From eede6aa4474a0c7db432af34408cf914eef645ea Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 14 Oct 2026 15:11:18 +0000 Subject: [PATCH] Commit the Miller value only when the degree needs it After the polynomial identity the weight is set by bytes, and most of the witness was the 88 committed Miller values. A step without a commitment drops its main relation: f' is the product itself, and the next committed relation checks both steps. schedule(cap) skips a commitment whenever S stays within the degree cap; 211 gives the fewest words: 35 Miller values and a quotient of 200 words, against 88 and 50. Q(z) is evaluated eight coefficients at a time by one shared function, so the larger quotient adds no unrolled code. Regtest spend: 107.4 kWU -> 90.8 kWU; cost 81.7 M -> 68.5 M mWU. Co-Authored-By: Claude Opus 5.5 --- README.md | 8 +- docs/benchmarks.md | 74 ++++---- docs/design.md | 14 +- examples/README.md | 4 +- examples/multiplier/leaves.json | 2 +- examples/multiplier/proof.json | 16 +- examples/multiplier/public.json | 2 +- examples/multiplier/verifier.simpl.gz | Bin 243628 -> 242352 bytes examples/multiplier/verifier.uprog | Bin 175836 -> 174759 bytes examples/multiplier/verifier.w0.json.gz | Bin 56117 -> 37009 bytes groth/pairing.py | 216 +++++++++++++++--------- tests/test_pairing.py | 5 +- 12 files changed, 206 insertions(+), 135 deletions(-) diff --git a/README.md b/README.md index 8d1ecae..6776668 100644 --- a/README.md +++ b/README.md @@ -25,10 +25,10 @@ Simplicity has no pairing jets, so we applied a bunch of tricks to fit Groth16 v | | | |---|---| -| program cost, one public signal | 81.7 M mWU | -| transaction | 107.4 kWU, 26.9 kvB | -| pruned program / witness | 61 kB / 45 kB | -| fee at 0.115 sat/vB | ~ 3,100 sat | +| program cost, one public signal | 68.5 M mWU | +| transaction | 90.8 kWU, 22.7 kvB | +| pruned program / witness | 60 kB / 30 kB | +| fee at 0.115 sat/vB | ~ 2,600 sat | The program's bytes earn more budget than it costs, so it needs no padding. The transaction is about a quarter of the 400 kWU standard limit. diff --git a/docs/benchmarks.md b/docs/benchmarks.md index 393e6e3..b7c28d8 100644 --- a/docs/benchmarks.md +++ b/docs/benchmarks.md @@ -4,30 +4,43 @@ All costs are Simplicity's static cost bound in milliweight units (mWU), as rust ## One spend -The toy circuit's verifier (one public signal bound to the transaction), spent on an Elements 23.3.1 regtest node in `9f4c1cb0…`: +The toy circuit's verifier (one public signal bound to the transaction), spent on an Elements 23.3.1 regtest node in `511a3da3…`: | | | |---|---| -| program cost | 81,743,632 mWU | -| budget: (witness stack bytes + 50) x 1000 | 106,812,000 mWU, no padding needed | -| pruned program | 61,411 bytes | -| witness (proof, Miller values, slopes, quotient, Merkle sibling) | 45,281 bytes | -| transaction | 107,423 WU, 26,856 vB | -| fee at 0.115 sat/vB, which Liquid testnet accepts | about 3,100 sat | +| program cost | 68,491,902 mWU | +| budget: (witness stack bytes + 50) x 1000 | 90,195,000 mWU, no padding needed | +| pruned program | 60,346 bytes | +| witness (proof, 35 Miller values, slopes, quotient, Merkle sibling) | 29,729 bytes | +| transaction | 90,804 WU, 22,701 vB | +| fee at 0.115 sat/vB, which Liquid testnet accepts | about 2,600 sat | ## Where the cost goes | part | mWU | share | |---|---:|---:| -| the loop: 88 steps, each a few relations at z | 66,612,776 | 81.5% | -| MSM of the public input: 51 windows of 5 bits | 8,645,785 | 10.6% | -| setup: on-curve checks, powers of z, c(z), the Frobenius images, the transcript header | 3,166,661 | 3.9% | -| reading the witness, passing it between the parts | 2,144,910 | 2.6% | -| end: the last relations, Q(z), the two hashes | 1,147,204 | 1.4% | +| the loop: 88 steps, each a few relations at z | 51,949,542 | 75.8% | +| MSM of the public input: 51 windows of 5 bits | 8,645,785 | 12.6% | +| end: the last relations, Q(z) (200 coefficients), the two hashes | 3,305,204 | 4.8% | +| setup: on-curve checks, powers of z, c(z), the Frobenius images, the transcript header | 3,166,661 | 4.6% | +| reading the witness, passing it between the parts | 1,398,414 | 2.0% | | transaction binding, range check | 26,296 | 0.0% | -| total | 81,743,632 | | +| total | 68,491,902 | | -Of the loop, 55.6 M is in the steps and 11.0 M in folding the loop state over the witness: the state carries 33 per-run values (the powers of z, the points at z, r) to every step. +Of the loop, 43.8 M is in the steps and 8.2 M in folding the loop state over the witness: the state carries 33 per-run values (the powers of z, the points at z, r) to every step. + +## Where the bytes go + +| | bytes | +|---|---:| +| line constants of γ and δ: 4 words per pair and step | 22,528 | +| program code | about 26,000 | +| MSM: selected table entries and pruned branches | about 11,400 | +| 35 committed Miller values | 13,440 | +| quotient Q, 200 coefficients | 6,400 | +| B's slopes, 88 steps | 5,632 | +| inverses proving xQ != x, 23 additions | 1,472 | +| MSM slopes, c, c^-1, the proof, the challenges | about 2,900 | ## Building blocks @@ -38,22 +51,23 @@ Of the loop, 55.6 M is in the steps and 11.0 M in folding the loop state over th | F_p product with a constant, or normalisation | 8,923 | | F_p addition | 2,579 | | 12-term dot product, one reduction | 116,744 | -| loop step: doubling (65 of them) | 609,273 | -| loop step: addition with c or c^-1 (21) | 698,901 | -| loop step: Frobenius addition (2) | 683,092 | +| loop step: doubling (51 of them) | 385,831 | +| loop step: doubling with a committed Miller value (14) | 608,873 | +| loop step: addition with c or c^-1, committed (21) | 698,901 | +| loop step: Frobenius addition (2) | 455,950 | +| Q(z): eight coefficients (25 calls) | 94,597 | | MSM window (select and add) | about 169,500 | -A doubling step evaluates the committed Miller value at z (a 12-term dot product), B's slope, the three lines, and adds its two relations to the Horner sum. Computed in the tower, the same step cost 3,644,556. +A doubling step evaluates B's slope and the three lines at z and adds its relations to the Horner sum; a committed one also evaluates the committed Miller value (a 12-term dot product). Computed in the tower, the same step cost 3,644,556. ## Before and after -| | tower Miller loop | one polynomial identity | +| | tower Miller loop (main) | one polynomial identity | |---|---:|---:| -| program cost | 371.8 M mWU (with 414 padding blocks) | 81.7 M mWU | -| pruned program | 161 kB | 61 kB | -| witness | 220 kB (212 kB of it padding) | 45 kB | -| transaction | 382.2 kWU, 95.6 kvB | 107.4 kWU, 26.9 kvB | -| room under the 400 kWU limit | one more public input | about 20 more | +| program cost | 371.8 M mWU (with 414 padding blocks) | 68.5 M mWU | +| pruned program | 161 kB | 60 kB | +| witness | 220 kB (212 kB of it padding) | 30 kB | +| transaction | 382.2 kWU, 95.6 kvB | 90.8 kWU, 22.7 kvB | ## Time @@ -61,7 +75,7 @@ A doubling step evaluates the committed Miller value at z (a 12-term dot product |---|---| | generate the program from a key (Python) | 2 s | | generate the witness: Miller values, residue witness, quotient (Python) | 2 s | -| `deploy compile`: parse the 1.1 MB program text, once per key | 2.6 min | +| `deploy compile`: parse the 1.1 MB program text, once per key | 2.5 min | | `deploy spend`: prune, execute, build the transaction | 1 s | | snarkjs setup / proof of the toy circuit | 1.3 min / 1 s | | `tests/test_groth16.py` | 3 min | @@ -82,7 +96,7 @@ python3 tests/test_groth16.py # the whole verifier python3 tests/test_binding.py # the verifier with the transaction binding ``` -`deploy spend` prints the cost, the sizes and the budget of a real transaction, and `tools/harness/target/release/costs ` the cost of every named definition of a program. The files in `examples/multiplier/` regenerate the program and the witness of the regtest spend `9f4c1cb0…` byte for byte: +`deploy spend` prints the cost, the sizes and the budget of a real transaction, and `tools/harness/target/release/costs ` the cost of every named definition of a program. The files in `examples/multiplier/` regenerate the program and the witness of the regtest spend `511a3da3…` byte for byte: ```sh E=examples/multiplier @@ -95,8 +109,8 @@ gunzip -c $E/verifier.simpl.gz | cmp - build/check.simpl && gunzip -c $E/verifie | network | transaction | what | |---|---|---| -| Elements 23.3.1 regtest | `0da39ffd…` | the walkthrough (`tools/regtest.sh`): proof spend, 107,421 WU, vsize 26,856 | -| Elements 23.3.1 regtest | `901f2f6a…` | the same kind of proof redirected to another address: rejected with `Assertion failed inside jet` | -| Elements 23.3.1 regtest | `9f4c1cb0…` | proof spend, the files in `examples/multiplier/` | -| Elements 23.3.1 regtest | `d8d87a6a…` | the key path of the same leaf, 721 WU | +| Elements 23.3.1 regtest | `beba377d…` | the walkthrough (`tools/regtest.sh`): proof spend, 90,804 WU, vsize 22,701 | +| Elements 23.3.1 regtest | `511a3da3…` | proof spend, the files in `examples/multiplier/` | +| Elements 23.3.1 regtest | a redirected proof | rejected with `Assertion failed inside jet` | +| Elements 23.3.1 regtest | `5b6b46f1…` | the key path of the same leaf, 721 WU | | Liquid testnet | [`f25a6e90…`](https://blockstream.info/liquidtestnet/tx/f25a6e90417c427f25c1d6883869825f30ec111a357b0e550bf24a20725d3cdb), block 2,659,534 | an earlier verifier (tower Miller loop, without the G2 check): 381,492 WU, fee 11,000 sat | diff --git a/docs/design.md b/docs/design.md index 914bdcf..e644e7d 100644 --- a/docs/design.md +++ b/docs/design.md @@ -117,12 +117,14 @@ The Groth16 equation e(A, B) = e(α, β) e(L, γ) e(C, δ) holds if and only if The Miller loop runs along the signed-binary (NAF) digits of 6x + 2, the loop length of BN254's optimal ate pairing, and ends with two additions by Frobenius images of each G2 point: 88 steps, 65 doublings and 23 additions. The three pairs (-A, B), (L, γ) and (C, δ) advance in lockstep, so the accumulator f is squared once per step for all of them. The fourth pair, (α, β), belongs to the verifying key; its Miller value is a constant of the program. -Each step multiplies f (squared, for a doubling) by three lines, one per pair, each of the form `yP + c3 xP w + c4 w^3` with c3 = -λ and c4 = λ xT - yT, where T is the pair's running G2 point and λ the slope of the step. The prover writes the new value f' into the witness, and the step becomes a relation between polynomials: +Each step multiplies f (squared, for a doubling) by three lines, one per pair, each of the form `yP + c3 xP w + c4 w^3` with c3 = -λ and c4 = λ xT - yT, where T is the pair's running G2 point and λ the slope of the step. When the prover writes the new value f' into the witness, the step becomes a relation between polynomials: ``` f^2 l_B l_γ l_δ - f' = 0 (mod M) ``` +The prover does not commit after every step. Without a commitment, f' is the product itself, a polynomial of higher degree that the next relation carries along, and that next relation checks two steps at once. A committed value costs 12 witness words; a higher degree costs words of the quotient (below). The generator skips a commitment whenever the identity's degree stays within 211, the cap with the fewest words in total: 35 of the 88 Miller values are committed. + γ and δ belong to the key, so their slopes and lines are computed when the program is generated: each step receives c3 and c4 for both, four words each. B comes from the proof. The prover writes only B's slopes; B's points are never written down. The new point of a doubling is x' = λ^2 - 2x, y' = λ (x - x') - y, a polynomial in the slopes, and the program checks each slope instead: ``` @@ -148,7 +150,7 @@ The loop also proves that B is in G2. Its last point is T = [6x+2]B + ψ(B) - ψ ### One identity, checked at one point -That makes 203 relations R_k, each a polynomial in w of degree at most 61. The prover combines them with the powers of a challenge r into S = Σ r^k R_k, divides S by M, and writes the quotient Q (50 coefficients) into the witness. The program then picks a point z, evaluates every polynomial at z, and checks +That makes 150 relations R_k, each a polynomial in w of degree at most 211. The prover combines them with the powers of a challenge r into S = Σ r^k R_k, divides S by M, and writes the quotient Q (200 coefficients) into the witness. The program then picks a point z, evaluates every polynomial at z, and checks ``` S(z) = Q(z) M(z) @@ -156,13 +158,13 @@ S(z) = Q(z) M(z) This is randomized extension-field arithmetic: [Garaga](https://hackmd.io/@feltroidprime/B1eyHHXNT) checks each F_p12 product with its own quotient, and Novakovic and Eagen (section 6) combine all relations into one quotient, as here. Two things are specific to this verifier: the challenges come from SHA-256 inside the program, and B's F_p2 arithmetic is part of the same identity, so B's points never enter the witness. -The program never multiplies in F_p12. A committed value f' becomes the number f'(z), a 12-term dot product with the powers of z; a line becomes a number; a step's relation becomes a few F_p products, added to a running Horner sum of S(z). A doubling step costs 0.61 M mWU, against 3.6 M for computing it in the tower. +The program never multiplies in F_p12. A committed value f' becomes the number f'(z), a 12-term dot product with the powers of z; a line becomes a number; a step's relation becomes a few F_p products, added to a running Horner sum of S(z). A doubling step costs 0.39 M mWU, or 0.61 M with a commitment, against 3.6 M for computing it in the tower. Q(z) is evaluated eight coefficients at a time by one shared function, so its code is in the program once. -r and z come from SHA-256 (Fiat-Shamir): r hashes the public inputs, the proof, c, c^(-1), s and everything the loop committed, and z hashes r and Q. The program gets r and z as witness words, uses them throughout, and at the end compares them with the two hashes it has computed along the way. If some relation does not hold, S is nonzero modulo M except for at most 203 values of r, and then S - Q M is a nonzero polynomial of degree at most 61, which vanishes at z with probability at most 61/p. A cheating prover must find a hash output in a set of relative size about 2^-246. +r and z come from SHA-256 (Fiat-Shamir): r hashes the public inputs, the proof, c, c^(-1), s and everything the loop committed, and z hashes r and Q. The program gets r and z as witness words, uses them throughout, and at the end compares them with the two hashes it has computed along the way. If some relation does not hold, S is nonzero modulo M except for at most 150 values of r, and then S - Q M is a nonzero polynomial of degree at most 211, which vanishes at z with probability at most 211/p. A cheating prover must find a hash output in a set of relative size about 2^-245. The committed words are checked to be below 2^254 (their two top bits are zero), so that the 12-term dot products fit the bound of the field arithmetic; every canonical word passes. The program normalises the challenges, whose words are arbitrary 256-bit hashes. -The witness grows to 45 kB, mostly the 88 committed Miller values, and the program shrinks to 61 kB, since the line data of γ and δ is now four words per pair and step instead of eighteen. +The witness is 30 kB: the committed Miller values (13 kB), the quotient (6 kB), B's slopes (6 kB) and the rest of the proof. The program is 60 kB; the line data of γ and δ, four words per pair and step, is 22 kB of it. ## Public input and transaction binding (curve.py, merkle.py, groth16.py) @@ -172,7 +174,7 @@ To need only one, a circuit for this verifier exposes a single public signal: th ## Budget and padding (groth16.py) -The program costs 81.7 M mWU. Its budget is 1,000 mWU per byte of the input's witness stack, so the 61 kB program and the 45 kB witness already earn about 107 M mWU, and no padding is needed: the input weighs about 107 kWU. +The program costs 68.5 M mWU. Its budget is 1,000 mWU per byte of the input's witness stack, so the 60 kB program and the 30 kB witness already earn about 90 M mWU, and no padding is needed: the transaction weighs about 91 kWU. Programs that cost more than their bytes earn can still be padded. Elements does not relay transactions with an annex, the field rust-simplicity would use for padding, so the blocks of 512 zero bytes go into the Simplicity witness. A witness value the program never reads would get the wrong type, so each block is hashed into a throwaway SHA-256 context, which costs 25.5 k mWU per block. `Groth16.padding` sizes the padding from the program's cost, and `deploy spend` reports whether the budget is covered. diff --git a/examples/README.md b/examples/README.md index 4979fe5..726a3b8 100644 --- a/examples/README.md +++ b/examples/README.md @@ -16,13 +16,13 @@ tools/harness/target/release/hand examples/fp_mul.simpl examples/fp_mul.w0.json ## multiplier/ -The verifier of `circuits/multiplier.circom` (a·b = c, bound to the spending transaction) and the files of one spend on an Elements regtest node, transaction `9f4c1cb0…`, which proves c = 21. +The verifier of `circuits/multiplier.circom` (a·b = c, bound to the spending transaction) and the files of one spend on an Elements regtest node, transaction `511a3da3…`, which proves c = 21. | file | | |---|---| | `verification_key.json` | the snarkjs key | | `verifier.simpl.gz` | the generated program, 1.1 MB of text | -| `verifier.uprog` | its consensus encoding before pruning (`deploy compile`), CMR `c910831b…26b0be62` | +| `verifier.uprog` | its consensus encoding before pruning (`deploy compile`), CMR `e502d43b…47fb0d6b` | | `proof.json`, `public.json` | the snarkjs proof and its public signal | | `leaves.json` | the two leaves behind the public signal: the transaction's sighash and c | | `verifier.w0.json.gz` | the witness: proof, Miller values, slopes, quotient, Merkle sibling | diff --git a/examples/multiplier/leaves.json b/examples/multiplier/leaves.json index e7860cf..09e32d2 100644 --- a/examples/multiplier/leaves.json +++ b/examples/multiplier/leaves.json @@ -1 +1 @@ -{"sighash": "2430abe796bfee9e30559fe742228b1d36004de4a4b0c4cb945366aa31bb7321", "values": ["21"]} \ No newline at end of file +{"sighash": "cdac35b192915c6ebbc240dd8c4f8d22cf4e5e13087fb23e9978fbc6ce2cb05c", "values": ["21"]} \ No newline at end of file diff --git a/examples/multiplier/proof.json b/examples/multiplier/proof.json index daa673f..74faff6 100644 --- a/examples/multiplier/proof.json +++ b/examples/multiplier/proof.json @@ -1,17 +1,17 @@ { "pi_a": [ - "7614975239070624198332854401873649505470572626547896149741070312897843358930", - "4611774705360608746977331987834566924868049144042721246704404533437671643615", + "21566684453736135585297791733111880454710809102449812304555056051874966362175", + "20439253185853980828249216164238736927967952351267181741770936990431369438778", "1" ], "pi_b": [ [ - "5261944696180792681624872483670754604546279463032739224986573158204266522465", - "12317633613943012002930990405081707670179665685030284658975824394534056369377" + "2376322599524250606318122939551750522956957733513865503231378596442919457301", + "21315688201840938592689832640776347449862151534123330070129388622190619729110" ], [ - "21154522702036096038089906596923283702887853601323724616219725041146024442675", - "17356759660496773857362515270681502939595449805612756466401971378462211378698" + "3943731764001132601666642157228626600487124330574975950047211703846244120826", + "9258774679406286764135352324685156081411875551338919970350745361914487123535" ], [ "1", @@ -19,8 +19,8 @@ ] ], "pi_c": [ - "3199714410403655289905644902552955024418647404663178093197136563332313053728", - "11760199718826279720608869179394965276413967892246556778442807843086576418841", + "4638708091212589675127653121818387886945714069030086334439062558372181182618", + "8670616252775378861952872650202232831163409290031897847958971087012591991932", "1" ], "protocol": "groth16", diff --git a/examples/multiplier/public.json b/examples/multiplier/public.json index fb4754e..15dd29d 100644 --- a/examples/multiplier/public.json +++ b/examples/multiplier/public.json @@ -1,3 +1,3 @@ [ - "11319441843641788037619681105067091827983231747872427158058865524502604764122" + 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The prover writes the Miller -value after every step into the witness, and each step becomes a few relations R(w) = 0 (mod M): +value after some of the steps (COMMIT) into the witness, and each step becomes a few relations +R(w) = 0 (mod M): doubling f^2 l_B l_gamma l_delta - f', lambda y - 3/2 x^2 addition f g l_B l_gamma l_delta - f', lambda (xQ - x) - (yQ - y), (xQ - x) d - 1 at the end x - x(-psi^3(B)), y - y(-psi^3(B)), f pi(c^-1) pi^2(c) pi^3(c^-1) - T_s, c c^-1 - 1 -f, f' are the committed Miller values, g = c^-1 or c (the residue witness, as in Novakovic and Eagen), +f, f' are the Miller values, g = c^-1 or c (the residue witness, as in Novakovic and Eagen), l the lines at P = -A, L, C. Only B's slopes lambda are committed: x' = lambda^2 - 2x and y' = lambda (x - x') - y are polynomials in them. A slope that satisfies its relation is the true one (y is never 0, and d, the inverse of xQ - x, proves xQ != x), so every step is the group law. The last three relations are the G2 membership of B and the residue check; T_s = (W27^s e(alpha, beta))^-1. +After a step whose value is not committed, f' is the product itself, so the next relation carries a +polynomial of higher degree; the first relation of a step is dropped then. The prover combines the relations with powers of a challenge r, S = sum r^k R_k, and supplies the quotient Q = S / M. The program evaluates everything at one point z and checks S(z) = Q(z) M(z). It multiplies only in F_p: an F_p12 product becomes one product of two evaluations. Soundness. r is the SHA-256 of the public inputs, the proof and everything committed, and z the -SHA-256 of r and Q. If some R_k is not 0 modulo M, S is not 0 modulo M except for at most 203 values of -r; then S - Q M is a nonzero polynomial of degree at most 61, which vanishes at z with probability at most -61 / p. The program gets r and z as witness words and compares them with the two hashes at the end. +SHA-256 of r and Q. If some R_k is not 0 modulo M, S is not 0 modulo M except for at most 150 values of +r; then S - Q M is a nonzero polynomial of degree at most 211, which vanishes at z with probability at +most 211 / p. The program gets r and z as witness words and compares them with the two hashes at the +end. `relations` is written once over an abstract ring: the prover runs it on polynomials to get S and Q, the tests run it on numbers to compare with the program. @@ -111,9 +115,12 @@ class DegRing: elem = staticmethod(lambda coeffs: 11) -def relations(R, v, lines, table): +def relations(R, v, lines, table, commit=None): """The relations R_k in order, over the ring R. v: the values (Prover.witness); lines: fixed_lines of - gamma and delta; table: T_s for s = 0, 1, 2 (direct coefficients).""" + gamma and delta; table: T_s for s = 0, 1, 2 (direct coefficients); commit: the steps whose Miller + value is committed (COMMIT). After any other step, f' is the product itself, a polynomial of + higher degree.""" + commit = COMMIT if commit is None else commit X = R.X X3 = R.mul(R.mul(X, X), X) U = R.sub(R.mul(X3, X3), R.k(9)) # u = w^6 - 9 @@ -147,8 +154,10 @@ def line(Pt, c3, c4): # yP + c3 xP w + c4 w^3: the tower line c0 + c3 w + c4 v w lB = line(v['P'], R.sub(R.k(0), lam), m) lg = line(v['L'], phi(lines[0][i][0]), phi(lines[0][i][1])) ld = line(v['C'], phi(lines[1][i][0]), phi(lines[1][i][1])) - f_next = R.elem(v['f'][i]) - rels.append(R.sub(R.mul(R.mul(R.mul(f2, lB), lg), ld), f_next)) + f_next = R.mul(R.mul(R.mul(f2, lB), lg), ld) + if commit[i]: + rels.append(R.sub(f_next, R.elem(v['f'][i]))) + f_next = R.elem(v['f'][i]) x, y, f = x2, R.sub(m, R.mul(lam, x2)), f_next rels.append(R.sub(x, phi(xb.conj() * bn254.FROB3[2]))) rels.append(R.sub(y, phi(yb.conj() * -bn254.FROB3[3]))) @@ -174,16 +183,29 @@ def horner(R, rels, r): return acc -def _shape(): +def _shape(commit): """The number of relations and the degree of S (DegRing reads only the structure).""" z = bn254.Fp2(0) v = dict(P=(0, 0), B=bn254.G2_GEN, C=(0, 0), L=(0, 0), c=[0] * 12, ci=[0] * 12, s=0, lam=[z] * N_STEPS, f=[[0] * 12] * N_STEPS, d={i: z for i in range(N_STEPS)}) - rels = relations(DegRing, v, [[(z, z)] * N_STEPS] * 2, [[0] * 12] * 3) + rels = relations(DegRing, v, [[(z, z)] * N_STEPS] * 2, [[0] * 12] * 3, commit) return len(rels), max(rels) -N_RELS, DEG_S = _shape() # 203 relations, degree 61 +def schedule(cap): + """The steps whose Miller value is committed: left to right, a step is skipped whenever S stays + within the degree cap. A committed value costs 12 words of witness, a degree of S one word of Q.""" + commit = [True] * N_STEPS + for i in range(N_STEPS): + commit[i] = False + if _shape(commit)[1] > cap: + commit[i] = True + return commit + + +DEG_CAP = 211 # the cap with the fewest committed words: 36 Miller values, Q of 200 words (vs 88 and 50) +COMMIT = schedule(DEG_CAP) +N_RELS, DEG_S = _shape(COMMIT) # 150 relations, degree 211 N_Q = DEG_S - 11 # coefficients of Q @@ -209,9 +231,9 @@ def blocks(n): return out -def leaf_words(kind): - """lambda (2 words), f' (12), and d (2) for an addition.""" - return 16 if kind in ADDS else 14 +def leaf_words(kind, commit): + """lambda (2 words), f' (12) if committed, d (2) for an addition.""" + return 2 + 12 * commit + 2 * (kind in ADDS) class Prover: @@ -244,7 +266,7 @@ def witness(self, pairs, inputs, residue=None, check=True): T, _ = bn254.affine_step(T, lams[i], pts[kind]) for k in range(3): f = f.mul_by_034(*lines[k][i]) - fs.append(poly.from_tower(f)) + fs.append(poly.from_tower(f) if COMMIT[i] else None) v = dict(P=nA, B=B, C=C, L=L, c=poly.from_tower(c), ci=poly.from_tower(ci), s=s, lam=lams, f=fs, d=ds) return self.seal(v, inputs, check) @@ -255,7 +277,8 @@ def seal(self, v, inputs, check=True): fp2 = lambda a: mont([a.c0, a.c1]) nA, B, C = v['P'], v['B'], v['C'] w = dict(points=mont([nA[0], nA[1], B[0].c0, B[0].c1, B[1].c0, B[1].c1, C[0], C[1]]), c=mont(v['c']), ci=mont(v['ci']), s=v['s'], - leaves=[fp2(v['lam'][i]) + mont(v['f'][i]) + (fp2(v['d'][i]) if i in v['d'] else []) for i in range(N_STEPS)]) + leaves=[fp2(v['lam'][i]) + (mont(v['f'][i]) if COMMIT[i] else []) + (fp2(v['d'][i]) if i in v['d'] else []) + for i in range(N_STEPS)]) t = wbytes(inputs) + wbytes(w['points']) + s_byte(v['s']) + wbytes(w['c'] + w['ci']) + b''.join(map(wbytes, w['leaves'])) w['r'] = int.from_bytes(hashlib.sha256(t).digest(), 'big') q, rem = poly.divmod_m(horner(PolyRing, relations(PolyRing, v, self.lines, self.table), from_mont(w['r']))) @@ -295,11 +318,15 @@ def __init__(self, fp, gamma, delta, const, b=None): self.sample_bounds() be = self.be st = self.state_shape() - self.STEP_D = be.function('STEP_D', [[st, leaf_shape('D')], words(8)], self.step_body('D')) - self.STEP_A = be.function('STEP_A', [[st, leaf_shape('A+')], [words(3), words(8)]], self.step_body('A')) - self.STEP_F = be.function('STEP_F', [[st, leaf_shape('F1')], [words(2), words(8)]], self.step_body('F')) + self.steps = {} + for kind, args in (('D', None), ('A', words(3)), ('F', words(2))): + for commit in sorted({COMMIT[k] for k in range(N_STEPS) if STEPS[k][0] == kind}): + shape = [[st, leaf_shape({'D': 'D', 'A': 'A+', 'F': 'F1'}[kind], commit)], words(8) if args is None else [args, words(8)]] + name = f'STEP_{kind}' + ('' if commit else '_') + self.steps[(kind, commit)] = be.function(name, shape, self.step_body(kind, commit)) + self.QCHUNK = be.function('QCHUNK', [[words(2), words(7)], words(8)], self.qchunk_body) if is_sym(be): - self.costs = {fn.t: fn.c for fn in (self.STEP_D, self.STEP_A, self.STEP_F)} + self.costs = {fn.t: fn.c for fn in list(self.steps.values()) + [self.QCHUNK]} def one(self): return self.fp.const(1) def m1(self): return self.fp.const(P - 1) @@ -345,9 +372,10 @@ def sha_final(self, h): def state_shape(self): return [words(4), E('', SHA_CTX_W, 0), words(len(RUN))] - def step_body(self, kind): - """Body of STEP_D, STEP_A or STEP_F: ((state, leaf), args) -> state. Evaluates lambda, f' and - the lines at z, adds the step's relations to acc and absorbs the leaf into h.""" + def step_body(self, kind, commit): + """Body of a step function: ((state, leaf), args) -> state. Evaluates lambda and the lines at z, + and f' (a committed value at z, or the product), and adds the step's relations to acc in the + order of `relations`; absorbs the leaf into h.""" fp, be, b = self.fp, self.be, self.b rd, one, m1 = self.rd, self.one, self.m1 add = kind != 'D' @@ -355,30 +383,33 @@ def step_body(self, kind): def body(accs): (st, leaf), args = accs (e, acc, x, y), h, run = st - fw = flat(leaf[1]) + fw = flat(leaf[1]) if commit else [] data = args if kind == 'D' else args[1] run = dict(zip(RUN, run)) c = {'e': Val(e, b['e']), 'acc': Val(acc, b['acc']), 'x': Val(x, b['x']), 'y': Val(y, b['y']), - 'l': [Val(w, WORD_MAX) for w in leaf[0]], 'f': [Val(w, F_HI) for w in fw], - 'dg': [Val(w, P - 1) for w in data[:4]], 'dd': [Val(w, P - 1) for w in data[4:]], - 'zp': [Val(run[f'z{k}'], KB) for k in range(1, 12)]} - for nm in ('z3', 'zeta', 'z3zeta', 'nxpz', 'yp', 'xlz', 'xlzz', 'yl', 'xcz', 'xczz', 'yc', 'r', 'r2') + (('r3',) if add else ()): + 'l': [Val(w, WORD_MAX) for w in leaf[0]], + 'dg': [Val(w, P - 1) for w in data[:4]], 'dd': [Val(w, P - 1) for w in data[4:]]} + if commit: + c['f'] = [Val(w, F_HI) for w in fw] + c['zp'] = [Val(run[f'z{k}'], KB) for k in range(1, 12)] + for nm in ('z3', 'zeta', 'z3zeta', 'nxpz', 'yp', 'xlz', 'xlzz', 'yl', 'xcz', 'xczz', 'yc', 'r') + (('r2',) if add or commit else ()) + (('r3',) if add and commit else ()): c[nm] = Val(run[nm], KB) if add: c['xq'], c['yq'] = Val(args[0][0], KB), Val(args[0][1], KB) - c['d'] = [Val(w, WORD_MAX) for w in leaf[2]] + c['d'] = [Val(w, WORD_MAX) for w in leaf[-1]] if kind == 'A': c['g'] = Val(args[0][2], KB) at_zeta = lambda c, a: rd([(a[1], c['zeta']), (a[0], one())]) # a0 + a1 zeta def first(c): - f = c['f'] - fz = rd([(f[0], one())] + [(f[k], c['zp'][k - 1]) for k in range(1, 12)]) - out = [at_zeta(c, c['l']), fp.normalize(fz)] + out = [at_zeta(c, c['l'])] + if commit: + f = c['f'] + out.append(fp.normalize(rd([(f[0], one())] + [(f[k], c['zp'][k - 1]) for k in range(1, 12)]))) return out + ([at_zeta(c, c['d']), fp.sub(c['xq'], c['x'])] if add else []) minus_x3 = (lambda c: [(c['x'], m1()), (c['xq'], m1())]) if add else (lambda c: [(c['x'], fp.const(P - 2))]) - steps = [('let', ['lam', 'e2'] + (['dz', 'dx'] if add else []), first), - ('drop', ['l', 'f', 'zp'] + (['d'] if add else [])), + steps = [('let', ['lam'] + (['e2'] if commit else []) + (['dz', 'dx'] if add else []), first), + ('drop', ['l'] + (['f', 'zp'] if commit else []) + (['d'] if add else [])), ('let', ['x2', 'm'], lambda c: [rd([(c['lam'], c['lam'])] + minus_x3(c)), rd([(c['lam'], c['x']), (c['y'], m1())])])] def final(c, _): @@ -389,39 +420,59 @@ def final(c, _): ld = rd([(c['yc'], one()), (c['xcz'], c['dd'][0]), (c['xczz'], c['dd'][1]), (c['z3'], c['dd'][2]), (c['z3zeta'], c['dd'][3])]) if kind == 'D': f2 = rd([(c['e'], c['e'])]) - terms = [(c['acc'], c['r2']), (rd([(lam, c['y']), (c['x'], fp.mul_const(c['x'], -THREE_HALVES))]), c['r'])] + rels = [rd([(lam, c['y']), (c['x'], fp.mul_const(c['x'], -THREE_HALVES))])] else: f2 = rd([(c['e'], c['g'])]) if kind == 'A' else c['e'] - slope = rd([(lam, c['dx']), (c['yq'], m1()), (c['y'], one())]) - inverse = rd([(c['dx'], c['dz']), (one(), m1())]) - terms = [(c['acc'], c['r3']), (slope, c['r2']), (inverse, c['r'])] - acc2 = rd(terms + [(rd([(rd([(f2, lB)]), lg)]), ld), (c['e2'], m1())]) - outs = [c['e2'], acc2, x2, y2] + rels = [rd([(lam, c['dx']), (c['yq'], m1()), (c['y'], one())]), rd([(c['dx'], c['dz']), (one(), m1())])] + prod = rd([(rd([(f2, lB)]), lg)]) # times l_delta below + # Horner: acc r^n + the step's relations, the last one with weight 1 + r_pow = lambda j: one() if j == 0 else c['r' if j == 1 else f'r{j}'] + n = len(rels) + commit + terms = [(c['acc'], r_pow(n))] + [(v, r_pow(n - 1 - k)) for k, v in enumerate(rels)] + if commit: + acc2 = rd(terms + [(prod, ld), (c['e2'], m1())]) + e2 = c['e2'] + else: + acc2 = rd(terms) + e2 = rd([(prod, ld)]) + outs = [e2, acc2, x2, y2] for nm, v in zip(('e', 'acc', 'x', 'y'), outs): b[nm + '_out'] = max(b.get(nm + '_out', 0), v.hi) return fp.pack(outs) new = chain(fp, c, steps, final) - h2 = self.sha_add(h, [leaf[0], leaf[1][0], leaf[1][1]] + leaf[2:]) + h2 = self.sha_add(h, [leaf[0]] + ([leaf[1][0], leaf[1][1]] if commit else []) + ([leaf[-1]] if add else [])) out = be.pack([new, h2, be.subtree(list(run.values()))]) - return be.seq(be.checks([self.top2_zero(w) for w in fw]), out) + return be.seq(be.checks([self.top2_zero(w) for w in fw]), out) if commit else out return body + def qchunk_body(self, accs): + """Body of QCHUNK: ((qz, z^8), (z .. z^7)), 8 coefficients of Q -> qz z^8 + their polynomial at z. + Q(z) is this step from the top chunk down; the coefficients are checked below 2^254.""" + ((qz, z8), zp), q = accs + one, b = self.one(), self.b + out = self.rd([(Val(qz, b['qz']), Val(z8, KB)), (Val(q[0], F_HI), one)] + + [(Val(q[k], F_HI), Val(zp[k - 1], KB)) for k in range(1, 8)]) + b['qz_out'] = max(b.get('qz_out', 0), out.hi) + return self.be.seq(self.be.checks([self.top2_zero(w) for w in q]), out.e) + def sample_bounds(self): - """The bounds of the loop state: a fixed point of the three step bodies, found on the numeric - backend by widening the inputs (1% slack) until the outputs fit.""" + """The bounds of the loop state: a fixed point of the step bodies, found on the numeric backend by + widening the inputs (1% slack) until the outputs fit.""" b = self.b - b.update(e=KB, acc=KB, x=KB, y=KB) + b.update(e=KB, acc=KB, x=KB, y=KB, qz=KB) st = [[0] * 4, b'', [0] * len(RUN)] - leaf = [[0, 0], [[0] * 8, [0] * 4]] for _ in range(30): - for nm in ('e', 'acc', 'x', 'y'): + for nm in ('e', 'acc', 'x', 'y', 'qz'): b[nm + '_out'] = 0 - self.step_body('D')([[st, leaf], [0] * 8]) - self.step_body('A')([[st, leaf + [[0, 0]]], [[0] * 3, [0] * 8]]) - self.step_body('F')([[st, leaf + [[0, 0]]], [[0] * 2, [0] * 8]]) - if all(b[nm + '_out'] <= b[nm] for nm in ('e', 'acc', 'x', 'y')): + for kind, n_args in (('D', 0), ('A', 3), ('F', 2)): + for commit in (False, True): + leaf = [[0, 0]] + ([[[0] * 8, [0] * 4]] if commit else []) + ([[0, 0]] if n_args else []) + self.step_body(kind, commit)([[st, leaf], [[0] * n_args, [0] * 8] if n_args else [0] * 8]) + self.qchunk_body([[[0, 0], [0] * 7], [0] * 8]) + names = ('e', 'acc', 'x', 'y', 'qz') + if all(b[nm + '_out'] <= b[nm] for nm in names): return - for nm in ('e', 'acc', 'x', 'y'): + for nm in names: b[nm] = max(b[nm], b[nm + '_out'] * 101 // 100) raise RuntimeError('no fixed point for the loop bounds') @@ -430,11 +481,12 @@ def step_value(self, k, st, env): be = self.be kind = STEPS[k] data = [be.word(v) for v in self.data[k]] + fn = self.steps[(kind[0], COMMIT[k])] if kind == 'D': - return be.call(self.STEP_D, [env, data]) + return be.call(fn, [env, data]) run = dict(zip(RUN, st[2])) args = [be.subtree(run[nm]) if is_sym(be) else run[nm] for nm in Q_KEYS[kind]] - return be.call(self.STEP_A if kind in ('A+', 'A-') else self.STEP_F, [env, [args, data]]) + return be.call(fn, [env, [args, data]]) def loop(self, state0, leaves): be = self.be @@ -442,7 +494,7 @@ def loop(self, state0, leaves): for k, leaf in enumerate(leaves): state0 = self.step_value(k, state0, [state0, leaf]) return state0 - widths = [256 * leaf_words(kind) for kind in STEPS] + widths = [256 * leaf_words(kind, COMMIT[k]) for k, kind in enumerate(STEPS)] return be.comp(be.pair(state0, leaves), fold_leaves(be, self.state_shape(), widths, lambda k: lambda accs: self.step_value(k, accs[0], be.iden()))) @@ -556,8 +608,13 @@ def final(self, state, end, q, rz): (e, acc, x, y), h, run = state run = {nm: Val(w, KB) for nm, w in zip(RUN, run)} b = self.b - c = {'e': Val(e, b['e']), 'acc': Val(acc, b['acc']), 'x': Val(x, b['x']), 'y': Val(y, b['y']), - 'q': [Val(w, F_HI) for w in flat(q)], 'zp': [run[f'z{k}'] for k in range(1, 12)]} + chunks = [half for blk in q for half in ([blk] if len(blk) == 8 else [blk[:8], blk[8:]])] + ensure(len(blocks(N_Q)) == len(q) and all(len(ch) == 8 for ch in chunks), 'Q is read in chunks of 8 words') + qz = be.word(0) + for ch in reversed(chunks): + zs = [run[f'z{k}'].e for k in range(1, 8)] + qz = be.call(self.QCHUNK, [[[qz, run['z8'].e], zs], be.subtree(ch) if is_sym(be) else ch]) + c = {'e': Val(e, b['e']), 'acc': Val(acc, b['acc']), 'x': Val(x, b['x']), 'y': Val(y, b['y']), 'qz': Val(qz, b['qz'])} c.update({nm: Val(w, hi) for nm, w, hi in zip(END, end, self.end_hi)}) c.update({nm: run[nm] for nm in ('r', 'r2', 'r3', 'z6')}) @@ -565,28 +622,19 @@ def horner_end(c): g2 = rd([(c['x'], c['r2']), (c['X3'], fp.mul_const(c['r2'], -1)), (c['y'], c['r']), (c['Y3'], fp.mul_const(c['r'], -1))]) return rd([(c['acc'], c['r3']), (g2, one()), (c['e'], c['pis']), (c['ts'], m1())]) - def q_chunks(c): - q, zp = c['q'], c['zp'] - return [rd([(q[j], one())] + [(q[j + k], zp[k - 1]) for k in range(1, min(12, len(q) - j))]) for j in range(0, len(q), 12)] steps = [('let', ['s', 'z12'], lambda c: [horner_end(c), rd([(c['z6'], c['z6'])])]), - ('drop', ['acc', 'x', 'y', 'e', 'r', 'r2', 'r3'] + END), - ('let', ['chunks'], lambda c: [q_chunks(c)]), - ('drop', ['zp', 'q'])] + ('drop', ['acc', 'x', 'y', 'e', 'r', 'r2', 'r3'] + END)] def check(c, _): - qz = c['chunks'][-1] - for part in reversed(c['chunks'][:-1]): - qz = rd([(qz, c['z12']), (part, one())]) minus_m = fp.lin_const([(c['z12'], -1), (c['z6'], 18), (one(), -82)]) - return cv.zero_check(rd([(c['s'], one()), (qz, minus_m)])).e + return cv.zero_check(rd([(c['s'], one()), (c['qz'], minus_m)])).e identity = chain(fp, c, steps, check) r_hash = self.sha_final(h) z_hash = self.sha_final(self.sha_add(self.sha_add(self.sha_init(), [rz[0]]), list(q))) if not is_sym(be): ensure(r_hash == rz[0] and z_hash == rz[1], 'challenge') return 0 - return be.checks([identity, be.verify(be.eq256(r_hash, rz[0])), be.verify(be.eq256(z_hash, rz[1]))] + - [self.top2_zero(w) for w in flat(q)]) + return be.checks([identity, be.verify(be.eq256(r_hash, rz[0])), be.verify(be.eq256(z_hash, rz[1]))]) def program(self, inputs, pts, c, ci, sel, rz, leaves, q): """The whole check. inputs: {'L': (x, y) Vals, 'words': the public input accessors}; the rest as @@ -596,7 +644,7 @@ def program(self, inputs, pts, c, ci, sel, rz, leaves, q): if not is_sym(be): st0, end = s0 return self.final(self.loop(st0, leaves), end, q, rz) - lw = sum(256 * leaf_words(k) for k in STEPS) + lw = sum(256 * leaf_words(kind, COMMIT[k]) for k, kind in enumerate(STEPS)) qshape = [words(n) for n in blocks(N_Q)] parts = {} @@ -645,35 +693,41 @@ def bounds(): # --------------------------------------------------------------------------------------------------- # The witness of the check: ((points, (c, c^-1)), ((s, (r, z)), (leaves, Q))). c, c^-1: 8 + 4 words; # a leaf: (lambda, (f' 8 + 4 words)) or ((lambda, f'), d); Q in hashing blocks. -def leaf_shape(kind): - return [words(2), [words(8), words(4)]] + ([words(2)] if kind in ADDS else []) +def leaf_shape(kind, commit): + """(lambda, f', d): f' (8 + 4 words) if committed, d for an addition.""" + return [words(2)] + ([[words(8), words(4)]] if commit else []) + ([words(2)] if kind in ADDS else []) def witness_shape(): bit = lambda: E('', 1, 0) c = lambda: [words(8), words(4)] - return [[words(8), [c(), c()]], [[[bit(), bit()], words(2)], [[leaf_shape(k) for k in STEPS], [words(n) for n in blocks(N_Q)]]]] + leaves = [leaf_shape(kind, COMMIT[k]) for k, kind in enumerate(STEPS)] + return [[words(8), [c(), c()]], [[[bit(), bit()], words(2)], [leaves, [words(n) for n in blocks(N_Q)]]]] def witness_type(): T = lambda n: Tbalanced(['2^256'] * n) c = f'({T(8)} * {T(4)})' - leaf = lambda k: f'(({T(2)} * {c}) * {T(2)})' if k in ADDS else f'({T(2)} * {c})' - return (f'(({T(8)} * ({c} * {c})) * (((2 * 2) * {T(2)}) * ' - f'({Tbalanced([leaf(k) for k in STEPS])} * {Tbalanced([T(n) for n in blocks(N_Q)])})))') + leaf = lambda kind, commit: Tbalanced([T(2)] + ([c] if commit else []) + ([T(2)] if kind in ADDS else [])) + leaves = Tbalanced([leaf(kind, COMMIT[k]) for k, kind in enumerate(STEPS)]) + return f'(({T(8)} * ({c} * {c})) * (((2 * 2) * {T(2)}) * ({leaves} * {Tbalanced([T(n) for n in blocks(N_Q)])})))' def witness_bits(): - return 256 * (8 + 24 + 2 + N_Q + sum(leaf_words(k) for k in STEPS)) + 2 + return 256 * (8 + 24 + 2 + N_Q + sum(leaf_words(kind, COMMIT[k]) for k, kind in enumerate(STEPS))) + 2 def _split(w): - leaf = lambda lf: [lf[:2], [lf[2:10], lf[10:14]]] + ([lf[14:]] if len(lf) > 14 else []) + def leaf(k, lf): + parts = [lf[:2]] + if COMMIT[k]: + parts.append([lf[2:10], lf[10:14]]) + return parts + ([lf[-2:]] if STEPS[k] in ADDS else []) qs, i = [], 0 for n in blocks(N_Q): qs.append(w['q'][i:i + n]) i += n - return w['points'], [w['c'][:8], w['c'][8:]], [w['ci'][:8], w['ci'][8:]], [leaf(lf) for lf in w['leaves']], qs + return w['points'], [w['c'][:8], w['c'][8:]], [w['ci'][:8], w['ci'][8:]], [leaf(k, lf) for k, lf in enumerate(w['leaves'])], qs def witness_num(w): @@ -687,7 +741,7 @@ def witness_value(w): pts, c, ci, leaves, qs = _split(w) W = lambda ws: wbalanced([wv(256, x) for x in ws]) pair2 = lambda a: wpair(W(a[0]), W(a[1])) - leaf = lambda lf: balanced([W(lf[0]), pair2(lf[1])] + [W(x) for x in lf[2:]], wpair) + leaf = lambda lf: balanced([W(lf[0])] + [pair2(x) if len(x) == 2 and isinstance(x[0], list) else W(x) for x in lf[1:]], wpair) return wpair(wpair(W(pts), wpair(pair2(c), pair2(ci))), wpair(wpair(wpair(wbit(w['s'] >> 1), wbit(w['s'] & 1)), W([w['r'], w['z']])), wpair(wbalanced([leaf(lf) for lf in leaves]), wbalanced([W(x) for x in qs])))) diff --git a/tests/test_pairing.py b/tests/test_pairing.py index f3a8803..434af2c 100644 --- a/tests/test_pairing.py +++ b/tests/test_pairing.py @@ -57,12 +57,13 @@ def witness(L, w): def tampered(prover, w, how): """The honest values with one change, sealed again.""" v = {k: (list(x) if isinstance(x, list) else dict(x) if isinstance(x, dict) else x) for k, x in w['values'].items()} - v['f'] = [list(f) for f in v['f']] + v['f'] = [None if f is None else list(f) for f in v['f']] add = next(i for i, k in enumerate(PC.STEPS) if k in PC.ADDS) if how == 'slope': v['lam'][10] = v['lam'][10] + 1 elif how == 'Miller value': - v['f'][40][3] += 1 + k = next(i for i in range(40, PC.N_STEPS) if PC.COMMIT[i]) + v['f'][k][3] += 1 elif how == 'inverse': v['d'][add] = v['d'][add] + 1 elif how == 's':