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Typically, though, we will use them via gadgets . This indirection is useful because, for reasons of efficiency and limitations imposed by PLONKish circuits, the chip interfaces will often be dependent on low-level implementation details. The gadget interface can provide a more convenient and stable API that abstracts away from extraneous detail. For example, consider a hash function such as SHA-256. The interface of a chip supporting SHA-256 might be dependent on internals of the hash function design such as the separation between message schedule and compression function. The corresponding gadget interface can provide a more convenient and familiar update/finalize API, and can also handle parts of the hash function that do not need chip support, such as padding. This is similar to how accelerated instructions for cryptographic primitives on CPUs are typically accessed via software libraries, rather than directly. Gadgets can also provide modular and reusable abstractions for circuit programming at a higher level, similar to their use in libraries such as libsnark and bellman . As well as abstracting functions , they can also abstract types , such as elliptic curve points or integers of specific sizes.","breadcrumbs":"Concepts » Gadgets » Gadgets","id":"10","title":"Gadgets"},"100":{"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 » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Variable-base scalar multiplication","id":"100","title":"Variable-base scalar multiplication"},"101":{"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:bool_check(ki)=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). The helper bool_check(x)=x⋅(1−x). 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))bool_check(ki)=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+1bool_check(ki)=0(xA,i−1,yA,i−1)=((xA,i,yA,i)+(xT,yT))+(xA,i,yA,i) If the least significant bit k0=1, we set B=O, otherwise we set B=−T. Then we return A+B 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 » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Constraint program for optimized double-and-add (incomplete addition)","id":"101","title":"Constraint program for optimized double-and-add (incomplete addition)"},"102":{"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⋮λ2,130q1hi100⋮0q2hi011⋮0q3hi000⋮1zloz130z129z128⋮z5z4xAloxA,129xA,128⋮xA,5xA,4xA,3λ1loyA,129λ1,129λ1,128⋮λ1,5λ1,4yA,3λ2loλ2,129λ2,128⋮λ2,5λ2,4q1lo100⋮00q2lo011⋮10q3lo000⋮01 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. q1=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. Degree3Constraintq1⋅(yA,nwitnessed−yA,n)=0 where yA,nyA,nwitnessed=2(λ1,n+λ2,n)⋅(xA,n−(λ1,n2−xA,n−xT)), is witnessed. q2=1 This gate is used on all rows corresponding to the for loop except the last. Degree223433Constraintq2⋅(xT,cur−xT,next)=0q2⋅(yT,cur−yT,next)=0q2⋅bool_check(ki)=0, where ki=zi−2zi+1q2⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0q2⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0q2⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1)=0 where yA,iyA,i−1=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)), q3=1 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. Degree3433Constraintq3⋅bool_check(ki)=0, where ki=zi−2zi+1q3⋅(λ1,i⋅(xA,i−xT,i)−yA,i+(2ki−1)⋅yT,i)=0q3⋅(λ2,i2−xA,i−1−λ1,i2+xT,i)=0q3⋅(λ2,i⋅(xA,i−xA,i−1)−yA,i−yA,i−1witnessed)=0 where yA,iyA,i−1witnessed=2(λ1,i+λ2,i)⋅(xA,i−(λ1,i2−xA,i−xT)), is witnessed.","breadcrumbs":"Design » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Circuit design","id":"102","title":"Circuit design"},"103":{"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 » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check","id":"103","title":"Overflow check"},"104":{"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 » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Optimized check for k∈[tq,p+tq)","id":"104","title":"Optimized check for k∈[tq,p+tq)"},"105":{"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: Degree22335Constraintq_muloverflow⋅(s−(α+k254⋅2130))=0q_muloverflow⋅(z0−α−tq)=0q_muloverflow⋅(k254⋅(z130−2124))=0q_muloverflow⋅(k254⋅(s−i=0∑1292i⋅si)/2130)=0q_muloverflow⋅((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 » Gadgets » Elliptic curve cryptography » Variable-base scalar multiplication » Overflow check constraints","id":"105","title":"Overflow check constraints"},"106":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » Sinsemilla","id":"106","title":"Sinsemilla"},"107":{"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 » Gadgets » Sinsemilla » Overview","id":"107","title":"Overview"},"108":{"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 » Gadgets » Sinsemilla » Description","id":"108","title":"Description"},"109":{"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 » Gadgets » Sinsemilla » Use as a commitment scheme","id":"109","title":"Use as a commitment scheme"},"11":{"body":"You're probably here because you want to write circuits? Excellent! This section will guide you through the process of creating circuits with halo2.","breadcrumbs":"User Documentation » User Documentation","id":"11","title":"User Documentation"},"110":{"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 » Gadgets » Sinsemilla » Efficient implementation","id":"110","title":"Efficient implementation"},"111":{"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 » Gadgets » Sinsemilla » Constraint program","id":"111","title":"Constraint program"},"112":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » PLONK / Halo 2 constraints","id":"112","title":"PLONK / Halo 2 constraints"},"113":{"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 » Gadgets » Sinsemilla » Message decomposition","id":"113","title":"Message decomposition"},"114":{"body":"The Sinsemilla circuit makes use of 210 pre-computed random generators. These are loaded into a lookup table: tableidx012⋮210−1tablexxP[0]xP[1]xP[2]⋮xP[210−1]tableyyP[0]yP[1]yP[2]⋮yP[210−1]","breadcrumbs":"Design » Gadgets » Sinsemilla » Generator lookup table","id":"114","title":"Generator lookup table"},"115":{"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′qS111111111110qS211110111120qS300000000020qS410000000000fixed_y_QyQ0000000000 xQ, z0, z0′, etc. would be copied in using equality constraints.","breadcrumbs":"Design » Gadgets » Sinsemilla » Layout","id":"115","title":"Layout"},"116":{"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. 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. Degree4635ConstraintqS4⋅(2⋅yQ−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","breadcrumbs":"Design » Gadgets » Sinsemilla » Optimized Sinsemilla gate","id":"116","title":"Optimized Sinsemilla gate"},"117":{"body":"","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » MerkleCRH","id":"117","title":"MerkleCRH"},"118":{"body":"SinsemillaHash is used in the MerkleCRHOrchard hash function . The input to SinsemillaHash is: l⋆∣∣left⋆∣∣right⋆, where: l⋆=I2LEBSP10(l)=I2LEBSP10(MerkleDepthOrchard−1−layer), left⋆=I2LEBSPℓMerkleOrchard(left), right⋆=I2LEBSPℓMerkleOrchard(right), with ℓMerkleOrchard=255. left and right are allowed to be non-canonical 255-bit encodings. We break these inputs into the following MessagePieces: a (250 bits)b (20 bits)c (250 bits)=a0∣∣a1=l⋆∣∣(bits 0..=239 of left )=b0∣∣b1∣∣b2=(bits 240..=249 of left)∣∣(bits 250..=254 of left)∣∣(bits 0..=4 of right)=bits 5..=254 of right a,b,c are constrained by the SinsemillaHash to be 250 bits, 20 bits, and 250 bits respectively. In a custom gate, we check this message decomposition by enforcing the following constraints: a0=l z1,a, the index-1 running sum output of SinsemillaHash(a), is copied into the gate. z1,a has been constrained by the SinsemillaHash to be 240 bits. We recover the subpieces a0,a1 using a,z1,a: z1,a⟹a0=210a−a0=a1=a−z1,a⋅210. l+1 is loaded into a fixed column at each layer of the hash. It is used both as a gate selector, and to fix the value of l. We check that a0=(l+1)−1. Note: The reason for using l+1 instead of l is that l=0 when layer=31 (hashing two leaves). We cannot have a zero-valued selector, since a constraint gated by a zero-valued selector is never checked. b1+25⋅b2=z1,b z1,b, the index-1 running sum output of SinsemillaHash(b), is copied into the gate. z1,b has been constrained by the SinsemillaHash to be 10 bits. We witness the subpieces b1,b2 outside this gate, and constrain them each to be 5 bits. Inside the gate, we check that b1+25⋅b2=z1,b. We also recover the subpiece b0 using (b,z1,b): z1,b⟹b0=210b−b0..=10=b−(z1,b⋅210). We have now derived or witnessed every subpiece, and range-constrained every subpiece: a0 (10 bits), derived as a0=a−210⋅z1,a; a1 (240 bits), equal to z1,a; b0 (10 bits), derived as b0=b−210⋅z1,b; b1 (5 bits) is witnessed and constrained outside the gate; b2 (5 bits) is witnessed and constrained outside the gate; b1+25⋅b2 is constrained to equal z1,b, and we use them to reconstruct the original field element inputs: left=a1+2240⋅b0+2254⋅b1 right=b2+25⋅c","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Message decomposition","id":"118","title":"Message decomposition"},"119":{"body":"The Orchard circuit spans 10 advice columns while the Sinsemilla chip only uses 5 advice columns. We distribute the path hashing evenly across two Sinsemilla chips to make better use of the available circuit area. Since the output from the previous layer hash is copied into the next layer hash, we maintain continuity even when moving from one chip to the other.","breadcrumbs":"Design » Gadgets » Sinsemilla » MerkleCRH » Circuit components","id":"119","title":"Circuit components"},"12":{"body":"The halo2 crate includes several utilities to help you design and implement your circuits.","breadcrumbs":"User Documentation » Developer tools » Developer tools","id":"12","title":"Developer tools"},"120":{"body":"Given a field element α, these gadgets decompose it into W K-bit windows α=k0+2K⋅k1+22K⋅k2+⋯+2(W−1)K⋅kW−1 where each ki a K-bit value. This is done using a running sum zi,i∈[0..W). We initialize the running sum z0=α, and compute subsequent terms zi+1=2Kzi−ki. This gives us: z0z1z2↓zW=α=k0+2K⋅k1+22K⋅k2+23K⋅k3+⋯,=(z0−k0)/2K=k1+2K⋅k2+22K⋅k3+⋯,=(z1−k1)/2K=k2+2K⋅k3+⋯,⋮ (in strict mode)=(zW−1−kW−1)/2K=0 (because zW−1=kW−1)","breadcrumbs":"Design » Gadgets » Decomposition » Decomposition","id":"120","title":"Decomposition"},"121":{"body":"Strict mode constrains the running sum output zW to be zero, thus range-constraining the field element to be within W⋅K bits. In strict mode, we are also assured that zW−1=kW−1 gives us the last window in the decomposition.","breadcrumbs":"Design » Gadgets » Decomposition » Strict mode","id":"121","title":"Strict mode"},"122":{"body":"This gadget makes use of a K-bit lookup table to decompose a field element α into K-bit words. Each K-bit word ki=zi−2K⋅zi+1 is range-constrained by a lookup in the K-bit table. The region layout for the lookup decomposition uses a single advice column z, and two selectors qlookup and qrunning. zz0z1⋮zn−1znqlookup11⋮10qrunning11⋮10","breadcrumbs":"Design » Gadgets » Decomposition » Lookup decomposition","id":"122","title":"Lookup decomposition"},"123":{"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. The short variant of the lookup decomposition introduces a qbitshift selector. The same advice column z has here been renamed to word for clarity: wordαα′2K−nqlookup110qrunning000qbitshift010 where α′=α⋅2K−n. Note that 2K−n is assigned to a fixed column at keygen, and copied in at proving time. This is used in the gate enabled by the qbitshift selector to check that α was shifted correctly: Degree2Constraintqbitshift⋅(α′−(α⋅2K−n))","breadcrumbs":"Design » Gadgets » Decomposition » Short range check","id":"123","title":"Short range check"},"124":{"body":"Since the lookup decomposition and its short variant both make use of the same lookup table, we combine their lookup input expressions into a single one: qlookup⋅(qrunning⋅(zi−2K⋅zi+1)+(1−qrunning)⋅word) where zi and word are the same cell (but distinguished here for clarity of usage).","breadcrumbs":"Design » Gadgets » Decomposition » Combined lookup expression","id":"124","title":"Combined lookup expression"},"125":{"body":"For a short range (for instance, [0,range) where range≤8), we can range-constrain each word using a degree-range polynomial constraint instead of a lookup: range_check(word,range)=word⋅(1−word)⋯(range−1−word).","breadcrumbs":"Design » Gadgets » Decomposition » Short range decomposition","id":"125","title":"Short range decomposition"},"126":{"body":"","breadcrumbs":"Design » Gadgets » SHA-256 » SHA-256","id":"126","title":"SHA-256"},"127":{"body":"SHA-256 is specified in NIST FIPS PUB 180-4 . Unlike the specification, we use ⊞ for addition modulo 232, and + for field addition. ⊕ is used for XOR.","breadcrumbs":"Design » Gadgets » SHA-256 » Specification","id":"127","title":"Specification"},"128":{"body":"SHA-256 maintains state in eight 32-bit variables. It processes input as 512-bit blocks, but internally splits these blocks into 32-bit chunks. We therefore designed the SHA-256 gadget to consume input in 32-bit chunks.","breadcrumbs":"Design » Gadgets » SHA-256 » Gadget interface","id":"128","title":"Gadget interface"},"129":{"body":"The SHA-256 gadget requires a chip with the following instructions: # extern crate halo2_proofs;\n# use halo2_proofs::plonk::Error;\n# use std::fmt;\n#\n# trait Chip: Sized {}\n# trait Layouter {}\nconst BLOCK_SIZE: usize = 16;\nconst DIGEST_SIZE: usize = 8; pub trait Sha256Instructions: Chip { /// Variable representing the SHA-256 internal state. type State: Clone + fmt::Debug; /// Variable representing a 32-bit word of the input block to the SHA-256 compression /// function. type BlockWord: Copy + fmt::Debug; /// Places the SHA-256 IV in the circuit, returning the initial state variable. fn initialization_vector(layouter: &mut impl Layouter) -> Result; /// Starting from the given initial state, processes a block of input and returns the /// final state. fn compress( layouter: &mut impl Layouter, initial_state: &Self::State, input: [Self::BlockWord; BLOCK_SIZE], ) -> Result; /// Converts the given state into a message digest. fn digest( layouter: &mut impl Layouter, state: &Self::State, ) -> Result<[Self::BlockWord; DIGEST_SIZE], Error>;\n} TODO: Add instruction for computing padding. This set of instructions was chosen to strike a balance between the reusability of the instructions, and the scope for chips to internally optimise them. In particular, we considered splitting the compression function into its constituent parts (Ch, Maj etc), and providing a compression function gadget that implemented the round logic. However, this would prevent chips from using relative references between the various parts of a compression round. Having an instruction that implements all compression rounds is also similar to the Intel SHA extensions, which provide an instruction that performs multiple compression rounds.","breadcrumbs":"Design » Gadgets » SHA-256 » Chip instructions","id":"129","title":"Chip instructions"},"13":{"body":"halo2_proofs::dev::MockProver is a tool for debugging circuits, as well as cheaply verifying their correctness in unit tests. The private and public inputs to the circuit are constructed as would normally be done to create a proof, but MockProver::run instead creates an object that will test every constraint in the circuit directly. It returns granular error messages that indicate which specific constraint (if any) is not satisfied.","breadcrumbs":"User Documentation » Developer tools » Mock prover","id":"13","title":"Mock prover"},"130":{"body":"This chip implementation is based around a single 16-bit lookup table. It requires a minimum of 216 circuit rows, and is therefore suitable for use in larger circuits. We target a maximum constraint degree of 9. That will allow us to handle constraining carries and \"small pieces\" to a range of up to {0..7} in one row.","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » 16-bit table chip for SHA-256","id":"130","title":"16-bit table chip for SHA-256"},"131":{"body":"There are 64 compression rounds. Each round takes 32-bit values A,B,C,D,E,F,G,H as input, and performs the following operations: Ch(E,F,G)Maj(A,B,C)Σ0(A)Σ1(E)H′EnewAnew========(E∧F)⊕(¬E∧G)(A∧B)⊕(A∧C)⊕(B∧C)count(A,B,C)≥2(A⋙2)⊕(A⋙13)⊕(A⋙22)(E⋙6)⊕(E⋙11)⊕(E⋙25)H+Ch(E,F,G)+Σ1(E)+Kt+Wtreduce6(H′+D)reduce7(H′+Maj(A,B,C)+Σ0(A)) where reducei must handle a carry 0≤carry = MyCircuit { a: None, lookup_table, }; // Create the area you want to draw on. // Use SVGBackend if you want to render to .svg instead. use plotters::prelude::*; let root = BitMapBackend::new(\"layout.png\", (1024, 768)).into_drawing_area(); root.fill(&WHITE).unwrap(); let root = root .titled(\"Example Circuit Layout\", (\"sans-serif\", 60)) .unwrap(); halo2_proofs::dev::CircuitLayout::default() // You can optionally render only a section of the circuit. .view_width(0..2) .view_height(0..16) // You can hide labels, which can be useful with smaller areas. .show_labels(false) // Render the circuit onto your area! // The first argument is the size parameter for the circuit. .render(5, &circuit, &root) .unwrap();\n} Columns are laid out from left to right as instance, advice and fixed. The order of columns is otherwise without meaning. Instance columns have a white background. Advice columns have a red background. Fixed columns have a blue background. Regions are shown as labelled green boxes (overlaying the background colour). A region may appear as multiple boxes if some of its columns happen to not be adjacent. Cells that have been assigned to by the circuit will be shaded in grey. If any cells are assigned to more than once (which is usually a mistake), they will be shaded darker than the surrounding cells.","breadcrumbs":"User Documentation » Developer tools » Circuit layout","id":"15","title":"Circuit layout"},"150":{"body":"v1 v1 of the σ1 gate takes in a word that's split into (10,7,2,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(10),b(7),c(2),d(13)). b(7) is further split into (2,2,3)-bit chunks b(7)lo,b(7)mid,b(7)hi. We witness the spread versions of the small chunks. We already have spread(a(10)) and spread(d(13)) from the message scheduling. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1 a0 a1 a2 a3 a4 a5 a6 0 {0,1,2,3,4,5} R0even spread(R0even) b(7)lo spread(b(7)lo) b(7)mid spread(b(7)mid) 1 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(a(10)) spread(d(13)) b(7) 0 {0,1,2,3,4,5} R1even spread(R1even) c(2) spread(c(2)) b(7)hi spread(b(7)hi) 0 {0,1,2,3,4,5} R1odd spread(R1odd) Constraints: s_low_sigma_1 (σ1 v1 constraint): LHS−RHS=0 LHS=spread(R0even)+2⋅spread(R0odd)+232⋅spread(R1even)+233⋅spread(R1odd) RHS=429b(7)hi430c(2)++49d(13)427b(7)mid427b(7)hi+++47c(2)425b(7)lo425b(7)mid+++44b(7)hi415a(10)423b(7)lo+++42b(7)mid42d(13)413a(10)+++b(7)loc(2)d(13)++ check that b was properly split into subsections for 7-bit pieces. Wb(7)lo+22Wb(7)mid+24Wb(7)hi−W=0 2-bit range check and 2-bit spread check on b(7)lo,b(7)mid,c(2) 3-bit range check and 3-bit spread check on b(7)hi v2 v2 of the σ1 gate takes in a word that's split into (3,4,3,7,1,1,13)-bit chunks (already constrained by message scheduling). We refer to these chunks respectively as (a(3),b(4),c(3),d(7),e(1),f(1),g(13)). We already have spread(d(7)),spread(g(13)) from the message scheduling. The 1-bit e(1),f(1) remain unchanged by the spread operation and can be used directly. We further split b(4) into two 2-bit chunks b(4)lo,b(4)hi. We witness the spread versions of the small chunks. (X⋙17)⊕(X⋙19)⊕(X≫10) is equivalent to (X⋘15)⊕(X⋘13)⊕(X≫10). s_low_sigma_1_v2 a0 a1 a2 a3 a4 a5 a6 a7 0 {0,1,2,3,4,5} R0even spread(R0even) b(4)lo spread(b(4)lo) b(4)hi spread(b(4)hi) 1 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(d(7)) spread(g(13)) b(4) e(1) 0 {0,1,2,3,4,5} R1even spread(R1even) a(3) spread(a(3)) c(3) spread(c(3)) f(1) 0 {0,1,2,3,4,5} R1odd spread(R1odd) Constraints: s_low_sigma_1_v2 (σ1 v2 constraint): LHS−RHS=0 LHS=spread(R0even)+2⋅spread(R0odd)+232⋅spread(R1even)+233⋅spread(R1odd) RHS=425d(7)431f(1)++422c(3)430e(1)++420b(4)hi423d(7)++49g(13)418b(4)lo420c(3)+++48f(1)415a418b(4)hi+++47e(1)42g(13)416b(4)lo+++d(7)41f(1)413a+++e(1)g(13)+ check that b was properly split into subsections for 4-bit pieces. Wb(4)lo+22Wb(4)hi−W=0 2-bit range check and 2-bit spread check on b(4)lo,b(4)hi 3-bit range check and 3-bit spread check on a(3),c(3)","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » σ_1 gate","id":"150","title":"σ_1 gate"},"151":{"body":"Small range constraints Let constrainn(x)=∏i=0n(x−i). Constraining this expression to equal zero enforces that x is in [0..n]. 2-bit range check (a−3)(a−2)(a−1)(a)=0 sr2 a0 1 a 2-bit spread l1(a)+4∗l2(a)+5∗l3(a)−a′=0 ss2 a0 a1 1 a a' with interpolation polynomials: l0(a)=(−3)(−2)(−1)(a−3)(a−2)(a−1) (spread(00)=0000) l1(a)=(−2)(−1)(1)(a−3)(a−2)(a) (spread(01)=0001) l2(a)=(−1)(1)(2)(a−3)(a−1)(a) (spread(10)=0100) l3(a)=(1)(2)(3)(a−2)(a−1)(a) (spread(11)=0101) 3-bit range check (a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a)=0 sr3 a0 1 a 3-bit spread l1(a)+4∗l2(a)+5∗l3(a)+16∗l4(a)+17∗l5(a)+20∗l6(a)+21∗l7(a)−a′=0 ss3 a0 a1 1 a a' with interpolation polynomials: l0(a)=(−7)(−6)(−5)(−4)(−3)(−2)(−1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1) (spread(000)=000000) l1(a)=(−6)(−5)(−4)(−3)(−2)(−1)(1)(a−7)(a−6)(a−5)(a−4)(a−3)(a−2)(a) (spread(001)=000001) l2(a)=(−5)(−4)(−3)(−2)(−1)(1)(2)(a−7)(a−6)(a−5)(a−4)(a−3)(a−1)(a) (spread(010)=000100) l3(a)=(−4)(−3)(−2)(−1)(1)(2)(3)(a−7)(a−6)(a−5)(a−4)(a−2)(a−1)(a) (spread(011)=000101) l4(a)=(−3)(−2)(−1)(1)(2)(3)(4)(a−7)(a−6)(a−5)(a−3)(a−2)(a−1)(a) (spread(100)=010000) l5(a)=(−2)(−1)(1)(2)(3)(4)(5)(a−7)(a−6)(a−4)(a−3)(a−2)(a−1)(a) (spread(101)=010001) l6(a)=(−1)(1)(2)(3)(4)(5)(6)(a−7)(a−5)(a−4)(a−3)(a−2)(a−1)(a) (spread(110)=010100) l7(a)=(1)(2)(3)(4)(5)(6)(7)(a−6)(a−5)(a−4)(a−3)(a−2)(a−1)(a) (spread(111)=010101) reduce_6 gate Addition (mod232) of 6 elements Input: E {eilo,eihi}i=05 carry Check: E=e0+e1+e2+e3+e4+e5(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0+a1+a2+a3+a4+a5+a6−a7=0 Carry gate (sc): 216a6ω−1+a6+[(a6−5)(a6−4)(a6−3)(a6−2)(a6−1)(a6)]=0 sa sc a0 a1 a2 a3 a4 a5 a6 a7 1 0 e0lo e1lo e2lo e3lo e4lo e5lo −carry∗216 Elo 1 1 e0hi e1hi e2hi e3hi e4hi e5hi carry Ehi Assume inputs are constrained to 16 bits. Addition gate (sa): a0ω−1+a1ω−1+a2ω−1+a0+a1+a2+a3ω−1−a3=0 Carry gate (sc): 216a3ω+a3ω−1=0 sa sc a0 a1 a2 a3 0 0 e0lo e1lo e2lo −carry∗216 1 1 e3lo e4lo e5lo Elo 0 0 e0hi e1hi e2hi carry 1 0 e3hi e4hi e5hi Ehi reduce_7 gate Addition (mod232) of 7 elements Input: E {eilo,eihi}i=06 carry Check: E=e0+e1+e2+e3+e4+e5+e6(mod32) Assume inputs are constrained to 16 bits. Addition gate (sa): a0+a1+a2+a3+a4+a5+a6+a7−a8=0 Carry gate (sc): 216a7ω−1+a7+[(a7−6)(a7−5)(a7−4)(a7−3)(a7−2)(a7−1)(a7)]=0 sa sc a0 a1 a2 a3 a4 a5 a6 a7 a8 1 0 e0lo e1lo e2lo e3lo e4lo e5lo e6lo −carry∗216 Elo 1 1 e0hi e1hi e2hi e3hi e4hi e5hi e6hi carry Ehi","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Helper gates","id":"151","title":"Helper gates"},"152":{"body":"For each block M∈{0,1}512 of the padded message, 64 words of 32 bits each are constructed as follows: the first 16 are obtained by splitting M into 32-bit blocks M=W0∣∣W1∣∣⋯∣∣W14∣∣W15; the remaining 48 words are constructed using the formula: Wi=σ1(Wi−2)⊞Wi−7⊞σ0(Wi−15)⊞Wi−16, for 16≤i<64. sw sd0 sd1 sd2 sd3 ss0 ss0_v2 ss1 ss1_v2 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0lo spread(W0lo) W0lo W0hi W0 σ0(W1)lo σ1(W14)lo W9lo 1 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W0hi spread(W0hi) W16 σ0(W1)hi σ1(W14)hi W9hi carry16 0 1 1 0 0 0 0 0 0 {0,1,2,3,4} W1d(14) spread(W1d(14)) W1lo W1hi W1 σ0(W2)lo σ1(W15)lo W10lo 1 0 0 0 0 0 0 0 0 {0,1,2} W1c(11) spread(W1c(11)) W1a(3) W1b(4) W17 σ0(W2)hi σ1(W15)hi W10hi carry17 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) W1b(4)lo spread(W1b(4)lo) W1b(4)hi spread(W1b(4)hi) 0 0 0 0 0 1 0 0 0 {0,1,2,3,4,5} R1odd spread(R0odd) spread(R1odd) spread(W1c(11)) spread(W1d(14)) W1b(4) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd spread(R1even) 0 0 W1a(3) spread(W1a(3)) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1odd) σ0v1R0 σ0v1R1 σ0v1R0even σ0v1R0odd .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 0 0 0 {0,1,2,3} W14g(13) spread(W14g(13)) W14a(3) W14c(3) 0 1 0 1 0 0 0 0 0 0 W14d(7) spread(W14d(7)) W14lo W14hi W14 σ0(W15)lo σ1(W28)lo W23lo 1 0 0 0 0 0 0 0 0 0 W14b(4) spread(W14b(4)) W14e(1) W14f(1) W30 σ0(W15)hi σ1(W28)hi W23hi carry30 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) W14b(4)lo spread(W14b(4)lo) W14b(4)hi spread(W14b(4)hi) 0 0 0 0 0 0 1 0 0 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(W14d(7)) spread(W14g(13)) W1b(14) W14e(1) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) W14a(3) spread(W14a(3)) W14c(3) spread(W14c(3)) W14f(1) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) σ0v2R0 σ0v2R1 σ0v2R0even σ0v2R0odd 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) W14b(4)lo spread(W14b(4)lo) W14b(4)hi spread(W14b(4)hi) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(d) spread(g) W14e(1) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) W14a(3) spread(W14a(3)) W14c(3) spread(W14c(3)) W14f(1) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) σ1v2R0 σ1v2R1 σ1v2R0even σ1v2R0odd .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 1 0 0 0 0 {0,1,2,3} W49d(13) spread(W49d(13)) W49lo W49hi W49 0 0 0 0 0 0 0 0 0 {0,1} W49a(10) spread(W49a(10)) W49c(2) W49b(7) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) W49b(7)lo spread(W49b(7)lo) W49b(7)mid spread(W49b(7)mid) 0 0 0 0 0 0 0 0 1 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(a) spread(d) W1b(49) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) W49c(2) spread(W49c(2)) W49b(7)hi spread(W49b(7)hi) 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) σ1v1R0 σ1v1R1 σ1v1R0even σ1v1R0odd .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62lo spread(W62lo) W62lo W62hi W62 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W62hi spread(W62hi) 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63lo spread(W63lo) W63lo W63hi W63 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} W63hi spread(W63hi) Constraints: sw: construct word using reduce4 sd0: decomposition gate for W0,W62,W63 Wlo+216Whi−W=0 sd1: decomposition gate for W1..13 (split into (3,4,11,14)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+218Wd(14)−W=0 sd2: decomposition gate for W14..48 (split into (3,4,3,7,1,1,13)-bit pieces) Wa(3)+23Wb(4)lo+25Wb(4)hi+27Wc(11)+210Wd(14)+217We(1)+218Wf(1)+219Wg(13)−W=0 sd3: decomposition gate for W49..61 (split into (10,7,2,13)-bit pieces) Wa(10)+210Wb(7)lo+212Wb(7)mid+215Wb(7)hi+217Wc(2)+219Wd(13)−W=0","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Message scheduling region","id":"152","title":"Message scheduling region"},"153":{"body":"+----------------------------------------------------------+\n| |\n| decompose E, |\n| Σ_1(E) |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_5() to get H' |\n| | |\n+----------------------------------------------------------+\n| decompose F, decompose G |\n| |\n| Ch(E,F,G) |\n| |\n+----------------------------------------------------------+\n| |\n| decompose A, |\n| Σ_0(A) |\n| |\n| |\n| +---------------------------------------+\n| | |\n| | reduce_7() to get A_new, |\n| | using H' |\n| | |\n+------------------+---------------------------------------+\n| decompose B, decompose C |\n| |\n| Maj(A,B,C) |\n| |\n| +---------------------------------------+\n| | reduce_6() to get E_new, |\n| | using H' |\n+------------------+---------------------------------------+ Initial round: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0d(7) spread(E0d(7)) E0b(5)lo spread(E0b(5)lo) E0b(5)hi spread(E0b(5)hi) E0lo spread(E0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0c(14) spread(E0c(14)) E0a(6)lo spread(E0a(6)lo) E0a(6)hi spread(E0a(6)hi) E0hi spread(E0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) spread(E0b(5)lo) spread(E0b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(E0d(7)) spread(E0c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) spread(E0a(6)lo) spread(E0a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0d(7) spread(F0d(7)) F0b(5)lo spread(F0b(5)lo) F0b(5)hi spread(F0b(5)hi) F0lo spread(F0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} F0c(14) spread(F0c(14)) F0a(6)lo spread(F0a(6)lo) F0a(6)hi spread(F0a(6)hi) F0hi spread(F0hi) 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} G0d(7) spread(G0d(7)) G0b(5)lo spread(G0b(5)lo) G0b(5)hi spread(G0b(5)hi) G0lo spread(G0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} G0c(14) spread(G0c(14)) G0a(6)lo spread(G0a(6)lo) G0a(6)hi spread(G0a(6)hi) G0hi spread(G0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even spread(P0even) spread(Elo) spread(Ehi) Q0odd K0lo H0lo W0lo 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd spread(P0odd) spread(P1odd) Σ1(E0)lo Σ1(E0)hi K0hi H0hi W0hi 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even spread(P1even) spread(Flo) spread(Fhi) Q1odd P1odd Hprime0lo Hprime0hi Hprime0carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd spread(P1odd) D0lo E1lo 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even spread(Q0even) spread(Eneglo) spread(Eneghi) spread(Elo) D0hi E1hi E1carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd spread(Q0odd) spread(Q1odd) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even spread(Q1even) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd spread(Q1odd) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0b(11) spread(A0b(11)) A0c(9)lo spread(A0c(9)lo) A0c(9)mid spread(A0c(9)mid) A0lo spread(A0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0d(10) spread(A0d(10)) A0a(2) spread(A0a(2)) A0c(9)hi spread(A0c(9)hi) A0hi spread(A0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} B0b(11) spread(B0b(11)) B0c(9)lo spread(B0c(9)lo) B0c(9)mid spread(B0c(9)mid) B0lo spread(B0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} B0d(10) spread(B0d(10)) B0a(2) spread(B0a(2)) B0c(9)hi spread(B0c(9)hi) B0hi spread(B0hi) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} C0b(11) spread(C0b(11)) C0c(9)lo spread(C0c(9)lo) C0c(9)mid spread(C0c(9)mid) C0lo spread(C0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} C0d(10) spread(C0d(10)) C0a(2) spread(C0a(2)) C0c(9)hi spread(C0c(9)hi) C0hi spread(C0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even spread(M0even) M1odd spread(A0lo) spread(A0hi) Hprime0lo Hprime0hi 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd spread(M0odd) spread(M1odd) spread(B0lo) spread(B0hi) Σ0(A0)lo A1lo A1carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even spread(M1even) spread(C0lo) spread(C0hi) Σ0(A0)hi A1hi 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd spread(M1odd) Steady-state: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 0 0 1 0 0 0 0 0 0 0 0 {0,1,2} F0d(7) spread(E0d(7)) E0b(5)lo spread(E0b(5)lo) E0b(5)hi spread(E0b(5)hi) E0lo spread(E0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} E0c(14) spread(E0c(14)) E0a(6)lo spread(E0a(6)lo) E0a(6)hi spread(E0a(6)hi) E0hi spread(E0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) spread(E0b(5)lo) spread(E0b(5)hi) 0 0 0 0 1 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(E0d(7)) spread(E0c(14)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) spread(E0a(6)lo) spread(E0a(6)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P0even spread(P0even) spread(Elo) spread(Ehi) Q0odd K0lo H0lo W0lo 0 0 0 0 0 0 0 1 0 0 1 {0,1,2,3,4,5} P0odd spread(P0odd) spread(P1odd) Σ1(E0)lo Σ1(E0)hi K0hi H0hi W0hi 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1even spread(P1even) spread(Flo) spread(Fhi) Q1odd P1odd Hprime0lo Hprime0hi Hprime0carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} P1odd spread(P1odd) D0lo E1lo 0 0 0 0 0 0 0 0 0 1 0 {0,1,2,3,4,5} Q0even spread(Q0even) spread(Eneglo) spread(Eneghi) spread(Elo) D0hi E1hi E1carry 0 0 0 0 0 0 1 0 0 0 0 {0,1,2,3,4,5} Q0odd spread(Q0odd) spread(Q1odd) spread(Ehi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1even spread(Q1even) spread(Glo) spread(Ghi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} Q1odd spread(Q1odd) 0 1 0 0 0 0 0 0 0 0 0 {0,1,2} A0b(11) spread(A0b(11)) A0c(9)lo spread(A0c(9)lo) A0c(9)mid spread(A0c(9)mid) A0lo spread(A0lo) 0 0 0 0 0 0 0 0 0 0 0 {0,1} A0d(10) spread(A0d(10)) A0a(2) spread(A0a(2)) A0c(9)hi spread(A0c(9)hi) A0hi spread(A0hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0even spread(R0even) spread(c(9)lo) spread(c(9)mid) 0 0 0 1 0 0 0 0 0 0 0 {0,1,2,3,4,5} R0odd spread(R0odd) spread(R1odd) spread(d(10)) spread(b(11)) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1even spread(R1even) spread(a(2)) spread(c(9)hi) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} R1odd spread(R1odd) 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M0even spread(M0even) M1odd spread(A0lo) spread(A0hi) Hprime0lo Hprime0hi 0 0 0 0 0 1 0 0 1 0 0 {0,1,2,3,4,5} M0odd spread(M0odd) spread(M1odd) spread(B0lo) spread(B0hi) Σ0(A0)lo A1lo A1carry 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1even spread(M1even) spread(C0lo) spread(C0hi) Σ0(A0)hi A1hi 0 0 0 0 0 0 0 0 0 0 0 {0,1,2,3,4,5} M1odd spread(M1odd) Final digest: s_digest sd_abcd sd_efgh ss0 ss1 s_maj s_ch_neg s_ch s_a_new s_e_new s_h_prime a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 1 0 0 0 0 0 0 0 0 0 0 0 0 0 A63lo A63hi A63 B63lo B63hi B63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C63lo C63hi C63 C63lo C63hi C63 1 0 0 0 0 0 0 0 0 0 0 0 0 0 E63lo E63hi E63 G63lo G63hi G63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F63lo F63hi F63 H63lo H63hi H63","breadcrumbs":"Design » Gadgets » SHA-256 » 16-bit table chip » Compression region","id":"153","title":"Compression region"},"154":{"body":"This section covers the background material required to understand the Halo 2 proving system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything could do with additional explanation, let us know !","breadcrumbs":"Background Material » Background Material","id":"154","title":"Background Material"},"155":{"body":"A fundamental component of many cryptographic protocols is the algebraic structure known as a field . Fields are sets of objects (usually numbers) with two associated binary operators + and × such that various field axioms hold. The real numbers R are an example of a field with uncountably many elements. Halo makes use of finite fields which have a finite number of elements. Finite fields are fully classified as follows: if F is a finite field, it contains ∣F∣=pk elements for some integer k≥1 and some prime p; any two finite fields with the same number of elements are isomorphic. In particular, all of the arithmetic in a prime field Fp is isomorphic to addition and multiplication of integers modulo p, i.e. in Zp. This is why we often refer to p as the modulus . We'll write a field as Fq where q=pk. The prime p is called its characteristic . In the cases where k>1 the field Fq is a k-degree extension of the field Fp. (By analogy, the complex numbers C=R(i) are an extension of the real numbers.) However, in Halo we do not use extension fields. Whenever we write Fp we are referring to what we call a prime field which has a prime p number of elements, i.e. k=1. Important notes: There are two special elements in any field: 0, the additive identity, and 1, the multiplicative identity. The least significant bit of a field element, when represented as an integer in binary format, can be interpreted as its \"sign\" to help distinguish it from its additive inverse (negation). This is because for some nonzero element a which has a least significant bit 0 we have that −a=p−a has a least significant bit 1, and vice versa. We could also use whether or not an element is larger than (p−1)/2 to give it a \"sign.\" Finite fields will be useful later for constructing polynomials and elliptic curves . Elliptic curves are examples of groups, which we discuss next.","breadcrumbs":"Background Material » Fields » Fields","id":"155","title":"Fields"},"156":{"body":"Groups are simpler and more limited than fields; they have only one binary operator ⋅ and fewer axioms. They also have an identity, which we'll denote as 1. Any non-zero element a in a group has an inverse b=a−1, which is the unique element b such that a⋅b=1. For example, the set of nonzero elements of Fp forms a group, where the group operation is given by multiplication on the field. (aside) Additive vs multiplicative notation If ⋅ is written as × or omitted (i.e. a⋅b written as ab), the identity as 1, and inversion as a−1, as we did above, then we say that the group is \"written multiplicatively\". If ⋅ is written as +, the identity as 0 or O, and inversion as −a, then we say it is \"written additively\". It's conventional to use additive notation for elliptic curve groups, and multiplicative notation when the elements come from a finite field. When additive notation is used, we also write [k]A=k timesA+A+⋯+A for nonnegative k and call this \"scalar multiplication\"; we also often use uppercase letters for variables denoting group elements. When multiplicative notation is used, we also write ak=k timesa×a×⋯×a and call this \"exponentiation\". In either case we call the scalar k such that [k]g=a or gk=a the \"discrete logarithm\" of a to base g. We can extend scalars to negative integers by inversion, i.e. [−k]A+[k]A=O or a−k×ak=1. The order of an element a of a finite group is defined as the smallest positive integer k such that ak=1 (in multiplicative notation) or [k]a=O (in additive notation). The order of the group is the number of elements. Groups always have a generating set , which is a set of elements such that we can produce any element of the group as (in multiplicative terminology) a product of powers of those elements. So if the generating set is g1..k, we can produce any element of the group as i=1∏kgiai. There can be many different generating sets for a given group. A group is called cyclic if it has a (not necessarily unique) generating set with only a single element — call it g. In that case we can say that g generates the group, and that the order of g is the order of the group. Any finite cyclic group G of order n is isomorphic to the integers modulo n (denoted Z/nZ), such that: the operation ⋅ in G corresponds to addition modulo n; the identity in G corresponds to 0; some generator g∈G corresponds to 1. Given a generator g, the isomorphism is always easy to compute in the Z/nZ→G direction; it is just a↦ga (or in additive notation, a↦[a]g). It may be difficult in general to compute in the G→Z/nZ direction; we'll discuss this further when we come to elliptic curves . If the order n of a finite group is prime, then the group is cyclic, and every non-identity element is a generator.","breadcrumbs":"Background Material » Fields » Groups","id":"156","title":"Groups"},"157":{"body":"We use the notation Fp× for the multiplicative group (i.e. the group operation is multiplication in Fp) over the set Fp−{0}. A quick way of obtaining the inverse in Fp× is a−1=ap−2. The reason for this stems from Fermat's little theorem , which states that ap=a(modp) for any integer a. If a is nonzero, we can divide by a twice to get ap−2=a−1. Let's assume that α is a generator of Fp×, so it has order p−1 (equal to the number of elements in Fp×). Therefore, for any element in a∈Fp× there is a unique integer i∈{0..p−2} such that a=αi. Notice that a×b where a,b∈Fp× can really be interpreted as αi×αj where a=αi and b=αj. Indeed, it holds that αi×αj=αi+j for all 0≤i,j = MyCircuit { a: None, lookup_table, }; // Generate the DOT graph string. let dot_string = halo2_proofs::dev::circuit_dot_graph(&circuit); // Now you can either handle it in Rust, or just // print it out to use with command-line tools. print!(\"{}\", dot_string);\n}","breadcrumbs":"User Documentation » Developer tools » Circuit structure","id":"16","title":"Circuit structure"},"160":{"body":"In a field Fp exactly half of all nonzero elements are squares; the remainder are non-squares or \"quadratic non-residues\". In order to see why, consider an α that generates the 2-order multiplicative subgroup of Fp× (this exists because p−1 is divisible by 2 since p is a prime greater than 2) and β that generates the t-order multiplicative subgroup of Fp× where p−1=2t. Then every element a∈Fp× can be written uniquely as αi⋅βj with i∈Z2 and j∈Zt. Half of all elements will have i=0 and the other half will have i=1. Let's consider the simple case where p≡3(mod4) and so t is odd (if t is even, then p−1 would be divisible by 4, which contradicts p being 3(mod4)). If a∈Fp× is a square, then there must exist b=αi⋅βj such that b2=a. But this means that a=(αi⋅βj)2=α2i⋅β2j=β2j. In other words, all squares in this particular field do not generate the 2-order multiplicative subgroup, and so since half of the elements generate the 2-order subgroup then at most half of the elements are square. In fact exactly half of the elements are square (since squaring each nonsquare element gives a unique square). This means we can assume all squares can be written as βm for some m, and therefore finding the square root is a matter of exponentiating by 2−1(modt). In the event that p≡1(mod4) then things get more complicated because 2−1(modt) does not exist. Let's write p−1 as 2k⋅t with t odd. The case k=0 is impossible, and the case k=1 is what we already described, so consider k≥2. α generates a 2k-order multiplicative subgroup and β generates the odd t-order multiplicative subgroup. Then every element a∈Fp× can be written as αi⋅βj for i∈Z2k and j∈Zt. If the element is a square, then there exists some b=a which can be written b=αi′⋅βj′ for i′∈Z2k and j′∈Zt. This means that a=b2=α2i′⋅β2j′, therefore we have i≡2i′(mod2k), and j≡2j′(modt). i would have to be even in this case because otherwise it would be impossible to have i≡2i′(mod2k) for any i′. In the case that a is not a square, then i is odd, and so half of all elements are squares. In order to compute the square root, we can first raise the element a=αi⋅βj to the power t to \"kill\" the t-order component, giving at=αit(mod2)k⋅βjt(modt)=αit(mod2)k and then raise this result to the power t−1(mod2k) to undo the effect of the original exponentiation on the 2k-order component: (αitmod2k)t−1(mod2k)=αi (since t is relatively prime to 2k). This leaves bare the αi value which we can trivially handle. We can similarly kill the 2k-order component to obtain βj⋅2−1(modt), and put the values together to obtain the square root. It turns out that in the cases k=2,3 there are simpler algorithms that merge several of these exponentiations together for efficiency. For other values of k, the only known way is to manually extract i by squaring until you obtain the identity for every single bit of i. This is the essence of the Tonelli-Shanks square root algorithm and describes the general strategy. (There is another square root algorithm that uses quadratic extension fields, but it doesn't pay off in efficiency until the prime becomes quite large.)","breadcrumbs":"Background Material » Fields » Square roots","id":"160","title":"Square roots"},"161":{"body":"In the previous sections we wrote p−1=2k⋅t with t odd, and stated that an element α∈Fp× generated the 2k-order subgroup. For convenience, let's denote n:=2k. The elements {1,α,…,αn−1} are known as the nth roots of unity . The primitive root of unity , ω, is an nth root of unity such that ωi=1 except when i≡0(modn). Important notes: If α is an nth root of unity, α satisfies αn−1=0. If α=1, then 1+α+α2+⋯+αn−1=0. Equivalently, the roots of unity are solutions to the equation Xn−1=(X−1)(X−α)(X−α2)⋯(X−αn−1). ω2n+i=−ωi (\"Negation lemma\") . Proof: ωn=1⟹ωn−1=0⟹(ωn/2+1)(ωn/2−1)=0. Since the order of ω is n, ωn/2=1. Therefore, ωn/2=−1. (ω2n+i)2=(ωi)2 (\"Halving lemma\") . Proof: (ω2n+i)2=ωn+2i=ωn⋅ω2i=ω2i=(ωi)2. In other words, if we square each element in the nth roots of unity, we would get back only half the elements, {(ωni)2}={ωn/2} (i.e. the 2nth roots of unity). There is a two-to-one mapping between the elements and their squares.","breadcrumbs":"Background Material » Fields » Roots of unity","id":"161","title":"Roots of unity"},"162":{"body":"Friedman, R. (n.d.) \"Cyclic Groups and Elementary Number Theory II\" (p. 5).","breadcrumbs":"Background Material » Fields » References","id":"162","title":"References"},"163":{"body":"Let A(X) be a polynomial over Fp with formal indeterminate X. As an example, A(X)=a0+a1X+a2X2+a3X3 defines a degree-3 polynomial. a0 is referred to as the constant term. Polynomials of degree n−1 have n coefficients. We will often want to compute the result of replacing the formal indeterminate X with some concrete value x, which we denote by A(x). In mathematics this is commonly referred to as \"evaluating A(X) at a point x\". The word \"point\" here stems from the geometrical usage of polynomials in the form y=A(x), where (x,y) is the coordinate of a point in two-dimensional space. However, the polynomials we deal with are almost always constrained to equal zero, and x will be an element of some field . This should not be confused with points on an elliptic curve , which we also make use of, but never in the context of polynomial evaluation. Important notes: Multiplication of polynomials produces a product polynomial that is the sum of the degrees of its factors. Polynomial division subtracts from the degree. deg(A(X)B(X))=deg(A(X))+deg(B(X)), deg(A(X)/B(X))=deg(A(X))−deg(B(X)). Given a polynomial A(X) of degree n−1, if we obtain n evaluations of the polynomial at distinct points then these evaluations perfectly define the polynomial. In other words, given these evaluations we can obtain a unique polynomial A(X) of degree n−1 via polynomial interpolation. [a0,a1,⋯,an−1] is the coefficient representation of the polynomial A(X). Equivalently, we could use its evaluation representation [(x0,A(x0)),(x1,A(x1)),⋯,(xn−1,A(xn−1))] at n distinct points. Either representation uniquely specifies the same polynomial. (aside) Horner's rule Horner's rule allows for efficient evaluation of a polynomial of degree n−1, using only n−1 multiplications and n−1 additions. It is the following identity: a0+a1X+a2X2+⋯+an−1Xn−1=a0+X(a1+X(a2+⋯+X(an−2+Xan−1))),","breadcrumbs":"Background Material » Polynomials » Polynomials","id":"163","title":"Polynomials"},"164":{"body":"The FFT is an efficient way of converting between the coefficient and evaluation representations of a polynomial. It evaluates the polynomial at the nth roots of unity {ω0,ω1,⋯,ωn−1}, where ω is a primitive nth root of unity. By exploiting symmetries in the roots of unity, each round of the FFT reduces the evaluation into a problem only half the size. Most commonly we use polynomials of length some power of two, n=2k, and apply the halving reduction recursively.","breadcrumbs":"Background Material » Polynomials » Fast Fourier Transform (FFT)","id":"164","title":"Fast Fourier Transform (FFT)"},"165":{"body":"In the coefficient representation, it takes O(n2) operations to multiply two polynomials A(X)⋅B(X)=C(X): A(X)B(X)C(X)=a0+a1X+a2X2+⋯+an−1Xn−1,=b0+b1X+b2X2+⋯+bn−1Xn−1,=a0⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+a1X⋅(b0+b1X+b2X2+⋯+bn−1Xn−1)+⋯+an−1Xn−1⋅(b0+b1X+b2X2+⋯+bn−1Xn−1), where each of the n terms in the first polynomial has to be multiplied by the n terms of the second polynomial. In the evaluation representation, however, polynomial multiplication only requires O(n) operations: ABC:{(x0,A(x0)),(x1,A(x1)),⋯,(xn−1,A(xn−1))},:{(x0,B(x0)),(x1,B(x1)),⋯,(xn−1,B(xn−1))},:{(x0,A(x0)B(x0)),(x1,A(x1)B(x1)),⋯,(xn−1,A(xn−1)B(xn−1))}, where each evaluation is multiplied pointwise. This suggests the following strategy for fast polynomial multiplication: Evaluate polynomials at all n points; Perform fast pointwise multiplication in the evaluation representation (O(n)); Convert back to the coefficient representation. The challenge now is how to evaluate and interpolate the polynomials efficiently. Naively, evaluating a polynomial at n points would require O(n2) operations (we use the O(n) Horner's rule at each point): ⎣⎡A(1)A(ω)A(ω2)⋮A(ωn−1)⎦⎤=⎣⎡111⋮11ωω2⋮ωn−11ω2ω2⋅2⋮ω2(n−1)………⋯1ωn−1ω2⋅(n−1)⋮ω(n−1)2⎦⎤⋅⎣⎡a0a1a2⋮an−1⎦⎤. For convenience, we will denote the matrices above as: A^=Vω⋅A. (A^ is known as the Discrete Fourier Transform of A; Vω is also called the Vandermonde matrix .)","breadcrumbs":"Background Material » Polynomials » Motivation: Fast polynomial multiplication","id":"165","title":"Motivation: Fast polynomial multiplication"},"166":{"body":"Our strategy is to divide a DFT of size n into two interleaved DFTs of size n/2. Given the polynomial A(X)=a0+a1X+a2X2+⋯+an−1Xn−1, we split it up into even and odd terms: AevenAodd=a0+a2X+⋯+an−2X2n−1,=a1+a3X+⋯+an−1X2n−1. To recover the original polynomial, we do A(X)=Aeven(X2)+XAodd(X2). Trying this out on points ωni and ωn2n+i, i∈[0..2n−1], we start to notice some symmetries: A(ωni)A(ωn2n+i)=Aeven((ωni)2)+ωniAodd((ωni)2),=Aeven((ωn2n+i)2)+ωn2n+iAodd((ωn2n+i)2)=Aeven((−ωni)2)−ωniAodd((−ωni)2)←(negation lemma)=Aeven((ωni)2)−ωniAodd((ωni)2). Notice that we are only evaluating Aeven(X) and Aodd(X) over half the domain {(ωn0)2,(ωn)2,⋯,(ωn2n−1)2}={ωn/2i},i=[0..2n−1] (halving lemma). This gives us all the terms we need to reconstruct A(X) over the full domain {ω0,ω,⋯,ωn−1}: which means we have transformed a length-n DFT into two length-2n DFTs. We choose n=2k to be a power of two (by zero-padding if needed), and apply this divide-and-conquer strategy recursively. By the Master Theorem [1] , this gives us an evaluation algorithm with O(nlog2n) operations, also known as the Fast Fourier Transform (FFT).","breadcrumbs":"Background Material » Polynomials » The (radix-2) Cooley-Tukey algorithm","id":"166","title":"The (radix-2) Cooley-Tukey algorithm"},"167":{"body":"So we've evaluated our polynomials and multiplied them pointwise. What remains is to convert the product from the evaluation representation back to coefficient representation. To do this, we simply call the FFT on the evaluation representation. However, this time we also: replace ωi by ω−i in the Vandermonde matrix, and multiply our final result by a factor of 1/n. In other words: A=n1Vω−1⋅A^. (To understand why the inverse FFT has a similar form to the FFT, refer to Slide 13-1 of [2] . The below image was also taken from [2] .)","breadcrumbs":"Background Material » Polynomials » Inverse FFT","id":"167","title":"Inverse FFT"},"168":{"body":"The Schwartz-Zippel lemma informally states that \"different polynomials are different at most points.\" Formally, it can be written as follows: Let p(x1,x2,⋯,xn) be a nonzero polynomial of n variables with degree d. Let S be a finite set of numbers with at least d elements in it. If we choose random α1,α1,⋯,αn from S, Pr[p(α1,α2,⋯,αn)=0]≤∣S∣d. In the familiar univariate case p(X), this reduces to saying that a nonzero polynomial of degree d has at most d roots. The Schwartz-Zippel lemma is used in polynomial equality testing. Given two multi-variate polynomials p1(x1,⋯,xn) and p2(x1,⋯,xn) of degrees d1,d2 respectively, we can test if p1(α1,⋯,αn)−p2(α1,⋯,αn)=0 for random α1,⋯,αn←S, where the size of S is at least ∣S∣≥(d1+d2). If the two polynomials are identical, this will always be true, whereas if the two polynomials are different then the equality holds with probability at most ∣S∣max(d1,d2).","breadcrumbs":"Background Material » Polynomials » The Schwartz-Zippel lemma","id":"168","title":"The Schwartz-Zippel lemma"},"169":{"body":"Consider the order-n multiplicative subgroup H with primitive root of unity ω. For all ωi∈H,i∈[n−1], we have (ωi)n=(ωn)i=(ω0)i=1. In other words, every element of H fulfils the equation ZH(X)=Xn−1=(X−ω0)(X−ω1)(X−ω2)⋯(X−ωn−1), meaning every element is a root of ZH(X). We call ZH(X) the vanishing polynomial over H because it evaluates to zero on all elements of H. This comes in particularly handy when checking polynomial constraints. For instance, to check that A(X)+B(X)=C(X) over H, we simply have to check that A(X)+B(X)−C(X) is some multiple of ZH(X). In other words, if dividing our constraint by the vanishing polynomial still yields some polynomial ZH(X)A(X)+B(X)−C(X)=H(X), we are satisfied that A(X)+B(X)−C(X)=0 over H.","breadcrumbs":"Background Material » Polynomials » Vanishing polynomial","id":"169","title":"Vanishing polynomial"},"17":{"body":"The cost-model binary takes high-level parameters for a circuit design, and estimates the verification cost, as well as resulting proof size. Usage: cargo run --example cost-model -- [OPTIONS] k Positional arguments: k 2^K bound on the number of rows. Optional arguments: -h, --help Print this message. -a, --advice R[,R..] An advice column with the given rotations. May be repeated. -i, --instance R[,R..] An instance column with the given rotations. May be repeated. -f, --fixed R[,R..] A fixed column with the given rotations. May be repeated. -g, --gate-degree D Maximum degree of the custom gates. -l, --lookup N,I,T A lookup over N columns with max input degree I and max table degree T. May be repeated. -p, --permutation N A permutation over N columns. May be repeated. For example, to estimate the cost of a circuit with three advice columns and one fixed column (with various rotations), and a maximum gate degree of 4: > cargo run --example cost-model -- -a 0,1 -a 0 -a-0,-1,1 -f 0 -g 4 11 Finished dev [unoptimized + debuginfo] target(s) in 0.03s Running `target/debug/examples/cost-model -a 0,1 -a 0 -a 0,-1,1 -f 0 -g 4 11`\nCircuit { k: 11, max_deg: 4, advice_columns: 3, lookups: 0, permutations: [], column_queries: 7, point_sets: 3, estimator: Estimator,\n}\nProof size: 1440 bytes\nVerification: at least 81.689ms","breadcrumbs":"User Documentation » Developer tools » Cost estimator","id":"17","title":"Cost estimator"},"170":{"body":"TODO: explain what a basis is in general (briefly). Polynomials are commonly written in the monomial basis (e.g. X,X2,...Xn). However, when working over a multiplicative subgroup of order n, we find a more natural expression in the Lagrange basis. Consider the order-n multiplicative subgroup H with primitive root of unity ω. The Lagrange basis corresponding to this subgroup is a set of functions {Li}i=0n−1, where Li(ωj)={10if i=j,otherwise. We can write this more compactly as Li(ωj)=δij, where δ is the Kronecker delta function. Now, we can write our polynomial as a linear combination of Lagrange basis functions, A(X)=i=0∑n−1aiLi(X),X∈H, which is equivalent to saying that p(X) evaluates to a0 at ω0, to a1 at ω1, to a2 at ω2,⋯, and so on. When working over a multiplicative subgroup, the Lagrange basis function has a convenient sparse representation of the form Li(X)=X−ωici⋅(Xn−1), where ci is the barycentric weight. (To understand how this form was derived, refer to [3] .) For i=0, we have c=1/n⟹L0(X)=n1X−1(Xn−1). Suppose we are given a set of evaluation points {x0,x1,⋯,xn−1}. Since we cannot assume that the xi's form a multiplicative subgroup, we consider also the Lagrange polynomials Li's in the general case. Then we can construct: Li(X)=j=i∏xi−xjX−xj,i∈[0..n−1]. Here, every X=xj=xi will produce a zero numerator term (xj−xj), causing the whole product to evaluate to zero. On the other hand, X=xi will evaluate to xi−xjxi−xj at every term, resulting in an overall product of one. This gives the desired Kronecker delta behaviour Li(xj)=δij on the set {x0,x1,⋯,xn−1}.","breadcrumbs":"Background Material » Polynomials » Lagrange basis functions","id":"170","title":"Lagrange basis functions"},"171":{"body":"Given a polynomial in its evaluation representation A:{(x0,A(x0)),(x1,A(x1)),⋯,(xn−1,A(xn−1))}, we can reconstruct its coefficient form in the Lagrange basis: A(X)=i=0∑n−1A(xi)Li(X), where X∈{x0,x1,⋯,x1−n}.","breadcrumbs":"Background Material » Polynomials » Lagrange interpolation","id":"171","title":"Lagrange interpolation"},"172":{"body":"Dasgupta, S., Papadimitriou, C. H., & Vazirani, U. V. (2008). \"Algorithms\" (ch. 2). New York: McGraw-Hill Higher Education. Golin, M. (2016). \"The Fast Fourier Transform and Polynomial Multiplication\" [lecture notes], COMP 3711H Design and Analysis of Algorithms, Hong Kong University of Science and Technology. Berrut, J. and Trefethen, L. (2004). \"Barycentric Lagrange Interpolation.\"","breadcrumbs":"Background Material » Polynomials » References","id":"172","title":"References"},"173":{"body":"In the section Inverses and groups we introduced the concept of groups . A group has an identity and a group operation. In this section we will write groups additively, i.e. the identity is O and the group operation is +. Some groups can be used as cryptographic groups . At the risk of oversimplifying, this means that the problem of finding a discrete logarithm of a group element P to a given base G, i.e. finding x such that P=[x]G, is hard in general.","breadcrumbs":"Background Material » Cryptographic groups » Cryptographic groups","id":"173","title":"Cryptographic groups"},"174":{"body":"The Pedersen commitment [ P99 ] is a way to commit to a secret message in a verifiable way. It uses two random public generators G,H∈G, where G is a cryptographic group of order p. A random secret r is chosen in Zq, and the message to commit to m is from any subset of Zq. The commitment is c=Commit(m,r)=[m]G+[r]H. To open the commitment, the committer reveals m and r, thus allowing anyone to verify that c is indeed a commitment to m. Notice that the Pedersen commitment scheme is homomorphic: Commit(m,r)+Commit(m′,r′)=[m]G+[r]H+[m′]G+[r′]H=[m+m′]G+[r+r′]H=Commit(m+m′,r+r′). Assuming the discrete log assumption holds, Pedersen commitments are also perfectly hiding and computationally binding: hiding : the adversary chooses messages m0,m1. The committer commits to one of these messages c=Commit(mb;r),b∈{0,1}. Given c, the probability of the adversary guessing the correct b is no more than 21. binding : the adversary cannot pick two different messages m0=m1, and randomness r0,r1, such that Commit(m0,r0)=Commit(m1,r1).","breadcrumbs":"Background Material » Cryptographic groups » Pedersen commitment","id":"174","title":"Pedersen commitment"},"175":{"body":"We can use a variant of the Pedersen commitment scheme to commit to multiple messages at once, m=(m1,⋯,mn). This time, we'll have to sample a corresponding number of random public generators G=(G0,⋯,Gn−1), along with a single random generator H as before (for use in hiding). Then, our commitment scheme is: Commit(m;r)=Commit((m0,⋯,mn−1);r)=[r]H+[m0]G0+⋯+[mn−1]Gn−1=[r]H+i=0∑n−1[mi]Gi. TODO: is this positionally binding?","breadcrumbs":"Background Material » Cryptographic groups » Vector Pedersen commitment","id":"175","title":"Vector Pedersen commitment"},"176":{"body":"An example of a protocol that uses cryptographic groups is Diffie--Hellman key agreement [ DH1976 ]. The Diffie--Hellman protocol is a method for two users, Alice and Bob, to generate a shared private key. It proceeds as follows: Alice and Bob publicly agree on two prime numbers, p and G, where p is large and G is a primitive root (modp). (Note that g is a generator of the group Fp×.) Alice chooses a large random number a as her private key. She computes her public key A=[a]G(modp), and sends A to Bob. Similarly, Bob chooses a large random number b as his private key. He computes his public key B=[b]G(modp), and sends B to Alice. Now both Alice and Bob compute their shared key K=[ab]G(modp), which Alice computes as K=[a]B(modp)=[a]([b]G)(modp), and Bob computes as K=[b]A(modp)=[b]([a]G)(modp). A potential eavesdropper would need to derive K=[ab]g(modp) knowing only g,p,A=[a]G, and B=[b]G: in other words, they would need to either get the discrete logarithm a from A=[a]G or b from B=[b]G, which we assume to be computationally infeasible in Fp×. More generally, protocols that use similar ideas to Diffie--Hellman are used throughout cryptography. One way of instantiating a cryptographic group is as an elliptic curve . Before we go into detail on elliptic curves, we'll describe some algorithms that can be used for any group.","breadcrumbs":"Background Material » Cryptographic groups » Diffie--Hellman","id":"176","title":"Diffie--Hellman"},"177":{"body":"","breadcrumbs":"Background Material » Cryptographic groups » Multiscalar multiplication","id":"177","title":"Multiscalar multiplication"},"178":{"body":"Reference: https://jbootle.github.io/Misc/pippenger.pdf","breadcrumbs":"Background Material » Cryptographic groups » TODO: Pippenger's algorithm","id":"178","title":"TODO: Pippenger's algorithm"},"179":{"body":"Elliptic curves constructed over finite fields are another important cryptographic tool. We use elliptic curves because they provide a cryptographic group , i.e. a group in which the discrete logarithm problem (discussed below) is hard. There are several ways to define the curve equation, but for our purposes, let Fp be a large (255-bit) field, and then let the set of solutions (x,y) to y2=x3+b for some constant b define the Fp-rational points on an elliptic curve E(Fp). These (x,y) coordinates are called \"affine coordinates\". Each of the Fp-rational points, together with a \"point at infinity\" O that serves as the group identity, can be interpreted as an element of a group. By convention, elliptic curve groups are written additively. \"Three points on a line sum to zero, which is the point at infinity.\" The group addition law is simple: to add two points together, find the line that intersects both points and obtain the third point, and then negate its y-coordinate. The case that a point is being added to itself, called point doubling, requires special handling: we find the line tangent to the point, and then find the single other point that intersects this line and then negate. Otherwise, in the event that a point is being \"added\" to its negation, the result is the point at infinity. The ability to add and double points naturally gives us a way to scale them by integers, called scalars . The number of points on the curve is the group order. If this number is a prime q, then the scalars can be considered as elements of a scalar field , Fq. Elliptic curves, when properly designed, have an important security property. Given two random elements G,H∈E(Fp) finding a such that [a]G=H, otherwise known as the discrete log of H with respect to G, is considered computationally infeasible with classical computers. This is called the elliptic curve discrete log assumption. If an elliptic curve group G has prime order q (like the ones used in Halo 2), then it is a finite cyclic group. Recall from the section on groups that this implies it is isomorphic to Z/qZ, or equivalently, to the scalar field Fq. Each possible generator G fixes the isomorphism; then an element on the scalar side is precisely the discrete log of the corresponding group element with respect to G. In the case of a cryptographically secure elliptic curve, the isomorphism is hard to compute in the G→Fq direction because the elliptic curve discrete log problem is hard. It is sometimes helpful to make use of this isomorphism by thinking of group-based cryptographic protocols and algorithms in terms of the scalars instead of in terms of the group elements. This can make proofs and notation simpler. For instance, it has become common in papers on proof systems to use the notation [x] to denote a group element with discrete log x, where the generator is implicit. We also used this idea in the \" distinct-x theorem \", in order to prove correctness of optimizations for elliptic curve scalar multiplication in Sapling, and an endomorphism-based optimization in Appendix C of the original Halo paper .","breadcrumbs":"Background Material » Elliptic curves » Elliptic curves","id":"179","title":"Elliptic curves"},"18":{"body":"Let's start with a simple circuit, to introduce you to the common APIs and how they are used. The circuit will take a public input c, and will prove knowledge of two private inputs a and b such that a2⋅b2=c.","breadcrumbs":"User Documentation » A simple example » A simple example","id":"18","title":"A simple example"},"180":{"body":"","breadcrumbs":"Background Material » Elliptic curves » Curve arithmetic","id":"180","title":"Curve arithmetic"},"181":{"body":"The simplest situation is doubling a point (x0,y0). Continuing with our example y2=x3+b, this is done first by computing the derivative λ=dxdy=2y3x2. To obtain expressions for (x1,y1)=(x0,y0)+(x0,y0), we consider x1−x0−y1−y0=λ⟹−y1=λ(x1−x0)+y0⟹y1=λ(x0−x1)−y0. To get the expression for x1, we substitute y=λ(x0−x)−y0 into the elliptic curve equation: y2=x3+b⟹(λ(x0−x)−y0)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0)(x−x0)(x−x1)←(known roots x0,x0,x1)=x3−(x0+x0+x1)x2+⋯. Comparing coefficients for the x2 term gives us λ2=x0+x0+x1⟹x1=λ2−2x0.","breadcrumbs":"Background Material » Elliptic curves » Point doubling","id":"181","title":"Point doubling"},"182":{"body":"This unfortunately requires an expensive inversion of 2y. We can avoid this by arranging our equations to \"defer\" the computation of the inverse, since we often do not need the actual affine (x′,y′) coordinate of the resulting point immediately after an individual curve operation. Let's introduce a third coordinate Z and scale our curve equation by Z3 like so: Z3y2=Z3x3+Z3b Our original curve is just this curve at the restriction Z=1. If we allow the affine point (x,y) to be represented by X=xZ, Y=yZ and Z=0 then we have the homogenous projective curve Y2Z=X3+Z3b. Obtaining (x,y) from (X,Y,Z) is as simple as computing (X/Z,Y/Z) when Z=0. (When Z=0, we are dealing with the point at infinity O:=(0:1:0).) In this form, we now have a convenient way to defer the inversion required by doubling a point. The general strategy is to express x′,y′ as rational functions using x=X/Z and y=Y/Z, rearrange to make their denominators the same, and then take the resulting point (X,Y,Z) to have Z be the shared denominator and X=x′Z,Y=y′Z. Projective coordinates are often, but not always, more efficient than affine coordinates. There may be exceptions to this when either we have a different way to apply Montgomery's trick, or when we're in the circuit setting where multiplications and inversions are about equally as expensive (at least in terms of circuit size). The following shows an example of doubling a point (X,Y,Z)=(xZ,yZ,Z) without an inversion. Substituting with X,Y,Z gives us λ=2y3x2=2(Y/Z)3(X/Z)2=2YZ3X2 and gives us x′y′=λ2−2x=λ2−Z2X=4Y2Z29X4−Z2X=4Y2Z29X4−8XY2Z=8Y3Z318X4YZ−16XY3Z2=λ(x−x′)−y=λ(ZX−4Y2Z29X4−8XY2Z)−ZY=2YZ3X2(ZX−4Y2Z29X4−8XY2Z)−ZY=2YZ23X3−8Y3Z327X6−24X3Y2Z−ZY=8Y3Z312X3Y2Z−8Y4Z2−27X6+24X3Y2Z Notice how the denominators of x′ and y′ are the same. Thus, instead of computing (x′,y′) we can compute (X,Y,Z) with Z=8Y3Z3 and X,Y set to the corresponding numerators such that X/Z=x′ and Y/Z=y′. This completely avoids the need to perform an inversion when doubling, and something analogous to this can be done when adding two distinct points.","breadcrumbs":"Background Material » Elliptic curves » Projective coordinates","id":"182","title":"Projective coordinates"},"183":{"body":"We now add two points with distinct x-coordinates, P=(x0,y0) and Q=(x1,y1), where x0=x1, to obtain R=P+Q=(x2,y2). The line PQ has slope λ=fracy1−y0x1−x0⟹y−y0=λ⋅(x−x0). Using the expression for PQ, we compute y-coordinate −y2 of −R as: −y2−y0=λ⋅(x2−x0)⟹y2=(x0−x2)−y0. Plugging the expression for PQ into the curve equation y2=x3+b yields y2=x3+b⟹(λ⋅(x−x0)+y0)2=x3+b⟹x3−λ2x2+⋯=0←(rearranging terms)=(x−x0)(x−x1)(x−x2)←(known roots x0,x1,x2)=x3−(x0+x1+x2)x2+⋯. Comparing coefficients for the x2 term gives us λ2=x0+x1+x2⟹x2=λ2−x0−x1. Important notes: There exist efficient formulae [1] for point addition that do not have edge cases (so-called \"complete\" formulae) and that unify the addition and doubling cases together. The result of adding a point to its negation using those formulae produces Z=0, which represents the point at infinity. In addition, there are other models like the Jacobian representation where (x,y)=(xZ2,yZ3,Z) where the curve is rescaled by Z6 instead of Z3, and this representation has even more efficient arithmetic but no unified/complete formulae. We can easily compare two curve points (X1,Y1,Z1) and (X2,Y2,Z2) for equality in the homogenous projective coordinate space by \"homogenizing\" their Z-coordinates; the checks become X1Z2=X2Z1 and Y1Z2=Y2Z1.","breadcrumbs":"Background Material » Elliptic curves » Point addition","id":"183","title":"Point addition"},"184":{"body":"Imagine that Fp has a primitive cube root of unity, or in other words that 3∣p−1 and so an element ζp generates a 3-order multiplicative subgroup. Notice that a point (x,y) on our example elliptic curve y2=x3+b has two cousin points: (ζpx,ζp2x), because the computation x3 effectively kills the ζ component of the x-coordinate. Applying the map (x,y)↦(ζpx,y) is an application of an endomorphism over the curve. The exact mechanics involved are complicated, but when the curve has a prime q number of points (and thus a prime \"order\") the effect of the endomorphism is to multiply the point by a scalar in Fq which is also a primitive cube root ζq in the scalar field.","breadcrumbs":"Background Material » Elliptic curves » Curve endomorphisms","id":"184","title":"Curve endomorphisms"},"185":{"body":"Given a point on the curve P=(x,y), we know that its negation −P=(x,−y) is also on the curve. To uniquely specify a point, we need only encode its x-coordinate along with the sign of its y-coordinate.","breadcrumbs":"Background Material » Elliptic curves » Curve point compression","id":"185","title":"Curve point compression"},"186":{"body":"As mentioned in the Fields section, we can interpret the least significant bit of a field element as its \"sign\", since its additive inverse will always have the opposite LSB. So we record the LSB of the y-coordinate as sign. Pallas and Vesta are defined over the Fp and Fq fields, which elements can be expressed in 255 bits. This conveniently leaves one unused bit in a 32-byte representation. We pack the y-coordinate sign bit into the highest bit in the representation of the x-coordinate: <----------------------------------- x --------------------------------->\nEnc(P) = [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ _] ... [_ _ _ _ _ _ _ _] [_ _ _ _ _ _ _ sign] ^ <-------------------------------------> ^ LSB 30 bytes MSB The \"point at infinity\" O that serves as the group identity, does not have an affine (x,y) representation. However, it turns out that there are no points on either the Pallas or Vesta curve with x=0 or y=0. We therefore use the \"fake\" affine coordinates (0,0) to encode O, which results in the all-zeroes 32-byte array.","breadcrumbs":"Background Material » Elliptic curves » Serialization","id":"186","title":"Serialization"},"187":{"body":"When deserializing a compressed curve point, we first read the most significant bit as ysign, the sign of the y-coordinate. Then, we set this bit to zero to recover the original x-coordinate. If x=0,y=0, we return the \"point at infinity\" O. Otherwise, we proceed to compute y=x3+b. Here, we read the least significant bit of y as sign. If sign == ysign, we already have the correct sign and simply return the curve point (x,y). Otherwise, we negate y and return (x,−y).","breadcrumbs":"Background Material » Elliptic curves » Deserialization","id":"187","title":"Deserialization"},"188":{"body":"Let Ep be an elliptic curve over a finite field Fp, where p is a prime. We denote this by Ep/Fp. and we denote the group of points of Ep over Fp, with order q=#E(Fp). For this curve, we call Fp the \"base field\" and Fq the \"scalar field\". We instantiate our proof system over the elliptic curve Ep/Fp. This allows us to prove statements about Fq-arithmetic circuit satisfiability. (aside) If our curve Ep is over Fp, why is the arithmetic circuit instead in Fq? The proof system is basically working on encodings of the scalars in the circuit (or more precisely, commitments to polynomials whose coefficients are scalars). The scalars are in Fq when their encodings/commitments are elliptic curve points in Ep/Fp. However, most of the verifier's arithmetic computations are over the base field Fp, and are thus efficiently expressed as an Fp-arithmetic circuit. (aside) Why are the verifier's computations (mainly) over Fp? The Halo 2 verifier actually has to perform group operations using information output by the circuit. Group operations like point doubling and addition use arithmetic in Fp, because the coordinates of points are in Fp. This motivates us to construct another curve with scalar field Fp, which has an Fp-arithmetic circuit that can efficiently verify proofs from the first curve. As a bonus, if this second curve had base field Eq/Fq, it would generate proofs that could be efficiently verified in the first curve's Fq-arithmetic circuit. In other words, we instantiate a second proof system over Eq/Fq, forming a 2-cycle with the first:","breadcrumbs":"Background Material » Elliptic curves » Cycles of curves","id":"188","title":"Cycles of curves"},"189":{"body":"Reference: https://github.com/zcash/pasta","breadcrumbs":"Background Material » Elliptic curves » TODO: Pallas-Vesta curves","id":"189","title":"TODO: Pallas-Vesta curves"},"19":{"body":"Firstly, we need to define the instructions that our circuit will rely on. Instructions are the boundary between high-level gadgets and the low-level circuit operations. Instructions may be as coarse or as granular as desired, but in practice you want to strike a balance between an instruction being large enough to effectively optimize its implementation, and small enough that it is meaningfully reusable. For our circuit, we will use three instructions: Load a private number into the circuit. Multiply two numbers. Expose a number as a public input to the circuit. We also need a type for a variable representing a number. Instruction interfaces provide associated types for their inputs and outputs, to allow the implementations to represent these in a way that makes the most sense for their optimization goals. trait NumericInstructions: Chip { /// Variable representing a number. type Num; /// Loads a number into the circuit as a private input. fn load_private(&self, layouter: impl Layouter, a: Option) -> Result; /// Loads a number into the circuit as a fixed constant. fn load_constant(&self, layouter: impl Layouter, constant: F) -> Result; /// Returns `c = a * b`. fn mul( &self, layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result; /// Exposes a number as a public input to the circuit. fn expose_public( &self, layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error>;\n}","breadcrumbs":"User Documentation » A simple example » Define instructions","id":"19","title":"Define instructions"},"190":{"body":"Sometimes it is useful to be able to produce a random point on an elliptic curve Ep/Fp corresponding to some input, in such a way that no-one will know its discrete logarithm (to any other base). This is described in detail in the Internet draft on Hashing to Elliptic Curves . Several algorithms can be used depending on efficiency and security requirements. The framework used in the Internet Draft makes use of several functions: hash_to_field: takes a byte sequence input and maps it to a element in the base field Fp map_to_curve: takes an Fp element and maps it to Ep.","breadcrumbs":"Background Material » Elliptic curves » Hashing to curves","id":"190","title":"Hashing to curves"},"191":{"body":"Reference: https://eprint.iacr.org/2019/403.pdf","breadcrumbs":"Background Material » Elliptic curves » TODO: Simplified SWU","id":"191","title":"TODO: Simplified SWU"},"192":{"body":"Renes, J., Costello, C., & Batina, L. (2016, May). \"Complete addition formulas for prime order elliptic curves.\" In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 403-428). Springer, Berlin, Heidelberg.","breadcrumbs":"Background Material » Elliptic curves » References","id":"192","title":"References"},"193":{"body":"We want to commit to some polynomial p(X)∈Fp[X], and be able to provably evaluate the committed polynomial at arbitrary points. The naive solution would be for the prover to simply send the polynomial's coefficients to the verifier: however, this requires O(n) communication. Our polynomial commitment scheme gets the job done using O(logn) communication.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Polynomial commitment using inner product argument","id":"193","title":"Polynomial commitment using inner product argument"},"194":{"body":"Given a parameter d=2k, we generate the common reference string σ=(G,G,H,Fp) defining certain constants for this scheme: G is a group of prime order p; G∈Gd is a vector of d random group elements; H∈G is a random group element; and Fp is the finite field of order p.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Setup","id":"194","title":"Setup"},"195":{"body":"The Pedersen vector commitment Commit is defined as Commit(σ,p(X);r)=⟨a,G⟩+[r]H, for some polynomial p(X)∈Fp[X] and some blinding factor r∈Fp. Here, each element of the vector ai∈Fp is the coefficient for the ith degree term of p(X), and p(X) is of maximal degree d−1.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Commit","id":"195","title":"Commit"},"196":{"body":"The modified inner product argument is an argument of knowledge for the relation {((P,x,v);(a,r)):P=⟨a,G⟩+[r]H,v=⟨a,b⟩}, where b=(1,x,x2,⋯,xd−1) is composed of increasing powers of the evaluation point x. This allows a prover to demonstrate to a verifier that the polynomial contained “inside” the commitment P evaluates to v at x, and moreover, that the committed polynomial has maximum degree d−1. The inner product argument proceeds in k=log2d rounds. For our purposes, it is sufficient to know about its final outputs, while merely providing intuition about the intermediate rounds. (Refer to Section 3 in the Halo paper for a full explanation.) Before beginning the argument, the verifier selects a random group element U and sends it to the prover. We initialize the argument at round k, with the vectors a(k):=a, G(k):=G and b(k):=b. In each round j=k,k−1,⋯,1: the prover computes two values Lj and Rj by taking some inner product of a(j) with G(j) and b(j). Note that are in some sense \"cross-terms\": the lower half of a is used with the higher half of G and b, and vice versa: LjRj=⟨alo(j),Ghi(j)⟩+[lj]H+[⟨alo(j),bhi(j)⟩]U=⟨ahi(j),Glo(j)⟩+[lj]H+[⟨ahi(j),blo(j)⟩]U the verifier issues a random challenge uj; the prover uses uj to compress the lower and higher halves of a(j), thus producing a new vector of half the original length a(j−1)=ahi(j)⋅uj−1+alo(j)⋅uj. The vectors G(j) and b(j) are similarly compressed to give G(j−1) and b(j−1). a(j−1), G(j−1) and b(j−1) are input to the next round j−1. Note that at the end of the last round j=1, we are left with a:=a(0), G:=G(0), b:=b(0), each of length 1. The intuition is that these final scalars, together with the challenges {uj} and \"cross-terms\" {Lj,Rj} from each round, encode the compression in each round. Since the prover did not know the challenges U,{uj} in advance, they would have been unable to manipulate the round compressions. Thus, checking a constraint on these final terms should enforce that the compression had been performed correctly, and that the original a satisfied the relation before undergoing compression. Note that G,b are simply rearrangements of the publicly known G,b, with the round challenges {uj} mixed in: this means the verifier can compute G,b independently and verify that the prover had provided those same values.","breadcrumbs":"Background Material » Polynomial commitment using inner product argument » Open (prover) and OpenVerify (verifier)","id":"196","title":"Open (prover) and OpenVerify (verifier)"},"197":{"body":"Alternative terms: Induction; Accumulation scheme; Proof-carrying data However, the computation of G requires a length-2k multiexponentiation ⟨G,s⟩, where s is composed of the round challenges u1,⋯,uk arranged in a binary counting structure. This is the linear-time computation that we want to amortise across a batch of proof instances. Instead of computing G, notice that we can express G as a commitment to a polynomial G=Commit(σ,g(X,u1,⋯,uk)), where g(X,u1,⋯,uk):=∏i=1k(ui+ui−1X2i−1) is a polynomial with degree 2k−1. Since G is a commitment, it can be checked in an inner product argument. The verifier circuit witnesses G and brings G,u1,⋯,uk out as public inputs to the proof π. The next verifier instance checks π using the inner product argument; this includes checking that G=Commit(g(X,u1,⋯,uk)) evaluates at some random point to the expected value for the given challenges u1,⋯,uk. Recall from the previous section that this check only requires logd work. At the end of checking π and G, the circuit is left with a new G′, along with the u1′,⋯,uk′ challenges sampled for the check. To fully accept π as valid, we should perform a linear-time computation of G′=⟨G,s′⟩. Once again, we delay this computation by witnessing G′ and bringing G,u1,⋯,uk out as public inputs to the proof π. This goes on from one proof instance to the next, until we are satisfied with the size of our batch of proofs. We finally perform a single linear-time computation, thus deciding the validity of the whole batch. We recall from the section Cycles of curves that we can instantiate this protocol over a two-cycle, where a proof produced by one curve is efficiently verified in the circuit of the other curve. However, some of these verifier checks can actually be efficiently performed in the native circuit; these are \"deferred\" to the next native circuit (see diagram below) instead of being immediately passed over to the other curve.","breadcrumbs":"Background Material » Recursion » Recursion","id":"197","title":"Recursion"},"2":{"body":"Requires Rust 1.56.1 or higher. Minimum supported Rust version can be changed in the future, but it will be done with a minor version bump.","breadcrumbs":"halo2 » Minimum Supported Rust Version","id":"2","title":"Minimum Supported Rust Version"},"20":{"body":"For our circuit, we will build a chip that provides the above numeric instructions for a finite field. /// The chip that will implement our instructions! Chips store their own\n/// config, as well as type markers if necessary.\nstruct FieldChip { config: FieldConfig, _marker: PhantomData,\n} Every chip needs to implement the Chip trait. This defines the properties of the chip that a Layouter may rely on when synthesizing a circuit, as well as enabling any initial state that the chip requires to be loaded into the circuit. impl Chip for FieldChip { type Config = FieldConfig; type Loaded = (); fn config(&self) -> &Self::Config { &self.config } fn loaded(&self) -> &Self::Loaded { &() }\n}","breadcrumbs":"User Documentation » A simple example » Define a chip implementation","id":"20","title":"Define a chip implementation"},"21":{"body":"The chip needs to be configured with the columns, permutations, and gates that will be required to implement all of the desired instructions. /// Chip state is stored in a config struct. This is generated by the chip\n/// during configuration, and then stored inside the chip.\n#[derive(Clone, Debug)]\nstruct FieldConfig { /// For this chip, we will use two advice columns to implement our instructions. /// These are also the columns through which we communicate with other parts of /// the circuit. advice: [Column; 2], /// This is the public input (instance) column. instance: Column, // We need a selector to enable the multiplication gate, so that we aren't placing // any constraints on cells where `NumericInstructions::mul` is not being used. // This is important when building larger circuits, where columns are used by // multiple sets of instructions. s_mul: Selector,\n} impl FieldChip { fn construct(config: >::Config) -> Self { Self { config, _marker: PhantomData, } } fn configure( meta: &mut ConstraintSystem, advice: [Column; 2], instance: Column, constant: Column, ) -> >::Config { meta.enable_equality(instance); meta.enable_constant(constant); for column in &advice { meta.enable_equality(*column); } let s_mul = meta.selector(); // Define our multiplication gate! meta.create_gate(\"mul\", |meta| { // To implement multiplication, we need three advice cells and a selector // cell. We arrange them like so: // // | a0 | a1 | s_mul | // |-----|-----|-------| // | lhs | rhs | s_mul | // | out | | | // // Gates may refer to any relative offsets we want, but each distinct // offset adds a cost to the proof. The most common offsets are 0 (the // current row), 1 (the next row), and -1 (the previous row), for which // `Rotation` has specific constructors. let lhs = meta.query_advice(advice[0], Rotation::cur()); let rhs = meta.query_advice(advice[1], Rotation::cur()); let out = meta.query_advice(advice[0], Rotation::next()); let s_mul = meta.query_selector(s_mul); // Finally, we return the polynomial expressions that constrain this gate. // For our multiplication gate, we only need a single polynomial constraint. // // The polynomial expressions returned from `create_gate` will be // constrained by the proving system to equal zero. Our expression // has the following properties: // - When s_mul = 0, any value is allowed in lhs, rhs, and out. // - When s_mul != 0, this constrains lhs * rhs = out. vec![s_mul * (lhs * rhs - out)] }); FieldConfig { advice, instance, s_mul, } }\n}","breadcrumbs":"User Documentation » A simple example » Configure the chip","id":"21","title":"Configure the chip"},"22":{"body":"/// A variable representing a number.\n#[derive(Clone)]\nstruct Number(AssignedCell); impl NumericInstructions for FieldChip { type Num = Number; fn load_private( &self, mut layouter: impl Layouter, value: Option, ) -> Result { let config = self.config(); layouter.assign_region( || \"load private\", |mut region| { region .assign_advice( || \"private input\", config.advice[0], 0, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn load_constant( &self, mut layouter: impl Layouter, constant: F, ) -> Result { let config = self.config(); layouter.assign_region( || \"load constant\", |mut region| { region .assign_advice_from_constant(|| \"constant value\", config.advice[0], 0, constant) .map(Number) }, ) } fn mul( &self, mut layouter: impl Layouter, a: Self::Num, b: Self::Num, ) -> Result { let config = self.config(); layouter.assign_region( || \"mul\", |mut region: Region<'_, F>| { // We only want to use a single multiplication gate in this region, // so we enable it at region offset 0; this means it will constrain // cells at offsets 0 and 1. config.s_mul.enable(&mut region, 0)?; // The inputs we've been given could be located anywhere in the circuit, // but we can only rely on relative offsets inside this region. So we // assign new cells inside the region and constrain them to have the // same values as the inputs. a.0.copy_advice(|| \"lhs\", &mut region, config.advice[0], 0)?; b.0.copy_advice(|| \"rhs\", &mut region, config.advice[1], 0)?; // Now we can assign the multiplication result, which is to be assigned // into the output position. let value = a.0.value().and_then(|a| b.0.value().map(|b| *a * *b)); // Finally, we do the assignment to the output, returning a // variable to be used in another part of the circuit. region .assign_advice( || \"lhs * rhs\", config.advice[0], 1, || value.ok_or(Error::Synthesis), ) .map(Number) }, ) } fn expose_public( &self, mut layouter: impl Layouter, num: Self::Num, row: usize, ) -> Result<(), Error> { let config = self.config(); layouter.constrain_instance(num.0.cell(), config.instance, row) }\n}","breadcrumbs":"User Documentation » A simple example » Implement chip traits","id":"22","title":"Implement chip traits"},"23":{"body":"Now that we have the instructions we need, and a chip that implements them, we can finally build our circuit! /// The full circuit implementation.\n///\n/// In this struct we store the private input variables. We use `Option` because\n/// they won't have any value during key generation. During proving, if any of these\n/// were `None` we would get an error.\n#[derive(Default)]\nstruct MyCircuit { constant: F, a: Option, b: Option,\n} impl Circuit for MyCircuit { // Since we are using a single chip for everything, we can just reuse its config. type Config = FieldConfig; type FloorPlanner = SimpleFloorPlanner; fn without_witnesses(&self) -> Self { Self::default() } fn configure(meta: &mut ConstraintSystem) -> Self::Config { // We create the two advice columns that FieldChip uses for I/O. let advice = [meta.advice_column(), meta.advice_column()]; // We also need an instance column to store public inputs. let instance = meta.instance_column(); // Create a fixed column to load constants. let constant = meta.fixed_column(); FieldChip::configure(meta, advice, instance, constant) } fn synthesize( &self, config: Self::Config, mut layouter: impl Layouter, ) -> Result<(), Error> { let field_chip = FieldChip::::construct(config); // Load our private values into the circuit. let a = field_chip.load_private(layouter.namespace(|| \"load a\"), self.a)?; let b = field_chip.load_private(layouter.namespace(|| \"load b\"), self.b)?; // Load the constant factor into the circuit. let constant = field_chip.load_constant(layouter.namespace(|| \"load constant\"), self.constant)?; // We only have access to plain multiplication. // We could implement our circuit as: // asq = a*a // bsq = b*b // absq = asq*bsq // c = constant*asq*bsq // // but it's more efficient to implement it as: // ab = a*b // absq = ab^2 // c = constant*absq let ab = field_chip.mul(layouter.namespace(|| \"a * b\"), a, b)?; let absq = field_chip.mul(layouter.namespace(|| \"ab * ab\"), ab.clone(), ab)?; let c = field_chip.mul(layouter.namespace(|| \"constant * absq\"), constant, absq)?; // Expose the result as a public input to the circuit. field_chip.expose_public(layouter.namespace(|| \"expose c\"), c, 0) }\n}","breadcrumbs":"User Documentation » A simple example » Build the circuit","id":"23","title":"Build the circuit"},"24":{"body":"halo2_proofs::dev::MockProver can be used to test that the circuit is working correctly. The private and public inputs to the circuit are constructed as we will do to create a proof, but by passing them to MockProver::run we get an object that can test every constraint in the circuit, and tell us exactly what is failing (if anything). // The number of rows in our circuit cannot exceed 2^k. Since our example // circuit is very small, we can pick a very small value here. let k = 4; // Prepare the private and public inputs to the circuit! let constant = Fp::from(7); let a = Fp::from(2); let b = Fp::from(3); let c = constant * a.square() * b.square(); // Instantiate the circuit with the private inputs. let circuit = MyCircuit { constant, a: Some(a), b: Some(b), }; // Arrange the public input. We expose the multiplication result in row 0 // of the instance column, so we position it there in our public inputs. let mut public_inputs = vec![c]; // Given the correct public input, our circuit will verify. let prover = MockProver::run(k, &circuit, vec![public_inputs.clone()]).unwrap(); assert_eq!(prover.verify(), Ok(())); // If we try some other public input, the proof will fail! public_inputs[0] += Fp::one(); let prover = MockProver::run(k, &circuit, vec![public_inputs]).unwrap(); assert!(prover.verify().is_err());","breadcrumbs":"User Documentation » A simple example » Testing the circuit","id":"24","title":"Testing the circuit"},"25":{"body":"You can find the source code for this example here .","breadcrumbs":"User Documentation » A simple example » Full example","id":"25","title":"Full example"},"26":{"body":"In normal programs, you can trade memory for CPU to improve performance, by pre-computing and storing lookup tables for some part of the computation. We can do the same thing in halo2 circuits! A lookup table can be thought of as enforcing a relation between variables, where the relation is expressed as a table. Assuming we have only one lookup argument in our constraint system, the total size of tables is constrained by the size of the circuit: each table entry costs one row, and it also costs one row to do each lookup. TODO","breadcrumbs":"User Documentation » Lookup tables » Lookup tables","id":"26","title":"Lookup tables"},"27":{"body":"","breadcrumbs":"User Documentation » Gadgets » Gadgets","id":"27","title":"Gadgets"},"28":{"body":"This section contains various ideas and snippets that you might find useful while writing halo2 circuits.","breadcrumbs":"User Documentation » Tips and tricks » Tips and tricks","id":"28","title":"Tips and tricks"},"29":{"body":"A common constraint used in R1CS circuits is the boolean constraint: b∗(1−b)=0. This constraint can only be satisfied by b=0 or b=1. In halo2 circuits, you can similarly constrain a cell to have one of a small set of values. For example, to constrain a to the range [0..5], you would create a gate of the form: a⋅(1−a)⋅(2−a)⋅(3−a)⋅(4−a)=0 while to constraint c to be either 7 or 13, you would use: (7−c)⋅(13−c)=0 The underlying principle here is that we create a polynomial constraint with roots at each value in the set of possible values we want to allow. In R1CS circuits, the maximum supported polynomial degree is 2 (due to all constraints being of the form a∗b=c). In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that higher-degree constraints are more expensive to use. Note that the roots don't have to be constants; for example (a−x)⋅(a−y)⋅(a−z)=0 will constrain a to be equal to one of {x,y,z} where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.","breadcrumbs":"User Documentation » Tips and tricks » Small range constraints","id":"29","title":"Small range constraints"},"3":{"body":"halo2 currently uses rayon for parallel computation. The RAYON_NUM_THREADS environment variable can be used to set the number of threads.","breadcrumbs":"halo2 » Controlling parallelism","id":"3","title":"Controlling parallelism"},"30":{"body":"We can use Lagrange interpolation to create a polynomial constraint that maps f(X)=Y for small sets of X∈{xi},Y∈{yi}. For instance, say we want to map a 2-bit value to a \"spread\" version interleaved with zeros. We first precompute the evaluations at each point: 00→000001→000110→010011→0101⟹⟹⟹⟹0→01→12→43→5 Then, we construct the Lagrange basis polynomial for each point using the identity: lj(X)=0≤m B\n^ +\n| |\n+ v\nD <---+ C E +---> F ^ + | | + v H <---+ G After adding constraint B≡E the above algorithm produces the cycle: A +---> B +-------------+\n^ |\n| |\n+ v\nD <---+ C <---+ E F ^ + | | + v H <---+ G","breadcrumbs":"Design » Proving system » Permutation argument » Algorithm","id":"46","title":"Algorithm"},"47":{"body":"If we did not check whether left and right were already in the same cycle, then we could end up undoing an equality constraint. For example, if we have the following constraints: a≡b b≡c c≡d b≡d and we tried to implement adding an equality constraint just using step 5 of the above algorithm, then we would end up constructing the cycle (a b) (c d), rather than the correct (a b c d).","breadcrumbs":"Design » Proving system » Permutation argument » Broken alternatives","id":"47","title":"Broken alternatives"},"48":{"body":"We need to check a permutation of cells in m columns, represented in Lagrange basis by polynomials v0,…,vm−1. We will label each cell in those m columns with a unique element of F×. Suppose that we have a permutation on these labels, σ(column:i,row:j)=(column:i′,row:j′). in which the cycles correspond to equality-constraint sets. If we consider the set of pairs {(label,value)}, then the values within each cycle are equal if and only if permuting the label in each pair by σ yields the same set: An example for a cycle (A B C D). The set before permuting the labels is {(A, 7), (B, 7), (C, 7), (D, 7)}, and the set after is {(D, 7), (A, 7), (B, 7), (C, 7)} which is the same. If one of the 7s is replaced by 3, then the set after permuting labels is not the same. Since the labels are distinct, set equality is the same as multiset equality, which we can check using a product argument. Let ω be a 2k root of unity and let δ be a T root of unity, where T⋅2S+1=p with T odd and k≤S. We will use δi⋅ωj∈F× as the label for the cell in the jth row of the ith column of the permutation argument. We represent σ by a vector of m polynomials si(X) such that si(ωj)=δi′⋅ωj′. Notice that the identity permutation can be represented by the vector of m polynomials IDi(ωj) such that IDi(ωj)=δi⋅ωj. We will use a challenge β to compress each (label,value) pair to value+β⋅label. Just as in the product argument we used for lookups , we also use a challenge γ to randomize each term of the product. Now given our permutation represented by s0,…,sm−1 over columns represented by v0,…,vm−1, we want to ensure that: i=0∏m−1j=0∏n−1(vi(ωj)+β⋅si(ωj)+γvi(ωj)+β⋅δi⋅ωj+γ)=1 Here vi(ωj)+β⋅δi⋅ωj represents the unpermuted (label,value) pair, and vi(ωj)+β⋅si(ωj) represents the permuted (σ(label),value) pair. Let ZP be such that ZP(ω0)=ZP(ωn)=1 and for 0≤j=r(λ) verifier challenges and let the ith challenge be sampled from Chi. The advantage metric AdvIPSRS(P,λ)=Pr[SRSPIP(λ)] of a state restoration prover P is defined with respect to the following game. GameSRSIPP(λ):win←false;tr←ϵpp←IP.Setup(1λ)(x,stP)←Pλ(pp)RunPλOSRS(stP)Return winOracleOSRS(τ=(a1,c1,...,ai−1,ci−1),ai):Ifτ∈trthenIfi≤rthenci←Chi;tr←tr∣∣(τ,ai,ci);ReturnciElse ifi=r+1thend←IP.V(pp,x,(τ,ai));tr←(τ,ai)Ifd=1then win←trueReturndReturn⊥ As shown in [GT20] (Theorem 1) state restoration soundness is tightly related to soundness after applying the Fiat-Shamir transformation. Knowledge Soundness We will show that our protocol satisfies a strengthened notion of knowledge soundness known as witness extended emulation . Informally, this notion states that for any successful prover algorithm there exists an efficient emulator that can extract a witness from it by rewinding it and supplying it with fresh randomness. However, we must slightly adjust our definition of witness extended emulation to account for the fact that our provers are state restoration provers and can rewind the verifier. Further, to avoid the need for rewinding the state restoration prover during witness extraction we study our protocol in the algebraic group model. Algebraic Group Model (AGM). An adversary Palg is said to be algebraic if whenever it outputs a group element X it also outputs a representation x∈Fn such that ⟨x,G⟩=X where G∈Gn is the vector of group elements that Palg has seen so far. Notationally, we write {X} to describe a group element X enhanced with this representation. We also write {X}iG to identify the component of the representation of X that corresponds with Gi. In other words, X=i=0∑n−1[{X}iG]Gi The algebraic group model allows us to perform so-called \"online\" extraction for some protocols: the extractor can obtain the witness from the representations themselves for a single (accepting) transcript. State Restoration Witness Extended Emulation Let IP be an interactive argument for relation R with r=r(λ) challenges. We define for all non-uniform algebraic provers Palg, extractors E, and computationally unbounded distinguishers D the advantage metric AdvIP,Rsr-wee(Palg,D,E,λ)=Pr[WEE-realIP,RP,D(λ)]−Pr[WEE-idealIP,RE,P,D(λ)] is defined with the respect to the following games. 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A public coin interactive argument (Setup,P,V) has perfect special honest-verifier zero knowledge (PSHVZK) if for all polynomial-time decidable relations R and for all (x,w)∈R and for all non-uniform polynomial-time adversaries A1,A2 there exists a probabilistic polynomial-time simulator S such that =Pr⎣⎡A1(σ,x,tr)=1∣∣pp←Setup(1λ);(x,w,ρ)←A2(pp);tr←⟨P(pp,x,w),V(pp,x,ρ)⟩⎦⎤Pr⎣⎡A1(σ,x,tr)=1∣∣pp←Setup(1λ);(x,w,ρ)←A2(pp);tr←S(pp,x,ρ)⎦⎤ where ρ is the internal randomness of the verifier. In this (common) definition of zero-knowledge the verifier is expected to act \"honestly\" and send challenges that correspond only with their internal randomness; they cannot adaptively respond to the prover based on the prover's messages. 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