From 0b201bef6c440dee338760b6fa3cff17894c43fd Mon Sep 17 00:00:00 2001
From: daira The nullifier design we use for Orchard is
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where:Nullifiers
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The nullifier design we use for Orchard is
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where:
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q. 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. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"66","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"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 q. 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. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"66","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"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 q. 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. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"66","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"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 q. 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. Constrain 0≤α⋅2K−n<2K to be within K bits using a K-bit lookup.","breadcrumbs":"Design » Circuit » Gadgets » Short range check","id":"66","title":"Short range check"},"7":{"body":"","breadcrumbs":"User Documentation » Creating notes","id":"7","title":"Creating notes"},"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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