Signature
乙太坊是否支持配對操作？
我需要建構一個使用雙線性配對操作來生成簽名的應用程序。如果我向乙太坊發送帶有簽名的消息，乙太坊可以驗證該消息嗎？EVM 是否支持配對操作？
這是使用 BLS 簽名的簽名驗證範例：
給定簽名 S 和公鑰 g^{x}，我們驗證 e(S ,g)=e(H(m),g^{x})。
是的，在地址 0x8，EVM 有一個預編譯合約，用於在 EIP 197 中描述的橢圓曲線 alt_bn128 上進行最優 ate 配對檢查。
我們如何在 Solidity 中驗證 BGLS 聚合簽名？有一個包含以下可能有用的程式碼的要點：
pragma solidity ^0.4.14; /* Example of how to verify BLS signatures and BGLS aggregate signatures in Ethereum. Signatures are generated using https://github.com/Project-Arda/bgls Code is based on https://github.com/jstoxrocky/zksnarks_example */ contract BLSExample { struct G1Point { uint X; uint Y; } // Encoding of field elements is: X[0] * z + X[1] struct G2Point { uint[2] X; uint[2] Y; } /// @return the generator of G1 function P1() internal returns (G1Point) { return G1Point(1, 2); } /// @return the generator of G2 function P2() internal returns (G2Point) { return G2Point( [11559732032986387107991004021392285783925812861821192530917403151452391805634, 10857046999023057135944570762232829481370756359578518086990519993285655852781], [4082367875863433681332203403145435568316851327593401208105741076214120093531, 8495653923123431417604973247489272438418190587263600148770280649306958101930] ); } //Example of BLS signature verification function verifyBLSTest() returns (bool) { bytes memory message = hex"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"; G1Point memory signature = G1Point(11181692345848957662074290878138344227085597134981019040735323471731897153462, 6479746447046570360435714249272776082787932146211764251347798668447381926167); G2Point memory v = G2Point( [18523194229674161632574346342370534213928970227736813349975332190798837787897, 5725452645840548248571879966249653216818629536104756116202892528545334967238], [3816656720215352836236372430537606984911914992659540439626020770732736710924, 677280212051826798882467475639465784259337739185938192379192340908771705870] ); G1Point memory h = hashToG1(message); return pairing2(negate(signature), P2(), h, v); } //Example of BGLS signature verification with 2 signers //Note that the messages differ in their last character. function verifyBGLS2() returns (bool) { uint numberOfSigners = 2; G1Point memory signature = G1Point(7985250684665362734034207174567341000146996823387166378141631317099216977152, 5471024627060516972461571110176333017668072838695251726406965080926450112048); bytes memory message0 = hex"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"; bytes memory message1 = hex"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"; G2Point memory v0 = G2Point( [15516709285352539082439213720585739724329002971882390582209636960597958801449, 19324541677661060388134143597417835654030498723817274130329567224531700170734], [16550775633156536193089672538964908973667410921848053632462693002610771214528, 10154483139478025296468271477739414260393126999813603835827647034319242387010] ); G2Point memory v1 = G2Point( [14125383697019450293340447180826714775062600193406387386692146468060627933203, 10886345395648455940547500614900453787797209052692168129177801883734751834552], [13494666809312056575532152175382485778895768300692817869062640713829304801648, 10580958449683540742032499469496205826101096579572266360455646078388895706251] ); G1Point memory h0 = hashToG1(message0); G1Point memory h1 = hashToG1(message1); G1Point[] memory a = new G1Point[](numberOfSigners + 1); G2Point[] memory b = new G2Point[](numberOfSigners + 1); a[0] = negate(signature); a[1] = h0; a[2] = h1; b[0] = P2(); b[1] = v0; b[2] = v1; return pairing(a, b); } //Example of BGLS signature verification with 3 signers //Note that the messages differ in their last character. function verifyBGLS3() returns (bool) { uint numberOfSigners = 3; G1Point memory signature = G1Point(385846518441062319503502284295243290270560187383398932887791670182362540842, 19731933537428695151702009864745685458233056709189425720845387511061953267292); bytes memory message0 = hex"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"; bytes memory message1 = hex"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"; bytes memory message2 = hex"7b0a2020226f70656e223a207b0a20202020227072696365223a2039353931372c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333134323430302c0a2020202020202269736f223a2022323031362d31322d33315430303a30303a30302e3030305a220a202020207d0a20207d2c0a202022636c6f7365223a207b0a20202020227072696365223a2039363736302c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333232383830302c0a2020202020202269736f223a2022323031372d30312d30315430303a30303a30302e3030305a220a202020207d0a20207d2c0a2020226c6f6f6b7570223a207b0a20202020227072696365223a2039363736302c0a20202020226b223a20312c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333232383830302c0a2020202020202269736f223a2022323031372d30312d30315430303a30303a30302e3030305a220a202020207d0a20207d0a7d0a6578616d706c652e636f6d2f6170692f32"; G2Point memory v0 = G2Point( [1787282038370667094324364195810339512415273589223814213215040505578200405366, 414568866548933554513940840943382696902163788831396286279770126458218272940], [6560020551439455112781785895092032589010633560844445112872109862153018855017, 19411093226570397520343120724285433000937737461010544490862811136406407315543] ); G2Point memory v1 = G2Point( [14831125462625540363404323739936082597729714855858291605999144010730542058037, 8342129546329626371616639780890580451066604883761980695690870205390518348707], [808186590373043742842665711030588185456231663895663328011864547134240543671, 1856705676948889458735296604372981546875220644939188415241687241562401814459] ); G2Point memory v2 = G2Point( [12507030828714819990408995725310388936101611986473926829733453468215798265704, 16402225253711577242710704509153100189802817297679524801952098990526969620006], [18717845356690477533392378472300056893077745517009561191866660997312973511514, 20124563173642533900823905467925868861151292863229012000403558815142682516349] ); G1Point memory h0 = hashToG1(message0); G1Point memory h1 = hashToG1(message1); G1Point memory h2 = hashToG1(message2); G1Point[] memory a = new G1Point[](numberOfSigners + 1); G2Point[] memory b = new G2Point[](numberOfSigners + 1); a[0] = negate(signature); a[1] = h0; a[2] = h1; a[3] = h2; b[0] = P2(); b[1] = v0; b[2] = v1; b[3] = v2; return pairing(a, b); } /// @return the result of computing the pairing check /// e(p1[0], p2[0]) * .... * e(p1[n], p2[n]) == 1 /// For example pairing([P1(), P1().negate()], [P2(), P2()]) should /// return true. function pairing(G1Point[] p1, G2Point[] p2) internal returns (bool) { require(p1.length == p2.length); uint elements = p1.length; uint inputSize = elements * 6; uint[] memory input = new uint[](inputSize); for (uint i = 0; i < elements; i++) { input[i * 6 + 0] = p1[i].X; input[i * 6 + 1] = p1[i].Y; input[i * 6 + 2] = p2[i].X[0]; input[i * 6 + 3] = p2[i].X[1]; input[i * 6 + 4] = p2[i].Y[0]; input[i * 6 + 5] = p2[i].Y[1]; } uint[1] memory out; bool success; assembly { success := call(sub(gas, 2000), 8, 0, add(input, 0x20), mul(inputSize, 0x20), out, 0x20) // Use "invalid" to make gas estimation work switch success case 0 {invalid} } require(success); return out[0] != 0; } /// Convenience method for a pairing check for two pairs. function pairing2(G1Point a1, G2Point a2, G1Point b1, G2Point b2) internal returns (bool) { G1Point[] memory p1 = new G1Point[](2); G2Point[] memory p2 = new G2Point[](2); p1[0] = a1; p1[1] = b1; p2[0] = a2; p2[1] = b2; return pairing(p1, p2); } function hashToG1(bytes message) internal returns (G1Point) { uint256 h = uint256(keccak256(message)); return mul(P1(), h); } function modPow(uint256 base, uint256 exponent, uint256 modulus) internal returns (uint256) { uint256[6] memory input = [32, 32, 32, base, exponent, modulus]; uint256[1] memory result; assembly { if iszero(call(not(0), 0x05, 0, input, 0xc0, result, 0x20)) { revert(0, 0) } } return result[0]; } /// @return the negation of p, i.e. p.add(p.negate()) should be zero. function negate(G1Point p) internal returns (G1Point) { // The prime q in the base field F_q for G1 uint q = 21888242871839275222246405745257275088696311157297823662689037894645226208583; if (p.X == 0 && p.Y == 0) return G1Point(0, 0); return G1Point(p.X, q - (p.Y % q)); } /// @return the sum of two points of G1 function add(G1Point p1, G1Point p2) internal returns (G1Point r) { uint[4] memory input; input[0] = p1.X; input[1] = p1.Y; input[2] = p2.X; input[3] = p2.Y; bool success; assembly { success := call(sub(gas, 2000), 6, 0, input, 0xc0, r, 0x60) // Use "invalid" to make gas estimation work switch success case 0 {invalid} } require(success); } /// @return the product of a point on G1 and a scalar, i.e. /// p == p.mul(1) and p.add(p) == p.mul(2) for all points p. function mul(G1Point p, uint s) internal returns (G1Point r) { uint[3] memory input; input[0] = p.X; input[1] = p.Y; input[2] = s; bool success; assembly { success := call(sub(gas, 2000), 7, 0, input, 0x80, r, 0x60) // Use "invalid" to make gas estimation work switch success case 0 {invalid} } require(success); } }上麵包含了整個 Solidity 文件，以幫助避免死連結。
如何在智能合約中實現加密算法中提到的solcrypto庫可能會有所幫助，包括 https://ethresear.ch/t/precompiled-snark-pairing-for-bls-signatures/3196/5中的註釋
要使用等效的 ECPAIRING，您可以執行以下操作：
c = pairing(G2, neg(S)) assert a * c == FQ12.one()聚合它們：
y = randint(1, p-1) # second secret key Q = multiply(G2, y) # second public key T = multiply(H_m, y) # second signature d = pairing(add(P, Q), double(H_m)) e = pairing(double(G2, add(S,T)) assert d == e要以 ECPAIRING 樣式驗證聚合：
d * pairing(double(G2) neg(add(S,T))) == FQ12.one()