Solidity

在 remix 中編譯 BLS 簽名時出錯？

  • June 10, 2019

Etherum 可以驗證 BLS 簽名，在這個答案中 Ethereum 是否支持配對操作？

有一個如何驗證 BLS 的可靠性範例。但是，當我嘗試在混音中測試程式碼時，更新：這是整個契約

          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);
}

}

我在混音中遇到以下 2 個錯誤：

browser/Pairing.sol:187:13: ParserError: Literal, identifier or instruction 
expected. if iszero(call(not(0), 0x05, 0, input, 0xc0, result, 0x20)) {

和

browser/Pairing.sol:187:16: ParserError: Expected token LParen got 
'Identifier'  if iszero(call(not(0), 0x05, 0, input, 0xc0, result, 0x20)) { 
 ^

你能幫我弄清楚如何解決這些錯誤嗎？

原始程式碼合約可在 github https://gist.github.com/BjornvdLaan/ca6dd4e3993e1ef392f363ec27fe74c4

更改 Remix 中的編譯器版本。如果您設置0.4.24或0.4.25（例如）程式碼會出現一些警告。如果要使用編譯器版本0.5.0或更高版本，則需要對程式碼進行一些修改。

修改程式碼0.5.x（帶有警告）：

pragma solidity ^0.5.0;

/*
  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 memory) {
   return G1Point(1, 2);
}

/// @return the generator of G2
function P2() internal returns (G2Point memory) {
   return G2Point(
       [11559732032986387107991004021392285783925812861821192530917403151452391805634,
       10857046999023057135944570762232829481370756359578518086990519993285655852781],

       [4082367875863433681332203403145435568316851327593401208105741076214120093531,
       8495653923123431417604973247489272438418190587263600148770280649306958101930]
   );
}

//Example of BLS signature verification
function verifyBLSTest() public returns (bool) {

   bytes memory message = hex"7b0a2020226f70656e223a207b0a20202020227072696365223a2039353931372c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333134323430302c0a2020202020202269736f223a2022323031362d31322d33315430303a30303a30302e3030305a220a202020207d0a20207d2c0a202022636c6f7365223a207b0a20202020227072696365223a2039363736302c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333232383830302c0a2020202020202269736f223a2022323031372d30312d30315430303a30303a30302e3030305a220a202020207d0a20207d2c0a2020226c6f6f6b7570223a207b0a20202020227072696365223a2039363736302c0a20202020226b223a20312c0a202020202274696d65223a207b0a20202020202022756e6978223a20313438333232383830302c0a2020202020202269736f223a2022323031372d30312d30315430303a30303a30302e3030305a220a202020207d0a20207d0a7d0a6578616d706c652e636f6d2f6170692f31";

   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() public 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() public 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"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";

   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[] memory p1, G2Point[] memory 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 memory a1, G2Point memory a2, G1Point memory b1, G2Point memory 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 memory message) internal returns (G1Point memory) {
   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 memory p) internal returns (G1Point memory) {
   // 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 memory p1, G1Point memory p2) internal returns (G1Point memory 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 memory p, uint s) internal returns (G1Point memory 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);
   }
}

引用自：https://ethereum.stackexchange.com/questions/71655

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