreHashDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFFaerie ResistanceROGH∧DLEROGH∧DLECollHash∧ROGH∧DLECollHash∧ROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEDLEDLECollF∧DLECollF∧DLEbrokenDLEDLEReason not to useNo SU for DL-breakingNo SU for DL-breakingCollHash for FRCollHash for FRPerf. (2 var-base)Perf. (1 var+1 fix-base)Perf. (1 var+1 fix-base)NP(OOB) not perfectNP(OOB) not perfectNP(OOB) not perfectCollF for FRCollF for FRbroken for FRPerf. (2 fix-base)Perf. (2 fix-base) In the above alternatives: Hash is a keyed circuit-efficient hash (such as Rescue). I is an fixed independent base, independent of G and any others returned by GH. Gv is a pair of fixed independent bases (independent of all others), where the specific choice of base depends on whether the note has zero value. H is a base unique to this output. For non-zero-valued notes, H=GH(ρ). As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. For zero-valued notes, H is constrained by the circuit to a fixed base independent of I and any others returned by GH.","breadcrumbs":"Design » Considered alternatives","id":"27","title":"Considered alternatives"},"28":{"body":"In order to satisfy the Balance security property, we require that the circuit must be able to enforce that only one nullifier is accepted for a given note. As in Sprout and Sapling, we achieve this by ensuring that the nullifier deterministically depends only on values committed to (directly or indirectly) by the note commitment. As in Sapling, this involves arguing that: There can be only one ivk for a given addr. This is true because the circuit checks that pkd=[ivk]gd, and the mapping ivk↦[ivk]gd is an injection for any gd. (ivk is in the base field of E, which must be smaller than its scalar field, as is the case for Pallas.) There can be only one nk for a given ivk. This is true because the circuit checks that ivk=ShortCommitrivkivk(ak,nk) where ShortCommit is binding (see Commitments ).","breadcrumbs":"Design » Rationale","id":"28","title":"Rationale"},"29":{"body":"Faerie Resistance requires that nullifiers be unique. This is primarily achieved by taking a unique value (checked for uniqueness by the public consensus rules) as an input to the nullifier. However, it is also necessary to ensure that the transformations applied to this value preserve its uniqueness. Meanwhile, to achieve Spend Unlinkability , we require that the nullifier does not reveal any information about the unique value it is derived from. The design alternatives fall into two categories in terms of how they balance these requirements: Publish a unique value ρ at note creation time, and blind that value within the nullifier computation. This is similar to the approach taken in Sprout and Sapling, which both implemented nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while Sapling uses BLAKE2s. Derive a unique base H from some unique value, publish that unique base at note creation time, and then blind the base (either additively or multiplicatively) during nullifier computation. For Spend Unlinkability , the only value unknown to the adversary is nk, and the cryptographic assumptions only involve the first term (other terms like cm or [rnf]I cannot be extracted directly from the observed nullifiers, but can be subtracted from them). We therefore ensure that the first term does not commit directly to the note (to avoid a DL-breaking adversary from immediately breaking SU ). We were considering using a design involving H with the goal of eliminating all usages of a PRF inside the circuit, for two reasons: Instantiating PRFF with a traditional hash function is expensive in the circuit. We didn't want to solely rely on an algebraic hash function satisfying PRFF to achieve Spend Unlinkability . However, those designs rely on both ROGH and DLE for Faerie Resistance , while still requiring DDHE for Spend Unlinkability . (There are two designs for which this is not the case, but they rely on DDHE† for Note Privacy (OOB) which was not acceptable). By contrast, several designs involving ρ (including the chosen design) have weaker assumptions for Faerie Resistance (only relying on DLE), and Spend Unlinkability does not require PRFF to hold: they can fall back on the same DDHE assumption as the H designs (along with an additional assumption about the output of F which is easily satisfied).","breadcrumbs":"Design » Use of ρ","id":"29","title":"Use of ρ"},"3":{"body":"","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"Most of the designs include either a multiplicative blinding term [θ]H, or an additive blinding term [rnf]I, in order to achieve perfect Note Privacy (OOB) (to an adversary who does not know the note). The chosen design is effectively using [ψ]G for this purpose; a DL-breaking adversary only learns Fnk(ρ)+ψ(modp). This reduces Note Privacy (OOB) from perfect to statistical, but given that ψ is from a distribution statistically close to uniform on [0,q), this is statistically close to better than 2−128. The benefit is that it does not require an additional scalar multiplication, making it more efficient inside the circuit. ψ's derivation has two motivations: Deriving from a random value rseed enables multiple derived values to be conveyed to the recipient within an action (such as the ephemeral secret esk, per ZIP 212 ), while keeping the note plaintext short. Mixing ρ into the derivation ensures that the sender can't repeat ψ across two notes, which could have enabled spend linkability attacks in some designs. The note that is committed to, and which the circuit takes as input, only includes ψ (i.e. the circuit does not check the derivation from rseed). However, an adversarial sender is still constrained by this derivation, because the recipient recomputes ψ during note decryption. If an action were created using an arbitrary ψ (for which the adversary did not have a corresponding rseed), the recipient would derive a note commitment that did not match the action's commitment field, and reject it (as in Sapling).","breadcrumbs":"Design » Use of ψ","id":"30","title":"Use of ψ"},"31":{"body":"The nullifier commits to the note value via cm for two reasons: It domain-separates nullifiers for zero-valued notes from other notes. This is necessary because we do not require zero-valued notes to exist in the commitment tree. Designs that bind the nullifier to Fnk(ρ) require CollF to achieve Faerie Resistance (and similarly where Hash is applied to a value derived from H). Adding cm to the nullifier avoids this assumption: all of the bases used to derive cm are fixed and independent of G, and so the nullifier can be viewed as a Pedersen hash where the input includes ρ directly. The Commitnf variants were considered to avoid directly depending on cm (which in its native type is a base field element, not a group element). We decided instead to follow Sapling by defining an intermediate representation of cm as a group element, that is only used in nullifier computation. The circuit already needs to compute cm, so this improves performance by removing We also considered variants that used a choice of fixed bases Gv to provide domain separation for zero-valued notes. The most performant design (similar to the chosen design) does not achieve Faerie Resistance for an adversary that knows the recipient's full viewing key (ψ could be brute-forced to cancel out Fnk(ρ), causing a collision), and the other variants require assuming CollF as mentioned above.","breadcrumbs":"Design » Use of cm","id":"31","title":"Use of cm"},"32":{"body":"Orchard signatures are an instantiation of RedDSA with a cofactor of 1. TODO: Should it be possible to sign partial transactions? If we're going to merge down all the signatures into a single one, and also want this, we need to ensure there's a feasible MPC.","breadcrumbs":"Design » Signatures","id":"32","title":"Signatures"},"33":{"body":"","breadcrumbs":"Design » Circuit","id":"33","title":"Circuit"},"34":{"body":"","breadcrumbs":"Design » Circuit » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"We will use formulae for curve arithmetic using affine coordinates on short Weierstrass curves, derived from section 4.1 of Hüseyin Hışıl's thesis . Inputs: P=(xp,yp),Q=(xq,yq) Output: R=P⸭Q=(xr,yr) The formulae from Hışıl's thesis are: x3=(x1−x2y1−y2)2−x1−x2 y3=x1−x2y1−y2⋅(x1−x3)−y1 Rename: (x1,y1) to (xq,yq) (x2,y2) to (xp,yp) (x3,y3) to (xr,yr). Let λ=xq−xpyq−yp=xp−xqyp−yq, which we implement as λ⋅(xp−xq)=yp−yq Also, xr=λ2−xq−xp yr=λ⋅(xq−xr)−yq which is equivalent to xr+xq+xp=λ2 Assuming xp=xq, ⟹⟹⟹(xr+xq+xp)⋅(xp−xq)2(xr+xq+xp)⋅(xp−xq)2yryr+yq(yr+yq)⋅(xp−xq)=====λ2⋅(xp−xq)2(λ⋅(xp−xq))2λ⋅(xq−xr)−yqλ⋅(xq−xr)λ⋅(xp−xq)⋅(xq−xr) Substituting for λ⋅(xp−xq), we get the constraints: (xr+xq+xp)⋅(xp−xq)2−(yp−yq)2=0 Note that this constraint is unsatisfiable for P⸭(−P) (when P=O), and so cannot be used with arbitrary inputs. (yr+yq)(xp−xq)−(yp−yq)(xq−xr)=0","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Incomplete addition","id":"35","title":"Incomplete addition"},"36":{"body":"OO(xp,yp)(x,y)(x,y)(xp,yp)++++++O(xq,yq)O(x,y)(x,−y)(xq,yq)=O=(xq,yq)=(xp,yp)=[2](x,y)=O=(xp,yp)⸭(xq,yq),if xp=xq Suppose that we represent O as (0,0). (0 is not an x-coordinate of a valid point because we would need y2=x3+5, and 5 is not square in Fq. Also 0 is not a y-coordinate of a valid point because −5 is not a cube in Fq.) P+Q(xp,yp)+(xq,yq)λxryr=R=(xr,yr)=xq−xpyq−yp=λ2−xp−xq=λ(xp−xr)−yp For the doubling case, Hışıl's thesis tells us that λ has to instead be computed as 2y3x2. Define inv0(x)={0,1/x,if x=0otherwise. Witness α,β,γ,δ,λ where: α=inv0(xq−xp) β=inv0(xp) γ=inv0(xq) δ={inv0(yq+yp),0,if xq=xpotherwise λ=⎩⎨⎧xq−xpyq−yp,2yp3xp20,if xq=xpif xq=xp∧yp=0otherwise.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Complete addition","id":"36","title":"Complete addition"},"37":{"body":"Degree456666444444Constraintqadd⋅(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))qadd⋅(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅(1−xp⋅β)⋅(xr−xq)qadd⋅(1−xp⋅β)⋅(yr−yq)qadd⋅(1−xq⋅γ)⋅(xr−xp)qadd⋅(1−xq⋅γ)⋅(yr−yp)qadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xrqadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr============000000000000Meaningxq=xp⟹λ=xq−xpyq−yp{xq=xp∧yp=0⟹λ=2yp3xp2xq=xp∧yp=0⟹xp=0xp=0∧xq=0∧xq=xp⟹xr=λ2−xp−xqxp=0∧xq=0∧xq=xp⟹yr=λ⋅(xp−xr)−ypxp=0∧xq=0∧yq=−yp⟹xr=λ2−xp−xqxp=0∧xq=0∧yq=−yp⟹yr=λ⋅(xp−xr)−ypxp=0⟹xr=xqxp=0⟹yr=yqxq=0⟹xr=xpxq=0⟹yr=ypxq=xp∧yq=−yp⟹xr=0xq=xp∧yq=−yp⟹yr=0 Max degree: 6","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraints","id":"37","title":"Constraints"},"38":{"body":"1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0).xq=0⟹(xr,yr)=(xp,yp). Propositions: (1)(2)(3)(4)(5)(6)xq=xp⟹λ=(yq−yp)/(xq−xp).(xq=xp)∧yp=0⟹λ=3xp2/2yp(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))⟹(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp)xp=0⟹(xr,yr)=(xq,yq)xq=0⟹(xr,yr)=(xp,yp)xq=xp∧yq=−yp⟹(xr,yr)=(0,0) Test cases: (xp,yp)+(xq,yq)=(xr,yr) (0,0)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because yp=0 holds because xp=0 holds because xp=0 only when xr=0,yr=0 holds because xq=0 only when xr=0,yr=0 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (x,y)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xq=0 holds because xp=0 because 0 is not a valid x-coordinate only when β=xp−1 holds because xq=0 only when (xr,yr)=(xp,yp) holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xp,yp) is the only solution (0,0)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xp=0 holds because xp=0 only when (xr,yr)=(xq,yq) holds because xq=0 because 0 is not a valid x-coordinate only when γ=xq−1 holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xq,yq) is the only solution (x,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate) only when λ=3xp2/2yp holds because xp=0∧xq=0andyq=−yp only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xp=0 only when γ=xq−1 holds because yq=−yp only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution (x,y)+(x,−y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate)only when λ=3xp2/2yp holds because xp=0∧xq=0 but yq=−yp∧xq=xp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (ζx,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xp=xp only when α=(xq−xp)−1 which is defined because xq=xp holds because (xp=0)∧(xq=0)∧(xq=xp) only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp only when δ=0 Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution All remaining cases (x,y)+(x′,y′) are identical to the case (ζx,y)+(x,y) when λαβγδ=(yq−yp)/(xq−xp)=(xq−xp)−1=xp−1=xq−1=0 because in all remaining cases, xq=xp,xp=0,xq=0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Analysis of constraints","id":"38","title":"Analysis of constraints"},"39":{"body":"There are 6 fixed bases in the Orchard protocol: KOrchard, used in deriving the nullifier; GOrchard, used in spend authorization; R base for NoteCommitOrchard; V and R bases for ValueCommitOrchard; and R base for Commitivk.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"39","title":"Fixed-base scalar multiplication"},"4":{"body":"","breadcrumbs":"Concepts » Preliminaries","id":"4","title":"Preliminaries"},"40":{"body":"In most cases, we multiply the fixed bases by 255−bit scalars from Fq. We decompose a full-width scalar α into 85 3-bit windows: α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23). The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255). If k0..84 is witnessed directly then no issue of canonicity arises. If the scalar is given as a base field element, then care must be taken to ensure a canonical representation, since 2255>p. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit. Degree9Constraintqscalar-fixed⋅1⋅(i=0∑7w−i)=0 At the point of witnessing the scalar, we range-constrain each 3-bit word of its decomposition. Degree9Constraintqwitness-scalar-fixed⋅range_check(ki,8)=0 where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"40","title":"Witness scalar"},"41":{"body":"Then, we precompute multiples of the fixed base B for each window. This takes the form of a window table: M[0..85)[0..8) such that: for the first 84 rows M[0..84)[0..8): M[w][k]=[(k+2)⋅(23)w]B in the last row M[84][0..8): M[w][k]=[k⋅(23)w−j=0∑8323j+1]B The additional (k+2) term lets us avoid adding the point at infinity in the case k=0. We offset these accumulated terms by subtracting them in the final window, i.e. we subtract j=0∑8323j+1. Note: Although an offset of (k+1) would naively suffice, it introduces an edge case when k0=7,k1=0. In this case, the window table entries evaluate to the same point: M[0][k0]=[(7+1)∗(23)0]B=[8]B, M[1][k1]=[(0+1)∗(23)1]B=[8]B. In fixed-base scalar multiplication, we sum the multiples of B at each window (except the last) using incomplete addition. Since the point doubling case is not handled by incomplete addition, we avoid it by using an offset of (k+2). For each window of fixed-base multiples M[w]=(M[w][0],⋯,M[w][7]),w∈[0..84): Define a Lagrange interpolation polynomial Lx(k) that maps k∈[0..8) to the x-coordinate of the multiple M[w][k], i.e. Lx(k)=⎩⎨⎧([(k+2)⋅(23)w]B)x([k⋅(23)w−j=0∑8323j+1]B)xfor w∈[0..84);for w=84; and Find a value zw such that zw+(M[w][k])y is a square u2 in the field, but the wrong-sign y-coordinate zw−(M[w][k])y does not produce a square. Repeating this for all 85 windows, we end up with: an 85×8 table Lx storing 8 coefficients interpolating the x−coordinate for each window. Each x-coordinate interpolation polynomial will be of the form Lx[w](k)=c0+c1⋅k+c2⋅k2+⋯+c7⋅k7, where k∈[0..8),w∈[0..85) and ck's are the coefficients for each power of k; and a length-85 array Z of zw's. We load these precomputed values into fixed columns whenever we do fixed-base scalar multiplication in the circuit.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Load fixed base","id":"41","title":"Load fixed base"},"42":{"body":"Given a decomposed scalar α and a fixed base B, we compute [α]B as follows: For each kw,w∈[0..85),kw∈[0..8) in the scalar decomposition, witness the x- and y-coordinates (xw,yw)=M[w][kw]. Check that (xw,yw) is on the curve: yw2=xw3+b. Witness uw such that yw+zw=uw2. For all windows but the last, use incomplete addition to sum the M[w][kw]'s, resulting in [α−k84⋅(23)84+j=0∑8323j+1]B. For the last window, use complete addition M[83][k83]+M[84][k84] and return the final result. Note: complete addition is required in the final step to correctly map [0]B to a representation of the point at infinity, (0,0); and also to handle a corner case for which the last step is a doubling. Constraints: Degree843Constraintqmul-fixed⋅(Lx[w](kw)−xw)=0qmul-fixed⋅(yw2−xw3−b)=0qmul-fixed⋅(uw2−yw−Z[w])=0 where b=5 (from the Pallas curve equation).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"42","title":"Fixed-base scalar multiplication"},"43":{"body":"This is used for ValueCommitOrchard. We want to compute ValueCommitrcvOrchard(vold−vnew)=[vold−vnew]V+[rcv]R, where −(264−1)≤vold−vnew≤264−1 vold and vnew are each already constrained to 64 bits (by their use as inputs to NoteCommitOrchard). Witness the sign s and magnitude m such that s∈{−1,1}m∈[0,264)vold−vnew=s⋅m Then compute P=[m]V, and conditionally negate P using (x,y)↦(x,s⋅y). We compute the window table in a similar way to full-width fixed-base scalar multiplication, but with only ceil(64/3)=22 windows: αshort=k0+k1⋅(23)1+⋯+k21⋅(23)21,ki∈[0..23). In addition to the 21 full-width 3-bit constraints, we have two additional constraints: Degree33Constraintqscalar-fixed-short⋅(k21⋅(1−k21))=0qscalar-fixed-short⋅(s2−1)=0CommentThe last window must be a single bit.The sign must be 1 or −1.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication with signed short exponent","id":"43","title":"Fixed-base scalar multiplication with signed short exponent"},"44":{"body":"We support using a base field element as the scalar in fixed-base multiplication. This is used in DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition. Decompose the base field element α into three-bit windows using a running sum z, where zi+1=(zi−ai)/23 for α=a0+23a1+...+23⋅84a84. (This running sum z is not to be confused with the Z array used to check the y-coordinate of a fixed-base window.) We set z0=α, which means: z1z2z84z85=(α−k0)/23, (subtract the lowest 3 bits)=k1+23k2+...+23⋅83k84,=(z1−k1)/23=k2+23k3+...+23⋅82k84,⋯,=k84=(z84−k84)/23=0. We must range-constrain the difference at each step of the running sum. We also constrain the final output of the running sum to be 0. Degree92Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅23,23)=0qdecompose-base-field⋅z85=0 We also enforce canonicity of this decomposition. That is, we want to check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq. (We use big-endian bit order for convenient input into the variable-base scalar multiplication algorithm.) k=k254⋅2254+k253⋅2253+⋯+k0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"47","title":"Witness scalar"},"48":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication","id":"48","title":"Variable-base scalar multiplication"},"49":{"body":"Define a running sum zj=∑i=jn(ki⋅2i−j), where n=254 and: zn+1=0,zn=kn,(most significant bit)z0=k. Initialize A254=[2]T. for i from 254 down to 4: (ki)(ki−1)=0zi=2zi+1+kixP,i=xTyP,i=(2ki−1)⋅yT(conditionally negate)λ1,i⋅(xA,i−xP,i)=yA,i−yP,iλ1,i2=xR,i+xA,i+xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)=2yA,iλ2,i2=xA,i−1+xR,i+xA,iλ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1, where xR,i=(λ1,i2−xA,i−xT). After substitution of xP,i,yP,i,xR,i,yA,i, and yA,i−1, this becomes: Initialize A254=[2]T. for i from 254 down to 4: // let ki=zi−2zi+1// let yA,i=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT))(ki)(ki−1)=0λ1,i⋅(xA,i−xT)=yA,i−(2ki−1)⋅yTλ2,i2=xA,i−1+λ1,i2−xT {λ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1,λ2,4⋅(xA,4−xA,3)=yA,4+yA,3witnessed,if i>4if i=4. Here, yA,3witnessed is assigned to a cell. This is unlike previous yA,i's, which were implicitly derived from λ1,i,λ2,i,xA,i,xT, but never actually assigned. The bits k3…1 are used in three further steps, using complete addition : for i from 3 down to 1: // let ki=zi−2zi+1(ki)(ki−1)=0(xA,i−1,yA,i−1)=((xA,i,yA,i)+(xT,yT))+(xA,i,yA,i) If the least significant bit is set k0=1, we return the accumulator A. Else, if k0=0, we return A−T (also using complete addition). Let B={(0,0),(xT,−yT), if k0=1, otherwise. Output (xA,0,yA,0)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraint program for optimized double-and-add (incomplete addition)","id":"49","title":"Constraint program for optimized double-and-add (incomplete addition)"},"5":{"body":"","breadcrumbs":"User Documentation","id":"5","title":"User Documentation"},"50":{"body":"We need six advice columns to witness (xT,yT,λ1,λ2,xA,i,zi). However, since (xT,yT) are the same, we can perform two incomplete additions in a single row, reusing the same (xT,yT). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xTxTxT⋮xTyTyTyT⋮yTzhiz255=0z254z253⋮z130xAhixA,254=2[T]xxA,253⋮xA,130xA,129λ1hiyA,254=2[T]yλ1,254λ1,253⋮λ1,130yA,129λ2hiλ2,254λ2,253⋮λ2,130qmulhi122⋮3zloz130z129z128⋮z5z4xAloxA,129xA,128⋮xA,5xA,4xA,3λ1loyA,129λ1,129λ1,128⋮λ1,5λ1,4yA,3λ2loλ2,129λ2,128⋮λ2,5λ2,4qmullo122⋮23 For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. qmul=1 This gate is only used on the first row (before the for loop). We check that λ1,λ2 are initialized to values consistent with the initial yA. Degree5Constraintqmulone⋅(yA,nwitnessed−yA,n)=0 where qmuloneyA,nyA,nwitnessed=qmul⋅(2−qmul)⋅(3−qmul),=2(λ1,n+λ2,n)⋅(xA,n−(λ1,n2−xA,n−xT)), is witnessed. qmul=2 This gate is used on all rows corresponding to the for loop except the last. Degree445655Constraintqmultwo⋅(xT,cur−xT,next)=0qmultwo⋅(yT,cur−yT,next)=0qmultwo⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmultwo⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmultwo⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmultwo⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1)=0 where qmultwoyA,iyA,i−1=qmul⋅(1−qmul)⋅(3−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)),=2(λ1,i−1+λ2,i−1)⋅(xA,i−1−(λ1,i−12−xA,i−1−xT)), qmul=3 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA has been witnessed correctly. Degree5655Constraintqmulthree⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmulthree⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmulthree⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmulthree⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1witnessed)=0 where qmulthreeyA,iyA,i−1witnessed=qmul⋅(1−qmul)⋅(2−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)), is witnessed.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Circuit design","id":"50","title":"Circuit design"},"51":{"body":"zi cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0=α+tq can overflow [0,p). Since overflow can only occur in the final step that constrains z0=2⋅z1+k0, we have z0=k(modp). It is then sufficient to also check that z0=α+tq(modp) (so that k=α+tq(modp)) and that k∈[tq,p+tq). These conditions together imply that k=α+tq as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0 do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check","id":"51","title":"Overflow check"},"52":{"body":"Since tp+tq<2130, we have [tq,p+tq)=[tq,tq+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq−2130,p+tq)). We may assume that k=α+tq(modp). Therefore, k∈[tq,p+tq)⇔⇔⇔(k∈[tq,tq+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(k∈[tq,tq+2130)∨k∈[2130,2254)))∧(k254=1⟹(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq−2130,p+tq) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq to give k+2130−p−tq∈[0,2130); observe that k+2130−p−tq is guaranteed to be in [2130−tp−tq,2131−tp−tq) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq(modp), observe that (k+2130−p−tq)modp=(α+tq+2130−p−tq)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq,p+tq)⇔⇔(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254=0⟹(α∈[0,2130)∨k253..130 are not all 0))∧(k254=1⟹(k253..130 are all 0∧(α+2130)modp∈[0,2130))) Constraining k253..130 to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi=∑h=in(kh⋅2h−i), so we have: z130z130z130−k254⋅2124===∑h=130254(kh⋅2h−130)k254⋅2254−130+∑h=130253(kh⋅2h−130)∑h=130253(kh⋅2h−130) So k253..130 are all 0 exactly when z130=k254⋅2124. Finally, we can merge the 130-bit decompositions for the k254=0 and k254=1 cases by checking that (α+k254⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Optimized check for k∈[tq,p+tq)","id":"52","title":"Optimized check for k∈[tq,p+tq)"},"53":{"body":"Let s=α+k254⋅2130. The constraints for the overflow check are: z0k254=1⟹(z130k254=0⟹(z130=α+tq(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130)) Define inv0(x)={0,1/x,if x=0otherwise. Witness η=inv0(z130), and decompose smodp as s129..0. Then the needed gates are: Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0 where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check constraints","id":"53","title":"Overflow check constraints"},"54":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » Sinsemilla","id":"54","title":"Sinsemilla"},"55":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Circuit » Gadgets » Overview","id":"55","title":"Overview"},"56":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi. let Acc0:=Q for i from 0 up to n−1: let Acci+1:=(Acci⸭P[mi+1])⸭Acci return Accn Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Circuit » Gadgets » Description","id":"56","title":"Description"},"57":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr(M)=Hash(M)⸭[r]H. Let ShortCommitr(M) be the x-coordinate of Commitr(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Circuit » Gadgets » Use as a commitment scheme","id":"57","title":"Use as a commitment scheme"},"58":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Circuit » Gadgets » Efficient implementation","id":"58","title":"Efficient implementation"},"59":{"body":"Let P={(j,xP[j],yP[j]) for j∈{0..2k−1}}. Input: m1..=n. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i,yA,i) corresponds to Acci in the protocol spec.) Output: (xA,n,yA,n). (xA,0,yA,0)=Q for i from 0 up to n−1: yP,i=yA,i−λ1,i⋅(xA,i−xP,i) xR,i=λ1,i2−xA,i−xP,i 2⋅yA,i=(λ1,i+λ2,i)⋅(xA,i−xR,i) (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i λ2,i⋅(xA,i−xA,i+1)=yA,i+yA,i+1","breadcrumbs":"Design » Circuit » Gadgets » Constraint program","id":"59","title":"Constraint program"},"6":{"body":"","breadcrumbs":"User Documentation » Creating keys and addresses","id":"6","title":"Creating keys and addresses"},"60":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » PLONK / Halo 2 constraints","id":"60","title":"PLONK / Halo 2 constraints"},"61":{"body":"We have an n-bit message m=m1+2km2+...+2k⋅(n−1)mn. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0=α and define zi+1:=2Kzi−mi+1. We will end up with zn=0. Rearranging gives us an expression for each word of the original message mi+1=zi−2k⋅zi+1, which we can look up in the table. In other words, zn−i=h=0∑i−12kh⋅mh+1. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2 selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1 that effectively forces zn to zero for the last step of each element, as indicated by qS2. This allows the cell that would have been zn to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Circuit » Gadgets » Message decomposition","id":"61","title":"Message decomposition"},"62":{"body":"Note: qS3 is synthesized from qS1 and qS2; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′xAxQxA,1xA,2⋮xA,n−1xA,0′xA,1′xA,2′⋮xA,n−1′xA,n′xPxP[m1]xP[m2]xP[m3]⋮xP[mn]xP[m1′]xP[m2′]xP[m3′]⋮xP[mn′]bitsz0z1z2⋮zn−1z0′z1′z2′⋮zn−1′λ1λ1,0λ1,1λ1,2⋮λ1,n−1λ1,0′λ1,1′λ1,2′⋮λ1,n−1′yA,nλ2λ2,0λ2,1λ2,2⋮λ2,n−1λ2,0′λ2,1′λ2,2′⋮λ2,n−1′qS111111111110qS211110111120qS300000000020fixed_yQyQ0000000000tableidx012⋮⋮⋮⋮⋮⋮⋮⋮tablexxP[0]xP[1]xP[2]⋮⋮⋮⋮⋮⋮⋮⋮tableyyP[0]yP[1]yP[2]⋮⋮⋮⋮⋮⋮⋮⋮ xQ, z0, z0′, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Circuit » Gadgets » Layout","id":"62","title":"Layout"},"63":{"body":"For i∈[0,n), letxR,iYA,iyP,imi+1qS3=====λ1,i2−xA,i−xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)YA,i/2−λ1,i⋅(xA,i−xP,i)zi−2k⋅(qS2,i−qS3,i)⋅zi+1qS2⋅(qS2−1) The Halo 2 circuit API can automatically substitute yP,i, xR,i, yA,i, and yA,i+1, so we don't need to do that manually. xA,0=xQ 2⋅yQ=YA,0 for i from 0 up to n−1: (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i 4⋅λ2,i⋅(xA,i−xA,i+1)=2⋅YA,i+(2−qS3)⋅YA,i+1+2qS3⋅yA,n Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. Degree4536ConstraintfixedyQ⋅(2⋅fixedyQ−YA,0)=0qS1,i⇒(mi+1,xP,i,yP,i)∈PqS1,i⋅(λ2,i2−(xA,i+1+xR,i+xA,i))qS1,i⋅(4⋅λ2,i⋅(xA,i−xA,i+1)−(2⋅YA,i+(2−qS3,i)⋅YA,i+1+2⋅qS3,i⋅yA,n))=0 By gating the lookup expression on qS1, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (qS1⋅mi+1,qS1⋅xP,i+(1−qS1)⋅xP,0,qS1⋅yP,i+(1−qS1)⋅yP,0) This increases the degree of the lookup argument to 6.","breadcrumbs":"Design » Circuit » Gadgets » Optimized Sinsemilla gate","id":"63","title":"Optimized Sinsemilla gate"},"64":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Firstly, express α=k0+2K⋅k1+22K⋅k2+⋯, where each ki a K-bit value. We initialize the running sum z0=s, and compute subsequent terms zi=2Kzi−1−ki−1. This gives us: z0z1z2=α=k0+2K⋅k1+22K⋅k2+23K⋅k3+⋯,=(z0−k0)/2K=k1+2K⋅k2+22K⋅k3+⋯,=(z1−k1)/2K=k2+2K⋅k3+⋯,⋮ Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table. This gadget takes in α,n and returns (z0,z1,⋯,zn).","breadcrumbs":"Design » Circuit » Gadgets » Lookup decomposition","id":"64","title":"Lookup decomposition"},"65":{"body":"Strict mode constrains the running sum output zn to be zero, thus range-constraining the field element to be within n⋅K bits.","breadcrumbs":"Design » Circuit » Gadgets » Strict mode","id":"65","title":"Strict mode"},"66":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. 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: This library is being actively developed and should not be used in production software. Requires Rust 1.51+.","breadcrumbs":"orchard Crates.io","id":"0","title":"orchard Crates.io"},"1":{"body":"The Orchard Book Crate documentation","breadcrumbs":"Documentation","id":"1","title":"Documentation"},"10":{"body":"","breadcrumbs":"Design","id":"10","title":"Design"},"11":{"body":"","breadcrumbs":"General design notes","id":"11","title":"General design notes"},"12":{"body":"Keep the design close to Sapling, while eliminating aspects we don't like.","breadcrumbs":"Requirements","id":"12","title":"Requirements"},"13":{"body":"Delegated proving with privacy from the prover. We know how to do this, but it would require a discrete log equality proof, and the most efficient way to do this would be to do RedDSA and this at the same time, which means more work for e.g. hardware wallets.","breadcrumbs":"Non-requirements","id":"13","title":"Non-requirements"},"14":{"body":"Should we have one memo per output, or one memo per transaction, or 0..n memos? Variable, or (1 or n), is a potential privacy leak. Need to consider the privacy issue related to light clients requesting individual memos vs being able to fetch all memos.","breadcrumbs":"Open issues","id":"14","title":"Open issues"},"15":{"body":"TODO: UDAs: arbitrary vs whitelisted","breadcrumbs":"Note structure","id":"15","title":"Note structure"},"16":{"body":"For Sapling, we have encountered multiple places where the specification uses typed variables to define the consensus rules, but the C++ implementation in zcashd relied on byte encodings to implement them. This resulted in subtly-different consensus rules being deployed than were intended, for example where a particular type was not round-trip encodable. In Orchard, we avoid this by defining the consensus rules in terms of the byte encodings of all variables, and being explicit about any types that are not round-trip encodable. This makes consensus compatibility between strongly-typed implementations (such as this crate) and byte-oriented implementations easier to achieve.","breadcrumbs":"Typed variables vs byte encodings","id":"16","title":"Typed variables vs byte encodings"},"17":{"body":"Orchard keys and payment addresses are structurally similar to Sapling. The main change is that Orchard keys use the Pallas curve instead of Jubjub, in order to enable the future use of the Pallas-Vesta curve cycle in the Orchard protocol. (We already use Vesta as the curve on which Halo 2 proofs are computed, but this doesn't yet require a cycle.) Using the Pallas curve and making the most efficient use of the Halo 2 proof system involves corresponding changes to the key derivation process, such as using Sinsemilla for Pallas-efficient commitments. We also take the opportunity to remove all uses of expensive general-purpose hashes (such as BLAKE2s) from the circuit. We make several structural changes, building on the lessons learned from Sapling: The nullifier private key nsk is removed. Its purpose in Sapling was as defense-in-depth, in case RedDSA was found to have weaknesses; an adversary who could recover ask would not be able to spend funds. In practice it has not been feasible to manage nsk much more securely than a full viewing key, as the computational power required to generate Sapling proofs has made it necessary to perform this step on the same device that is creating the overall transaction (rather than on a more constrained device like a hardware wallet). We are also more confident in RedDSA now. nk is now a field element instead of a curve point, making it more efficient to generate nullifiers. ovk is now derived from fvk, instead of being derived in parallel. This places it in a similar position within the key structure to ivk, and also removes an issue where two full viewing keys could be constructed that have the same ivk but different ovks. Users still have control over whether ovk is used when constructing a transaction. All diversifiers now result in valid payment addresses, due to group hashing into Pallas being specified to be infallible. This removes significant complexity from the use cases for diversified addresses. The fact that Pallas is a prime-order curve simplifies the protocol and removes the need for cofactor multiplication in key agreement. Unlike Sapling, we define public (including ephemeral) and private keys used for note encryption to exclude the zero point and the zero scalar. Without this change, the implementation of the Orchard Action circuit would need special cases for the zero point, since Pallas is a short Weierstrass rather than an Edwards curve. This also has the advantage of ensuring that the key agreement has \"contributory behaviour\" — that is, if either party contributes a random scalar, then the shared secret will be random to an observer who does not know that scalar and cannot break Diffie–Hellman. Other than the above, Orchard retains the same design rationale for its keys and addresses as Sapling. For example, diversifiers remain at 11 bytes, so that a raw Orchard address is the same length as a raw Sapling address. Orchard payment addresses do not have a stand-alone string encoding. Instead, we define \"unified addresses\" that can bundle together addresses of different types, including Orchard. Unified addresses have a Human-Readable Part of \"u\" on Mainnet, i.e. they will have the prefix \"u1\". For specifications of this and other formats (e.g. for Orchard viewing and spending keys), see section 5.6.4 of the NU5 protocol specification [#NU5-orchardencodings].","breadcrumbs":"Design » Keys and addresses","id":"17","title":"Keys and addresses"},"18":{"body":"When designing Sapling, we defined a BIP 32 -like mechanism for generating hierarchical deterministic wallets in ZIP 32 . We decided at the time to stick closely to the design of BIP 32, on the assumption that there were Bitcoin use cases that used both hardened and non-hardened derivation that we might not be aware of. This decision created significant complexity for Sapling: we needed to handle derivation separately for each component of the expanded spending key and full viewing key (whereas for transparent addresses there is only a single component in the spending key). Non-hardened derivation enables creating a multi-level path of child addresses below some parent address, without involving the parent spending key. The primary use case for this is HD wallets for transparent addresses, which use the following structure defined in BIP 44 : (H) BIP 44 (H) Coin type: Zcash (H) Account 0 (N) Normal addresses (N) Address 0 (N) Address 1... (N) Change addresses (N) Change address 0 (N) Change address 1... (H) Account 1... Shielded accounts do not require separating change addresses from normal addresses, because addresses are not revealed in transactions. Similarly, there is also no need to generate a fresh spending key for every transaction, and in fact this would cause a linear slow-down in wallet scanning. But for users who do want to generate multiple addresses per account, they can generate the following structure, which does not use non-hardened derivation: (H) ZIP 32 (H) Coin type: Zcash (H) Account 0 Diversified address 0 Diversified address 1... (H) Account 1... Non-hardened derivation is therefore only required for use-cases that require the ability to derive more than one child layer of addresses. However, in the years since Sapling was deployed, we have not seen any such use cases appear. Therefore, for Orchard we only define hardened derivation, and do so with a much simpler design than ZIP 32. All derivations produce an opaque binary spending key, from which the keys and addresses are then derived. As a side benefit, this makes key formats shorter. (The formats that will actually be used in practice for Orchard will correspond to the simpler Sapling formats in the protocol specification, rather than the longer and more complicated \"extended\" ones defined by ZIP 32.)","breadcrumbs":"Design » Hierarchical deterministic wallets","id":"18","title":"Hierarchical deterministic wallets"},"19":{"body":"In Sprout, we had a single proof that represented two spent notes and two new notes. This was necessary in order to faciliate spending multiple notes in a single transaction (to balance value, an output of one JoinSplit could be spent in the next one), but also provided a minimal level of arity-hiding: single-JoinSplit transactions all looked like 2-in 2-out transactions, and in multi-JoinSplit transactions each JoinSplit looked like a 1-in 1-out. In Sapling, we switched to using value commitments to balance the transaction, removing the min-2 arity requirement. We opted for one proof per spent note and one (much simpler) proof per output note, which greatly improved the performance of generating outputs, but removed any arity-hiding from the proofs (instead having the transaction builder pad transactions to 1-in, 2-out). For Orchard, we take a combined approach: we define an Orchard transaction as containing a bundle of actions, where each action is both a spend and an output. This provides the same inherent arity-hiding as multi-JoinSplit Sprout, but using Sapling value commitments to balance the transaction without doubling its size. TODO: Depending on the circuit cost, we may switch to having an action internally represent either a spend or an output. Externally spends and outputs would still be indistinguishable, but the transaction would be larger.","breadcrumbs":"Design » Actions","id":"19","title":"Actions"},"2":{"body":"Copyright 2020 The Electric Coin Company. You may use this package under the Bootstrap Open Source Licence, version 1.0, or at your option, any later version. See the file LICENSE-BOSL for the terms of the Bootstrap Open Source Licence, version 1.0. The purpose of the BOSL is to allow commercial improvements to the package while ensuring that all improvements are open source. See here for why the BOSL exists.","breadcrumbs":"License","id":"2","title":"License"},"20":{"body":"TODO: One memo per tx vs one memo per output","breadcrumbs":"Design » Memo fields","id":"20","title":"Memo fields"},"21":{"body":"As in Sapling, we require two kinds of commitment schemes in Orchard: HomomorphicCommit is a linearly homomorphic commitment scheme with perfect hiding, and strong binding reducible to DL. Commit and ShortCommit are commitment schemes with perfect hiding, and strong binding reducible to DL. By \"strong binding\" we mean that the scheme is collision resistant on the input and randomness. We instantiate HomomorphicCommit with a Pedersen commitment, and use it for value commitments: cv=HomomorphicCommitrcvcv(v) We instantiate Commit and ShortCommit with Sinsemilla, and use them for all other commitments: ivk=ShortCommitrivkivk(ak,nk) cm=Commitrcmcm(rest of note) This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient Sinsemilla instead of Bowe--Hopwood Pedersen hashes. Note that for ivk, we also deviate from Sapling in two ways: We use ShortCommit to derive ivk instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit, at the cost of requiring rivk to be part of the full viewing key. We define ivk as an integer in [1,qP); that is, we exclude ivk=0. For Sapling, we relied on BLAKE2s to make ivk=0 infeasible to produce, but it was still technically possible. For Orchard, we get this by construction: 0 is not a valid x-coordinate for any Pallas point. SinsemillaShortCommit internally maps points to field elements by replacing the identity (which has no affine coordinates) with 0. But SinsemillaCommit is defined using incomplete addition, and thus will never produce the identity.","breadcrumbs":"Design » Commitments","id":"21","title":"Commitments"},"22":{"body":"The commitment tree structure for Orchard is identical to Sapling: A single global commitment tree of fixed depth 32. Note commitments are appended to the tree in-order from the block. Valid Orchard anchors correspond to the global tree state at block boundaries (after all commitments from a block have been appended, and before any commitments from the next block have been appended). The only difference is that we instantiate MerkleCRHOrchard with Sinsemilla (whereas MerkleCRHSapling used a Bowe--Hopwood Pedersen hash).","breadcrumbs":"Design » Commitment tree","id":"22","title":"Commitment tree"},"23":{"body":"The fixed-depth incremental Merkle trees that we use (in Sprout and Sapling, and again in Orchard) require specifying an \"empty\" or \"uncommitted\" leaf - a value that will never be appended to the tree as a regular leaf. For Sprout (and trees composed of the outputs of bit-twiddling hash functions), we use the all-zeroes array; the probability of a real note having a colliding note commitment is cryptographically negligible. For Sapling, where leaves are u-coordinates of Jubjub points, we use the value 1 which is not the u-coordinate of any Jubjub point. Orchard note commitments are the x-coordinates of Pallas points; thus we take the same approach as Sapling, using a value that is not the x-coordinate of any Pallas point as the uncommitted leaf value. We use the value 2 for both Pallas and Vesta, because 23+5 is not a square in either Fp or Fq: sage: p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001\nsage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: EllipticCurve(GF(p), [0, 5]).count_points() == q\nTrue\nsage: EllipticCurve(GF(q), [0, 5]).count_points() == p\nTrue\nsage: Mod(13, p).is_square()\nFalse\nsage: Mod(13, q).is_square()\nFalse Note: There are also no Pallas points with x-coordinate 0, but we map the identity to (0,0) within the circuit. Although SinsemillaCommit cannot return the identity (the incomplete addition would return ⊥ instead), it would arguably be confusing to rely on that.","breadcrumbs":"Design » Uncommitted leaves","id":"23","title":"Uncommitted leaves"},"24":{"body":"We considered splitting the commitment tree into several sub-trees: Bundle tree, that accumulates the commitments within a single bundle (and thus a single transaction). Block tree, that accumulates the bundle tree roots within a single block. Global tree, that accumulates the block tree roots. Each of these trees would have had a fixed depth (necessary for being able to create proofs). Chains that integrated Orchard could have decoupled the limits on commitments-per-subtree from higher-layer constraints like block size, by enabling their blocks and transactions to be structured internally as a series of Orchard blocks or txs (e.g. a Zcash block would have contained a Vec, that each were appended in-order). The motivation for considering this change was to improve the lives of light client wallet developers. When a new note is received, the wallet derives its incremental witness from the state of the global tree at the point when the note's commitment is appended; this incremental state then needs to be updated with every subsequent commitment in the block in-order. Wallets can't get help from the server to create these for new notes without leaking the specific note that was received. We decided that this was too large a change from Sapling, and that it should be possible to improve the Incremental Merkle Tree implementation to work around the efficiency issues without domain-separating the tree.","breadcrumbs":"Design » Considered alternatives","id":"24","title":"Considered alternatives"},"25":{"body":"The nullifier design we use for Orchard is nf=ExtractP([(Fnk(ρ)+ψ)modp]G+cm), where: F is a keyed circuit-efficient PRF (such as Rescue or Poseidon). ρ is unique to this output. As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. Given that an action consists of a single spend and a single output, we set ρ to be the nullifier of the spent note. ψ is sender-controlled randomness. It is not required to be unique, and in practice is derived from both ρ and a sender-selected random value rseed: ψ=KDFψ(ρ,rseed). G is a fixed independent base. ExtractP extracts the x-coordinate of a Pallas curve point. This gives a note structure of (addr,v,ρ,ψ,rcm). The note plaintext includes rseed in place of ψ and rcm, and omits ρ (which is a public part of the action).","breadcrumbs":"Design » Nullifiers","id":"25","title":"Nullifiers"},"26":{"body":"We care about several security properties for our nullifiers: Balance: can I forge money? Note Privacy: can I gain information about notes only from the public block chain? This describes notes sent in-band. Note Privacy (OOB): can I gain information about notes sent out-of-band, only from the public block chain? In this case, we assume privacy of the channel over which the note is sent, and that the adversary does not have access to any notes sent to the same address which are then spent (so that the nullifier is on the block chain somewhere). Spend Unlinkability: given the incoming viewing key for an address, and not the full viewing key, can I (possibly the sender) detect spends of any notes sent to that address? We're giving ivk to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address. Faerie Resistance: can I perform a Faerie Gold attack (i.e. cause notes to be accepted that are unspendable)? We're giving the full viewing key to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address, and be able to derive every nullifier. We assume (and instantiate elsewhere) the following primitives: GH is a cryptographic hash into the group (such as BLAKE2s with simplified SWU), used to derive all fixed independent bases. E is an elliptic curve (such as Pallas). KDF is the note encryption key derivation function. For our chosen design, our desired security properties rely on the following assumptions: BalanceNote PrivacyNote Privacy (OOB)Spend UnlinkabilityFaerie ResistanceDLEHashDHEKDFNear perfect‡DDHE†∨PRFFDLE HashDHEKDF is computational Diffie-Hellman using KDF for the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker than DDHE but stronger than DLE. We omit ROGH as a security assumption because we only rely on the random oracle applied to fixed inputs defined by the protocol, i.e. to generate the fixed base G, not to attacker-specified inputs. † We additionally assume that for any input x, {Fnk(x):nk∈E} gives a scalar in an adequate range for DDHE. (Otherwise, F could be trivial, e.g. independent of nk.) ‡ Statistical distance <2−167.8 from perfect.","breadcrumbs":"Design » Security properties","id":"26","title":"Security properties"},"27":{"body":"⚠ Caution: be skeptical of the claims in this table about what problem(s) each security property depends on. They may not be accurate and are definitely not fully rigorous. The entries in this table omit the application of ExtractP, which is an optimization to halve the nullifier length. That optimization requires its own security analysis, but because it is a deterministic mapping, only Faerie Resistance could be affected by it. nf[nk][θ]H[nk]H+[rnf]IHash([nk][θ]H)Hash([nk]H+[rnf]I)[Fnk(ψ)][θ]H[Fnk(ψ)]H+[rnf]I[Fnk(ψ)]G+[θ]H[Fnk(ψ)]H+cm[Fnk(ρ,ψ)]G+cm[Fnk(ρ)]G+cm[Fnk(ρ,ψ)]Gv+[rnf]I[Fnk(ρ)]Gv+[rnf]I[(Fnk(ρ)+ψ)modp]Gv[Fnk(ρ,ψ)]G+Commitrnfnf(v,ρ)[Fnk(ρ)]G+Commitrnfnf(v,ρ)Note(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,rcm)(addr,v,H,rnf,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,rnf,ψ,rcm)(addr,v,H,θ,ψ,rcm)(addr,v,H,ψ,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)(addr,v,ρ,ψ,rcm)(addr,v,ρ,rnf,ψ,rcm)(addr,v,ρ,rnf,rcm)BalanceDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLEDLENote PrivacyHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFHashDHEKDFNote Priv OOBPerfectPerfectPerfectPerfectPerfectPerfectPerfectDDHE†DDHE†DDHE†PerfectPerfectNear perfect‡PerfectPerfectSpend UnlinkabilityDDHEDDHEDDHE∨PreHashDDHE∨PreHashDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFDDHE†∨PRFFFaerie ResistanceROGH∧DLEROGH∧DLECollHash∧ROGH∧DLECollHash∧ROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEROGH∧DLEDLEDLECollF∧DLECollF∧DLEbrokenDLEDLEReason not to useNo SU for DL-breakingNo SU for DL-breakingCollHash for FRCollHash for FRPerf. (2 var-base)Perf. (1 var+1 fix-base)Perf. (1 var+1 fix-base)NP(OOB) not perfectNP(OOB) not perfectNP(OOB) not perfectCollF for FRCollF for FRbroken for FRPerf. (2 fix-base)Perf. (2 fix-base) In the above alternatives: Hash is a keyed circuit-efficient hash (such as Rescue). I is an fixed independent base, independent of G and any others returned by GH. Gv is a pair of fixed independent bases (independent of all others), where the specific choice of base depends on whether the note has zero value. H is a base unique to this output. For non-zero-valued notes, H=GH(ρ). As with hSig in Sprout, ρ includes the nullifiers of any Orchard notes being spent in the same action. For zero-valued notes, H is constrained by the circuit to a fixed base independent of I and any others returned by GH.","breadcrumbs":"Design » Considered alternatives","id":"27","title":"Considered alternatives"},"28":{"body":"In order to satisfy the Balance security property, we require that the circuit must be able to enforce that only one nullifier is accepted for a given note. As in Sprout and Sapling, we achieve this by ensuring that the nullifier deterministically depends only on values committed to (directly or indirectly) by the note commitment. As in Sapling, this involves arguing that: There can be only one ivk for a given addr. This is true because the circuit checks that pkd=[ivk]gd, and the mapping ivk↦[ivk]gd is an injection for any gd. (ivk is in the base field of E, which must be smaller than its scalar field, as is the case for Pallas.) There can be only one nk for a given ivk. This is true because the circuit checks that ivk=ShortCommitrivkivk(ak,nk) where ShortCommit is binding (see Commitments ).","breadcrumbs":"Design » Rationale","id":"28","title":"Rationale"},"29":{"body":"Faerie Resistance requires that nullifiers be unique. This is primarily achieved by taking a unique value (checked for uniqueness by the public consensus rules) as an input to the nullifier. However, it is also necessary to ensure that the transformations applied to this value preserve its uniqueness. Meanwhile, to achieve Spend Unlinkability , we require that the nullifier does not reveal any information about the unique value it is derived from. The design alternatives fall into two categories in terms of how they balance these requirements: Publish a unique value ρ at note creation time, and blind that value within the nullifier computation. This is similar to the approach taken in Sprout and Sapling, which both implemented nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while Sapling uses BLAKE2s. Derive a unique base H from some unique value, publish that unique base at note creation time, and then blind the base (either additively or multiplicatively) during nullifier computation. For Spend Unlinkability , the only value unknown to the adversary is nk, and the cryptographic assumptions only involve the first term (other terms like cm or [rnf]I cannot be extracted directly from the observed nullifiers, but can be subtracted from them). We therefore ensure that the first term does not commit directly to the note (to avoid a DL-breaking adversary from immediately breaking SU ). We were considering using a design involving H with the goal of eliminating all usages of a PRF inside the circuit, for two reasons: Instantiating PRFF with a traditional hash function is expensive in the circuit. We didn't want to solely rely on an algebraic hash function satisfying PRFF to achieve Spend Unlinkability . However, those designs rely on both ROGH and DLE for Faerie Resistance , while still requiring DDHE for Spend Unlinkability . (There are two designs for which this is not the case, but they rely on DDHE† for Note Privacy (OOB) which was not acceptable). By contrast, several designs involving ρ (including the chosen design) have weaker assumptions for Faerie Resistance (only relying on DLE), and Spend Unlinkability does not require PRFF to hold: they can fall back on the same DDHE assumption as the H designs (along with an additional assumption about the output of F which is easily satisfied).","breadcrumbs":"Design » Use of ρ","id":"29","title":"Use of ρ"},"3":{"body":"","breadcrumbs":"Concepts","id":"3","title":"Concepts"},"30":{"body":"Most of the designs include either a multiplicative blinding term [θ]H, or an additive blinding term [rnf]I, in order to achieve perfect Note Privacy (OOB) (to an adversary who does not know the note). The chosen design is effectively using [ψ]G for this purpose; a DL-breaking adversary only learns Fnk(ρ)+ψ(modp). This reduces Note Privacy (OOB) from perfect to statistical, but given that ψ is from a distribution statistically close to uniform on [0,q), this is statistically close to better than 2−128. The benefit is that it does not require an additional scalar multiplication, making it more efficient inside the circuit. ψ's derivation has two motivations: Deriving from a random value rseed enables multiple derived values to be conveyed to the recipient within an action (such as the ephemeral secret esk, per ZIP 212 ), while keeping the note plaintext short. Mixing ρ into the derivation ensures that the sender can't repeat ψ across two notes, which could have enabled spend linkability attacks in some designs. The note that is committed to, and which the circuit takes as input, only includes ψ (i.e. the circuit does not check the derivation from rseed). However, an adversarial sender is still constrained by this derivation, because the recipient recomputes ψ during note decryption. If an action were created using an arbitrary ψ (for which the adversary did not have a corresponding rseed), the recipient would derive a note commitment that did not match the action's commitment field, and reject it (as in Sapling).","breadcrumbs":"Design » Use of ψ","id":"30","title":"Use of ψ"},"31":{"body":"The nullifier commits to the note value via cm for two reasons: It domain-separates nullifiers for zero-valued notes from other notes. This is necessary because we do not require zero-valued notes to exist in the commitment tree. Designs that bind the nullifier to Fnk(ρ) require CollF to achieve Faerie Resistance (and similarly where Hash is applied to a value derived from H). Adding cm to the nullifier avoids this assumption: all of the bases used to derive cm are fixed and independent of G, and so the nullifier can be viewed as a Pedersen hash where the input includes ρ directly. The Commitnf variants were considered to avoid directly depending on cm (which in its native type is a base field element, not a group element). We decided instead to follow Sapling by defining an intermediate representation of cm as a group element, that is only used in nullifier computation. The circuit already needs to compute cm, so this improves performance by removing We also considered variants that used a choice of fixed bases Gv to provide domain separation for zero-valued notes. The most performant design (similar to the chosen design) does not achieve Faerie Resistance for an adversary that knows the recipient's full viewing key (ψ could be brute-forced to cancel out Fnk(ρ), causing a collision), and the other variants require assuming CollF as mentioned above.","breadcrumbs":"Design » Use of cm","id":"31","title":"Use of cm"},"32":{"body":"Orchard signatures are an instantiation of RedDSA with a cofactor of 1. TODO: Should it be possible to sign partial transactions? If we're going to merge down all the signatures into a single one, and also want this, we need to ensure there's a feasible MPC.","breadcrumbs":"Design » Signatures","id":"32","title":"Signatures"},"33":{"body":"","breadcrumbs":"Design » Circuit","id":"33","title":"Circuit"},"34":{"body":"","breadcrumbs":"Design » Circuit » Gadgets","id":"34","title":"Gadgets"},"35":{"body":"We will use formulae for curve arithmetic using affine coordinates on short Weierstrass curves, derived from section 4.1 of Hüseyin Hışıl's thesis . Inputs: P=(xp,yp),Q=(xq,yq) Output: R=P⸭Q=(xr,yr) The formulae from Hışıl's thesis are: x3=(x1−x2y1−y2)2−x1−x2 y3=x1−x2y1−y2⋅(x1−x3)−y1 Rename: (x1,y1) to (xq,yq) (x2,y2) to (xp,yp) (x3,y3) to (xr,yr). Let λ=xq−xpyq−yp=xp−xqyp−yq, which we implement as λ⋅(xp−xq)=yp−yq Also, xr=λ2−xq−xp yr=λ⋅(xq−xr)−yq which is equivalent to xr+xq+xp=λ2 Assuming xp=xq, ⟹⟹⟹(xr+xq+xp)⋅(xp−xq)2(xr+xq+xp)⋅(xp−xq)2yryr+yq(yr+yq)⋅(xp−xq)=====λ2⋅(xp−xq)2(λ⋅(xp−xq))2λ⋅(xq−xr)−yqλ⋅(xq−xr)λ⋅(xp−xq)⋅(xq−xr) Substituting for λ⋅(xp−xq), we get the constraints: (xr+xq+xp)⋅(xp−xq)2−(yp−yq)2=0 Note that this constraint is unsatisfiable for P⸭(−P) (when P=O), and so cannot be used with arbitrary inputs. (yr+yq)(xp−xq)−(yp−yq)(xq−xr)=0","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Incomplete addition","id":"35","title":"Incomplete addition"},"36":{"body":"OO(xp,yp)(x,y)(x,y)(xp,yp)++++++O(xq,yq)O(x,y)(x,−y)(xq,yq)=O=(xq,yq)=(xp,yp)=[2](x,y)=O=(xp,yp)⸭(xq,yq),if xp=xq Suppose that we represent O as (0,0). (0 is not an x-coordinate of a valid point because we would need y2=x3+5, and 5 is not square in Fq. Also 0 is not a y-coordinate of a valid point because −5 is not a cube in Fq.) P+Q(xp,yp)+(xq,yq)λxryr=R=(xr,yr)=xq−xpyq−yp=λ2−xp−xq=λ(xp−xr)−yp For the doubling case, Hışıl's thesis tells us that λ has to instead be computed as 2y3x2. Define inv0(x)={0,1/x,if x=0otherwise. Witness α,β,γ,δ,λ where: α=inv0(xq−xp) β=inv0(xp) γ=inv0(xq) δ={inv0(yq+yp),0,if xq=xpotherwise λ=⎩⎨⎧xq−xpyq−yp,2yp3xp20,if xq=xpif xq=xp∧yp=0otherwise.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Complete addition","id":"36","title":"Complete addition"},"37":{"body":"Degree456666444444Constraintqadd⋅(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))qadd⋅(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)qadd⋅xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)qadd⋅(1−xp⋅β)⋅(xr−xq)qadd⋅(1−xp⋅β)⋅(yr−yq)qadd⋅(1−xq⋅γ)⋅(xr−xp)qadd⋅(1−xq⋅γ)⋅(yr−yp)qadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xrqadd⋅(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr============000000000000Meaningxq=xp⟹λ=xq−xpyq−yp{xq=xp∧yp=0⟹λ=2yp3xp2xq=xp∧yp=0⟹xp=0xp=0∧xq=0∧xq=xp⟹xr=λ2−xp−xqxp=0∧xq=0∧xq=xp⟹yr=λ⋅(xp−xr)−ypxp=0∧xq=0∧yq=−yp⟹xr=λ2−xp−xqxp=0∧xq=0∧yq=−yp⟹yr=λ⋅(xp−xr)−ypxp=0⟹xr=xqxp=0⟹yr=yqxq=0⟹xr=xpxq=0⟹yr=ypxq=xp∧yq=−yp⟹xr=0xq=xp∧yq=−yp⟹yr=0 Max degree: 6","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraints","id":"37","title":"Constraints"},"38":{"body":"1.2.3.4.5.6.a)b)c)d)a)b)a)b)a)b)(xq−xp)⋅((xq−xp)⋅λ−(yq−yp))=0At least one of or xq−xp=0(xq−xp)⋅λ−(yq−yp)=0must be satisfied for the constraint to be satisfied.If xq−xp=0, then (xq−xp)⋅λ−(yq−yp)=0If (xq−xp)⋅λ−(yq−yp)=0, then because xq−xp=0, by rearranging both sides we get λ=(yq−yp)/(xq−xp)and therefore:xq=xp⟹λ=(yq−yp)/(xq−xp).(1−(xq−xp)⋅α)⋅(2yp⋅λ−3xp2)=0At least one of or (1−(xq−xp)⋅α)=0(2yp⋅λ−3xp2)=0must be satisfied for the constraint to be satisfied.If xq=xp, then 1−(xq−xp)⋅α=0 has no solution for α, so it must be that 2yp⋅λ−3xp2=0.If yp=0 then xp=0, and the constraint is satisfied.If yp=0 by rearranging both sides we get λ=3xp2/2ypTherefore:(xq=xp)∧yp=0⟹λ=3xp2/2yp.xp⋅xq⋅(xq−xp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(xq−xp)⋅(λ⋅(xp−xr)−yp−yr)=0xp⋅xq⋅(yq+yp)⋅(λ2−xp−xq−xr)=0xp⋅xq⋅(yq+yp)⋅(λ⋅(xp−xr)−yp−yr)=0At least one of or or or xp=0xp=0(xq−xp)=0(λ2−xp−xq−xr)=0must be satisfied for constraint (a) to be satisfied.Let xp=0∧xq=0∧xq=xp.• Constraint (a) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (b) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Let xp=0∧xq=0∧yq=−yp.• Similarly, constraint (c) imposes that xr=λ2−xp−xq is satisfied.• Similarly, constraint (d) imposes that yr=λ⋅(xp−xr)−yp is satisfied.Therefore:⟹(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp).(1−xp⋅β)⋅(xr−xq)=0(1−xp⋅β)⋅(yr−yq)=0At least one of 1−xp⋅βor xr−xq=0=0must be satisfied for constraint (a) to be satisfied.If xp=0 then 1−xp⋅β=0 has no solutions for β,and so it must be that xr−xq=0.Similarly, constraint (b) imposes that yr−yq=0.Therefore:xp=0⟹(xr,yr)=(xq,yq).(1−xq⋅β)⋅(xr−xp)=0(1−xq⋅β)⋅(yr−yp)=0At least one of 1−xq⋅βor xr−xp=0=0must be satisfied for constraint (a) to be satisfied.If xq=0 then 1−xq⋅β=0 has no solutions for β,and so it must be that xr−xp=0.Similarly, constraint (b) imposes that yr−yp=0.Therefore:xq=0⟹(xr,yr)=(xp,yp).(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅xr=0(1−(xq−xp)⋅α−(yq+yp)⋅δ)⋅yr=0At least one of or 1−(xq−xp)⋅α−(yq+yp)⋅δ=0xr=0must be satisfied for constraint (a) to be satisfied.If xr=0, then it must be that 1−(xq−xp)⋅α−(yq+yp)⋅δ=0.However, if xq=xp∧yq=−yp, then there are no solutions for α and δ.Therefore: xq=xp∧yq=−yp⟹(xr,yr)=(0,0). Propositions: (1)(2)(3)(4)(5)(6)xq=xp⟹λ=(yq−yp)/(xq−xp).(xq=xp)∧yp=0⟹λ=3xp2/2yp(xp=0)∧(xq=0)∧((xq=xp)∨(yq=−yp))⟹(xr=λ2−xp−xq)∧(yr=λ⋅(xp−xr)−yp)xp=0⟹(xr,yr)=(xq,yq)xq=0⟹(xr,yr)=(xp,yp)xq=xp∧yq=−yp⟹(xr,yr)=(0,0) Test cases: (xp,yp)+(xq,yq)=(xr,yr) (0,0)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because yp=0 holds because xp=0 holds because xp=0 only when xr=0,yr=0 holds because xq=0 only when xr=0,yr=0 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (x,y)+(0,0) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xq=0 holds because xp=0 because 0 is not a valid x-coordinate only when β=xp−1 holds because xq=0 only when (xr,yr)=(xp,yp) holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xp,yp) is the only solution (0,0)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp because 0 is not a valid x-coordinate only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xq=xp because 0 is not a valid x-coordinate only when α=(xq−xp)−1 holds because xp=0 holds because xp=0 only when (xr,yr)=(xq,yq) holds because xq=0 because 0 is not a valid x-coordinate only when γ=xq−1 holds because yp=−yp because 0 is not a valid y-coordinate only when δ=(yq+yp)−1which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(xq,yq) is the only solution (x,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate) only when λ=3xp2/2yp holds because xp=0∧xq=0andyq=−yp only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xp=0 only when γ=xq−1 holds because yq=−yp only when δ=(yq+yp)−1 which is defined because 0 is not a valid y-coordinate Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution (x,y)+(x,−y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp holds because xq=xp∧yp=0 (because 0 is not a valid y-coordinate)only when λ=3xp2/2yp holds because xp=0∧xq=0 but yq=−yp∧xq=xp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp∧yq=−yp only when (xr,yr)=(0,0) Soundness: (xr,yr)=(0,0) is the only solution (ζx,y)+(x,y) Completeness: (1)(2)(3)(4)(5)(6) holds because xq=xp only when λ=(yq−yp)/(xq−xp) which is defined because xq=xp holds because xp=xp only when α=(xq−xp)−1 which is defined because xq=xp holds because (xp=0)∧(xq=0)∧(xq=xp) only when xr=λ2−xp−xq∧yr=λ∗(xp−xr)−yp holds because xp=0 only when β=xp−1 holds because xq=0 only when γ=xq−1 holds because xq=xp only when δ=0 Soundness: (xr,yr)=(λ2−xp−xq,λ∗(xp−xr)−yp) is the only solution All remaining cases (x,y)+(x′,y′) are identical to the case (ζx,y)+(x,y) when λαβγδ=(yq−yp)/(xq−xp)=(xq−xp)−1=xp−1=xq−1=0 because in all remaining cases, xq=xp,xp=0,xq=0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Analysis of constraints","id":"38","title":"Analysis of constraints"},"39":{"body":"There are 6 fixed bases in the Orchard protocol: KOrchard, used in deriving the nullifier; GOrchard, used in spend authorization; R base for NoteCommitOrchard; V and R bases for ValueCommitOrchard; and R base for Commitivk.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Fixed-base scalar multiplication","id":"39","title":"Fixed-base scalar multiplication"},"4":{"body":"","breadcrumbs":"Concepts » Preliminaries","id":"4","title":"Preliminaries"},"40":{"body":"We support fixed-base scalar multiplication with three types of scalars:","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Decompose scalar","id":"40","title":"Decompose scalar"},"41":{"body":"A 255-bit scalar from Fq. We decompose a full-width scalar α into 85 3-bit windows: α=k0+k1⋅(23)1+⋯+k84⋅(23)84,ki∈[0..23). The scalar multiplication will be computed correctly for k0..84 representing any integer in the range [0,2255). Degree9Constraintqscalar-fixed⋅(i=0∑7w−i)=0 We range-constrain each 3-bit word of the scalar decomposition using a polynomial range-check constraint: Degree9Constraintqdecompose-base-field⋅range_check(word,23)=0 where range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Full-width scalar","id":"41","title":"Full-width scalar"},"42":{"body":"We support using a base field element as the scalar in fixed-base multiplication. This occurs, for example, in the scalar multiplication for the nullifier computation of the Action circuit DeriveNullifiernk=ExtractP([(PRFnknfOrchard(ρ)+ψ)modqP]KOrchard+cm): here, the scalar [(PRFnknfOrchard(ρ)+ψ)modqP] is the result of a base field addition. Decompose the base field element α into three-bit windows, and range-constrain each window, using the short range decomposition gadget in strict mode, with W=85,K=3. If k0..84 is witnessed directly then no issue of canonicity arises. However, because the scalar is given as a base field element here, care must be taken to ensure a canonical representation, since 2255>p. That is, we must check that 0≤αq. Thus, given a scalar α, we witness the boolean decomposition of k=α+tq. (We use big-endian bit order for convenient input into the variable-base scalar multiplication algorithm.) k=k254⋅2254+k253⋅2253+⋯+k0.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Witness scalar","id":"49","title":"Witness scalar"},"5":{"body":"","breadcrumbs":"User Documentation","id":"5","title":"User Documentation"},"50":{"body":"We use an optimized double-and-add algorithm, copied from \"Faster variable-base scalar multiplication in zk-SNARK circuits\" with some variable name changes: Acc := [2] T\nfor i from n-1 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc\n}\nreturn (k_0 = 0) ? (Acc - T) : Acc It remains to check that the x-coordinates of each pair of points to be added are distinct. When adding points in a prime-order group, we can rely on Theorem 3 from Appendix C of the Halo paper , which says that if we have two such points with nonzero indices wrt a given odd-prime order base, where the indices taken in the range −(q−1)/2..(q−1)/2 are distinct disregarding sign, then they have different x-coordinates. This is helpful, because it is easier to reason about the indices of points occurring in the scalar multiplication algorithm than it is to reason about their x-coordinates directly. So, the required check is equivalent to saying that the following \"indexed version\" of the above algorithm never asserts: acc := 2\nfor i from n-1 down to 0 { p = k_{i+1} ? 1 : −1 assert acc ≠ ± p assert (acc + p) ≠ acc // X acc := (acc + p) + acc assert 0 < acc ≤ (q-1)/2\n}\nif k_0 = 0 { assert acc ≠ 1 acc := acc - 1\n} The maximum value of acc is: <--- n 1s ---> 1011111...111111\n= 1100000...000000 - 1 = 2n+1+2n−1 The assertion labelled X obviously cannot fail because p=0. It is possible to see that acc is monotonically increasing except in the last conditional. It reaches its largest value when k is maximal, i.e. 2n+1+2n−1. So to entirely avoid exceptional cases, we would need 2n+1+2n−1<(q−1)/2. But we can use n larger by c if the last c iterations use complete addition . The first i for which the algorithm using only incomplete addition fails is going to be 252, since 2252+1+2252−1>(q−1)/2. We need n=254 to make the wraparound technique above work. sage: q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001\nsage: 2^253 + 2^252 - 1 < (q-1)//2\nFalse\nsage: 2^252 + 2^251 - 1 < (q-1)//2\nTrue So the last three iterations of the loop (i=2..0) need to use complete addition , as does the conditional subtraction at the end. Writing this out using ⸭ for incomplete addition (as we do in the spec), we have: Acc := [2] T\nfor i from 253 down to 3 { P := k_{i+1} ? T : −T Acc := (Acc ⸭ P) ⸭ Acc\n}\nfor i from 2 down to 0 { P := k_{i+1} ? T : −T Acc := (Acc + P) + Acc // complete addition\n}\nreturn (k_0 = 0) ? (Acc + (-T)) : Acc // complete addition","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication","id":"50","title":"Variable-base scalar multiplication"},"51":{"body":"Define a running sum zj=∑i=jn(ki⋅2i−j), where n=254 and: zn+1=0,zn=kn,(most significant bit)z0=k. Initialize A254=[2]T. for i from 254 down to 4: (ki)(ki−1)=0zi=2zi+1+kixP,i=xTyP,i=(2ki−1)⋅yT(conditionally negate)λ1,i⋅(xA,i−xP,i)=yA,i−yP,iλ1,i2=xR,i+xA,i+xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)=2yA,iλ2,i2=xA,i−1+xR,i+xA,iλ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1, where xR,i=(λ1,i2−xA,i−xT). After substitution of xP,i,yP,i,xR,i,yA,i, and yA,i−1, this becomes: Initialize A254=[2]T. for i from 254 down to 4: // let ki=zi−2zi+1// let yA,i=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT))(ki)(ki−1)=0λ1,i⋅(xA,i−xT)=yA,i−(2ki−1)⋅yTλ2,i2=xA,i−1+λ1,i2−xT {λ2,i⋅(xA,i−xA,i−1)=yA,i+yA,i−1,λ2,4⋅(xA,4−xA,3)=yA,4+yA,3witnessed,if i>4if i=4. Here, yA,3witnessed is assigned to a cell. This is unlike previous yA,i's, which were implicitly derived from λ1,i,λ2,i,xA,i,xT, but never actually assigned. The bits k3…1 are used in three further steps, using complete addition : for i from 3 down to 1: // let ki=zi−2zi+1(ki)(ki−1)=0(xA,i−1,yA,i−1)=((xA,i,yA,i)+(xT,yT))+(xA,i,yA,i) If the least significant bit is set k0=1, we return the accumulator A. Else, if k0=0, we return A−T (also using complete addition). Let B={(0,0),(xT,−yT), if k0=1, otherwise. Output (xA,0,yA,0)+B. (Note that (0,0) represents O.)","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Constraint program for optimized double-and-add (incomplete addition)","id":"51","title":"Constraint program for optimized double-and-add (incomplete addition)"},"52":{"body":"We need six advice columns to witness (xT,yT,λ1,λ2,xA,i,zi). However, since (xT,yT) are the same, we can perform two incomplete additions in a single row, reusing the same (xT,yT). We split the scalar bits used in incomplete addition into hi and lo halves and process them in parallel. This means that we effectively have two for loops: the first, covering the hi half for i from 254 down to 130, with a special case at i=130; and the second, covering the lo half for the remaining i from 129 down to 4, with a special case at i=4. xTxTxT⋮xTyTyTyT⋮yTzhiz255=0z254z253⋮z130xAhixA,254=2[T]xxA,253⋮xA,130xA,129λ1hiyA,254=2[T]yλ1,254λ1,253⋮λ1,130yA,129λ2hiλ2,254λ2,253⋮λ2,130qmulhi122⋮3zloz130z129z128⋮z5z4xAloxA,129xA,128⋮xA,5xA,4xA,3λ1loyA,129λ1,129λ1,128⋮λ1,5λ1,4yA,3λ2loλ2,129λ2,128⋮λ2,5λ2,4qmullo122⋮23 For each hi and lo half, we have three sets of gates. Note that i is going from 255..=3; i is NOT indexing the rows. qmul=1 This gate is only used on the first row (before the for loop). We check that λ1,λ2 are initialized to values consistent with the initial yA. Degree5Constraintqmulone⋅(yA,nwitnessed−yA,n)=0 where qmuloneyA,nyA,nwitnessed=qmul⋅(2−qmul)⋅(3−qmul),=2(λ1,n+λ2,n)⋅(xA,n−(λ1,n2−xA,n−xT)), is witnessed. qmul=2 This gate is used on all rows corresponding to the for loop except the last. Degree445655Constraintqmultwo⋅(xT,cur−xT,next)=0qmultwo⋅(yT,cur−yT,next)=0qmultwo⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmultwo⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmultwo⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmultwo⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1)=0 where qmultwoyA,iyA,i−1=qmul⋅(1−qmul)⋅(3−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)),=2(λ1,i−1+λ2,i−1)⋅(xA,i−1−(λ1,i−12−xA,i−1−xT)), qmul=3 This gate is used on the final iteration of the for loop, handling the special case where we check that the output yA has been witnessed correctly. Degree5655Constraintqmulthree⋅ki⋅(ki−1)=0, where ki=zi−2zi+1qmulthree⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0qmulthree⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0qmulthree⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1witnessed)=0 where qmulthreeyA,iyA,i−1witnessed=qmul⋅(1−qmul)⋅(2−qmul),=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)), is witnessed.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Circuit design","id":"52","title":"Circuit design"},"53":{"body":"zi cannot overflow for any i≥1, because it is a weighted sum of bits only up to 2n−1=2253, which is smaller than p (and also q). However, z0=α+tq can overflow [0,p). Since overflow can only occur in the final step that constrains z0=2⋅z1+k0, we have z0=k(modp). It is then sufficient to also check that z0=α+tq(modp) (so that k=α+tq(modp)) and that k∈[tq,p+tq). These conditions together imply that k=α+tq as an integer, and so 2254+k=α(modq) as required. Note: the bits k254..0 do not represent a value reduced modulo q, but rather a representation of the unreduced α+tq.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check","id":"53","title":"Overflow check"},"54":{"body":"Since tp+tq<2130, we have [tq,p+tq)=[tq,tq+2130)∪[2130,2254)∪([2254,2254+2130)∩[p+tq−2130,p+tq)). We may assume that k=α+tq(modp). Therefore, k∈[tq,p+tq)⇔⇔⇔(k∈[tq,tq+2130)∨k∈[2130,2254))∨(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(k∈[tq,tq+2130)∨k∈[2130,2254)))∧(k254=1⟹(k∈[2254,2254+2130)∧k∈[p+tq−2130,p+tq))(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))Ⓐ Given k∈[2254,2254+2130), we prove equivalence of k∈[p+tq−2130,p+tq) and (α+2130)modp∈[0,2130) as follows: shift the range by 2130−p−tq to give k+2130−p−tq∈[0,2130); observe that k+2130−p−tq is guaranteed to be in [2130−tp−tq,2131−tp−tq) and therefore cannot overflow or underflow modulo p; using the fact that k=α+tq(modp), observe that (k+2130−p−tq)modp=(α+tq+2130−p−tq)modp=(α+2130)modp. (We can see in a different way that this is correct by observing that it checks whether αmodp∈[p−2130,p), so the upper bound is aligned as we would expect.) Now, we can continue optimizing from Ⓐ: k∈[tq,p+tq)⇔⇔(k254=0⟹(α∈[0,2130)∨k∈[2130,2254))∧(k254=1⟹(k∈[2254,2254+2130)∧(α+2130)modp∈[0,2130)))(k254=0⟹(α∈[0,2130)∨k253..130 are not all 0))∧(k254=1⟹(k253..130 are all 0∧(α+2130)modp∈[0,2130))) Constraining k253..130 to be all-0 or not-all-0 can be implemented almost \"for free\", as follows. Recall that zi=∑h=in(kh⋅2h−i), so we have: z130z130z130−k254⋅2124===∑h=130254(kh⋅2h−130)k254⋅2254−130+∑h=130253(kh⋅2h−130)∑h=130253(kh⋅2h−130) So k253..130 are all 0 exactly when z130=k254⋅2124. Finally, we can merge the 130-bit decompositions for the k254=0 and k254=1 cases by checking that (α+k254⋅2130)modp∈[0,2130).","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Optimized check for k∈[tq,p+tq)","id":"54","title":"Optimized check for k∈[tq,p+tq)"},"55":{"body":"Let s=α+k254⋅2130. The constraints for the overflow check are: z0k254=1⟹(z130k254=0⟹(z130=α+tq(modp)=2124∧smodp∈[0,2130))=0∨smodp∈[0,2130)) Define inv0(x)={0,1/x,if x=0otherwise. Witness η=inv0(z130), and decompose smodp as s129..0. Then the needed gates are: Degree22335Constraintqmuloverflow⋅(s−(α+k254⋅2130))=0qmuloverflow⋅(z0−α−tq)=0qmuloverflow⋅(k254⋅(z130−2124))=0qmuloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0qmuloverflow⋅((1−k254)⋅(1−z130⋅η)⋅(s−i=0∑1292i⋅si)/2130)=0 where (s−i=0∑1292i⋅si)/2130 can be computed by another running sum. Note that the factor of 1/2130 has no effect on the constraint, since the RHS is zero. Running sum range check We make use of a 10-bit lookup range check in the circuit to subtract the low 130 bits of s. The range check subtracts the first 13⋅10 bits of s, and right-shifts the result to give (s−i=0∑1292i⋅si)/2130.","breadcrumbs":"Design » Circuit » Gadgets » Elliptic curve cryptography » Overflow check constraints","id":"55","title":"Overflow check constraints"},"56":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » Sinsemilla","id":"56","title":"Sinsemilla"},"57":{"body":"Sinsemilla is a collision-resistant hash function and commitment scheme designed to be efficient in algebraic circuit models that support lookups , such as PLONK or Halo 2. The security properties of Sinsemilla are similar to Pedersen hashes; it is not designed to be used where a random oracle, PRF, or preimage-resistant hash is required. The only claimed security property of the hash function is collision-resistance for fixed-length inputs. Sinsemilla is roughly 4 times less efficient than the algebraic hashes Rescue and Poseidon inside a circuit, but around 19 times more efficient than Rescue outside a circuit. Unlike either of these hashes, the collision resistance property of Sinsemilla can be proven based on cryptographic assumptions that have been well-established for at least 20 years. Sinsemilla can also be used as a computationally binding and perfectly hiding commitment scheme. The general approach is to split the message into k-bit pieces, and for each piece, select from a table of 2k bases in our cryptographic group. We combine the selected bases using a double-and-add algorithm. This ends up being provably as secure as a vector Pedersen hash, and makes advantageous use of the lookup facility supported by Halo 2.","breadcrumbs":"Design » Circuit » Gadgets » Overview","id":"57","title":"Overview"},"58":{"body":"This section is an outline of how Sinsemilla works: for the normative specification, refer to §5.4.1.9 Sinsemilla Hash Function in the protocol spec. The incomplete point addition operator, ⸭, that we use below is also defined there. Let G be a cryptographic group of prime order q. We write G additively, with identity O, and using [m]P for scalar multiplication of P by m. Let k≥1 be an integer chosen based on efficiency considerations (the table size will be 2k). Let n be an integer, fixed for each instantiation, such that messages are kn bits, where 2n≤2q−1. We use zero-padding to the next multiple of k bits if necessary. Setup: Choose Q and P[0..2k−1] as 2k+1 independent, verifiably random generators of G, using a suitable hash into G, such that none of Q or P[0..2k−1] are O. In Orchard, we define Q to be dependent on a domain separator D. The protocol specification uses Q(D) in place of Q and S(m) in place of P[m]. Hash(M): Split M into n groups of k bits. Interpret each group as a k-bit little-endian integer mi. let Acc0:=Q for i from 0 up to n−1: let Acci+1:=(Acci⸭P[mi+1])⸭Acci return Accn Let ShortHash(M) be the x-coordinate of Hash(M). (This assumes that G is a prime-order elliptic curve in short Weierstrass form, as is the case for Pallas and Vesta.) It is slightly more efficient to express a double-and-add [2]A+R as (A+R)+A. We also use incomplete additions: it is shown in the Sinsemilla security argument that in the case where G is a prime-order short Weierstrass elliptic curve, an exceptional case for addition would lead to finding a discrete logarithm, which can be assumed to occur with negligible probability even for adversarial input.","breadcrumbs":"Design » Circuit » Gadgets » Description","id":"58","title":"Description"},"59":{"body":"Choose another generator H independently of Q and P[0..2k−1]. The randomness r for a commitment is chosen uniformly on [0,q). Let Commitr(M)=Hash(M)⸭[r]H. Let ShortCommitr(M) be the x-coordinate of Commitr(M). (This again assumes that G is a prime-order elliptic curve in short Weierstrass form.) Note that unlike a simple Pedersen commitment, this commitment scheme (Commit or ShortCommit) is not additively homomorphic.","breadcrumbs":"Design » Circuit » Gadgets » Use as a commitment scheme","id":"59","title":"Use as a commitment scheme"},"6":{"body":"","breadcrumbs":"User Documentation » Creating keys and addresses","id":"6","title":"Creating keys and addresses"},"60":{"body":"The aim of the design is to optimize the number of bits that can be processed for each step of the algorithm (which requires a doubling and addition in G) for a given table size. Using a single table of size 2k group elements, we can process k bits at a time.","breadcrumbs":"Design » Circuit » Gadgets » Efficient implementation","id":"60","title":"Efficient implementation"},"61":{"body":"Let P={(j,xP[j],yP[j]) for j∈{0..2k−1}}. Input: m1..=n. (The message words are 1-indexed here, as in the protocol spec , but we start the loop from i=0 so that (xA,i,yA,i) corresponds to Acci in the protocol spec.) Output: (xA,n,yA,n). (xA,0,yA,0)=Q for i from 0 up to n−1: yP,i=yA,i−λ1,i⋅(xA,i−xP,i) xR,i=λ1,i2−xA,i−xP,i 2⋅yA,i=(λ1,i+λ2,i)⋅(xA,i−xR,i) (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i λ2,i⋅(xA,i−xA,i+1)=yA,i+yA,i+1","breadcrumbs":"Design » Circuit » Gadgets » Constraint program","id":"61","title":"Constraint program"},"62":{"body":"","breadcrumbs":"Design » Circuit » Gadgets » PLONK / Halo 2 constraints","id":"62","title":"PLONK / Halo 2 constraints"},"63":{"body":"We have an n-bit message m=m1+2km2+...+2k⋅(n−1)mn. (Note that the message words are 1-indexed as in the protocol spec .) Initialise the running sum z0=α and define zi+1:=2Kzi−mi+1. We will end up with zn=0. Rearranging gives us an expression for each word of the original message mi+1=zi−2k⋅zi+1, which we can look up in the table. In other words, zn−i=h=0∑i−12kh⋅mh+1. For a little-endian decomposition as used here, the running sum is initialized to the scalar and ends at 0. For a big-endian decomposition as used in variable-base scalar multiplication , the running sum would start at 0 and end with recovering the original scalar. The running sum only applies to message words within a single field element, i.e. if n≥PrimeField::NUM_BITS then we will have several disjoint running sums. A longer message can be constructed by splitting the message words across several field elements, and then running several instances of the constraints below. An additional qS2 selector is set to 0 for the last step of each element, except for the last element where it is set to 2. In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for mi+1 that effectively forces zn to zero for the last step of each element, as indicated by qS2. This allows the cell that would have been zn to be used to reinitialize the running sum for the next element.","breadcrumbs":"Design » Circuit » Gadgets » Message decomposition","id":"63","title":"Message decomposition"},"64":{"body":"Note: qS3 is synthesized from qS1 and qS2; it is shown here only for clarity. Step012⋮n−10′1′2′⋮n−1′n′xAxQxA,1xA,2⋮xA,n−1xA,0′xA,1′xA,2′⋮xA,n−1′xA,n′xPxP[m1]xP[m2]xP[m3]⋮xP[mn]xP[m1′]xP[m2′]xP[m3′]⋮xP[mn′]bitsz0z1z2⋮zn−1z0′z1′z2′⋮zn−1′λ1λ1,0λ1,1λ1,2⋮λ1,n−1λ1,0′λ1,1′λ1,2′⋮λ1,n−1′yA,nλ2λ2,0λ2,1λ2,2⋮λ2,n−1λ2,0′λ2,1′λ2,2′⋮λ2,n−1′qS111111111110qS211110111120qS300000000020fixed_yQyQ0000000000tableidx012⋮⋮⋮⋮⋮⋮⋮⋮tablexxP[0]xP[1]xP[2]⋮⋮⋮⋮⋮⋮⋮⋮tableyyP[0]yP[1]yP[2]⋮⋮⋮⋮⋮⋮⋮⋮ xQ, z0, z0′, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Circuit » Gadgets » Layout","id":"64","title":"Layout"},"65":{"body":"For i∈[0,n), letxR,iYA,iyP,imi+1qS3=====λ1,i2−xA,i−xP,i(λ1,i+λ2,i)⋅(xA,i−xR,i)YA,i/2−λ1,i⋅(xA,i−xP,i)zi−2k⋅(qS2,i−qS3,i)⋅zi+1qS2⋅(qS2−1) The Halo 2 circuit API can automatically substitute yP,i, xR,i, yA,i, and yA,i+1, so we don't need to do that manually. xA,0=xQ 2⋅yQ=YA,0 for i from 0 up to n−1: (mi+1,xP,i,yP,i)∈P λ2,i2=xA,i+1+xR,i+xA,i 4⋅λ2,i⋅(xA,i−xA,i+1)=2⋅YA,i+(2−qS3)⋅YA,i+1+2qS3⋅yA,n Note that each term of the last constraint is multiplied by 4 relative to the constraint program given earlier. This is a small optimization that avoids divisions by 2. Degree4536ConstraintfixedyQ⋅(2⋅fixedyQ−YA,0)=0qS1,i⇒(mi+1,xP,i,yP,i)∈PqS1,i⋅(λ2,i2−(xA,i+1+xR,i+xA,i))qS1,i⋅(4⋅λ2,i⋅(xA,i−xA,i+1)−(2⋅YA,i+(2−qS3,i)⋅YA,i+1+2⋅qS3,i⋅yA,n))=0 By gating the lookup expression on qS1, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of 0) is: (qS1⋅mi+1,qS1⋅xP,i+(1−qS1)⋅xP,0,qS1⋅yP,i+(1−qS1)⋅yP,0) This increases the degree of the lookup argument to 6.","breadcrumbs":"Design » Circuit » Gadgets » Optimized Sinsemilla gate","id":"65","title":"Optimized Sinsemilla gate"},"66":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0+2K⋅k1+22K⋅k2+⋯+2(W−1)K⋅kW−1 where each ki a K-bit value. This is done using a running sum zi,i∈[0..W). We initialize the running sum z0=α, and compute subsequent terms zi+1=2Kzi−ki. This gives us: z0z1z2↓zW=α=k0+2K⋅k1+22K⋅k2+23K⋅k3+⋯,=(z0−k0)/2K=k1+2K⋅k2+22K⋅k3+⋯,=(z1−k1)/2K=k2+2K⋅k3+⋯,⋮ (in strict mode)=(zW−1−kW−1)/2K=0 (because zW−1=kW−1)","breadcrumbs":"Design » Circuit » Gadgets » Decomposition","id":"66","title":"Decomposition"},"67":{"body":"Strict mode constrains the running sum output zW to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1=kW−1 gives us the last window in the decomposition.","breadcrumbs":"Design » Circuit » Gadgets » Strict mode","id":"67","title":"Strict mode"},"68":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table.","breadcrumbs":"Design » Circuit » Gadgets » Lookup decomposition","id":"68","title":"Lookup decomposition"},"69":{"body":"Using two K-bit lookups, we can range-constrain a field element α to be n bits, where n≤K. To do this: Constrain 0≤α<2K to be within K bits using a K-bit lookup. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"69","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"70":{"body":"For a short range (for instance, 2K≤8), we can range-constrain each word using a polynomial constraint instead of a lookup: Degree2K+12Constraintqdecompose-base-field⋅range_check(zi−zi+1⋅2K,2K)=0qstrict⋅zW=0 where range_check(ki,range)=ki⋅(1−ki)⋯(range−1−ki).","breadcrumbs":"Design » Circuit » Gadgets » Short range decomposition","id":"70","title":"Short range decomposition"},"8":{"body":"","breadcrumbs":"User Documentation » Spending notes","id":"8","title":"Spending notes"},"9":{"body":"","breadcrumbs":"User Documentation » Integration into an existing chain","id":"9","title":"Integration into an existing 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