elliptic-curve-solidity

elliptic-curve-solidity is an open source implementation of Elliptic Curve arithmetic operations written in Solidity.

DISCLAIMER: This is experimental software. Use it at your own risk!

The solidity contracts have been generalized in order to support any elliptic curve based on prime numbers up to 256 bits.

elliptic-curve-solidity has been designed as a library with only pure functions aiming at decreasing gas consumption as much as possible. Additionally, gas consumption comparison can be found in the benchmark section. This library does not check whether the points passed as arguments to the library belong to the curve. However, the library exposes a method called isOnCurve that can be utilized before using the library functions.

It contains 2 solidity libraries:

1. EllipticCurve.sol: provides main elliptic curve operations in affine and Jacobian coordinates.
2. FastEcMul.sol: provides a fast elliptic curve multiplication by using scalar decomposition and wNAF scalar representation.

EllipticCurve library provides functions for:

• Modular
  • inverse
  • exponentiation
• Jacobian coordinates
  • addition
  • double
  • multiplication
• Affine coordinates
  • inverse
  • addition
  • subtraction
  • multiplication
• Auxiliary
  • conversion to affine coordinates
  • derive coordinate Y from compressed EC point
  • check if EC point is on curve

FastEcMul library provides support for:

• Scalar decomposition
• Simultaneous multiplication (computes 2 EC multiplications using wNAF scalar representation)

Supported curves

The elliptic-curve-solidity contract supports up to 256-bit curves. However, it has been extensively tested for the following curves:

• secp256k1
• secp224k1
• secp192k1
• secp256r1 (aka P256)
• secp192r1 (aka P192)
• secp224r1 (aka P224)

Known limitations:

• deriveY function do not work with the curves secp224r1 and secp224k1 because of the selected derivation algorithm. The computations for this curve are done with a modulo prime p such that p mod 4 = 1, thus a more complex algorithm is required (e.g. Tonelli-Shanks algorithm). Note that deriveY is just an auxiliary function, and thus does not limit the functionality of curve arithmetic operations.
• the library only supports elliptic curves with cofactor = 1 (all supported curves have a cofactor = 1).

Usage

EllipticCurve.sol library can be used directly by importing it.

The Secp256k1 example depicts how to use the library by providing a function to derive a public key from a secret key:

pragma solidity 0.6.12;

import "elliptic-curve-solidity/contracts/EllipticCurve.sol";

contract Secp256k1 {

  uint256 public constant GX = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798;
  uint256 public constant GY = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8;
  uint256 public constant AA = 0;
  uint256 public constant BB = 7;
  uint256 public constant PP = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F;

  function derivePubKey(uint256 privKey) public pure returns(uint256 qx, uint256 qy) {
    (qx, qy) = EllipticCurve.ecMul(
      privKey,
      GX,
      GY,
      AA,
      PP
    );
  }
}

The cost of a key derivation operation in Secp256k1 is around 550k gas.

·--------------------------------------------------|--------------------------·
|                      Gas                         · Block limit: 6721975 gas │
···················································|···························
|                 ·          100 gwei/gas           ·     592.30 usd/eth      │
··················|··········|··········|··········|············|··············
|  Method         ·  Min     ·  Max     ·  Avg     ·  # calls   ·  usd (avg)  │
··················|··········|··········|··········|············|··············
|  derivePubKey   ·  476146  ·  518863  ·  499884  ·        18  ·      29.61  │
··················|··········|··········|··········|············|··············

The cost of a simultaneous multiplication (using wNAF) consumes around 35% of the gas required by 2 EC multiplications.

Benchmark

Gas consumption and USD price estimation with a gas price of 100 Gwei, derived from ETH Gas Station:

·----------------------------------------|---------------------------|-------------|----------------------------·
|  Solc version: 0.6.12+commit.27d51765  ·  Optimizer enabled: true  ·  Runs: 200  ·  Block limit: 6718946 gas  │
·········································|···························|·············|·····························
|  Methods                               ·              100 gwei/gas               ·       613.52 usd/eth       │
··················|······················|·············|·············|·············|··············|··············
|  Contract       ·  Method              ·  Min        ·  Max        ·  Avg        ·  # calls     ·  usd (avg)  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  decomposeScalar     ·      55811  ·      65399  ·      61939  ·         134  ·       3.80  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  deriveY             ·      45275  ·      55545  ·      50410  ·           4  ·       3.09  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  ecAdd               ·      24305  ·      56323  ·      49119  ·         472  ·       3.01  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  ecInv               ·      22906  ·      23074  ·      22990  ·           2  ·       1.41  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  ecMul               ·      24911  ·     623087  ·     350939  ·         561  ·      21.53  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  ecSimMul            ·      76465  ·     488165  ·     243763  ·         125  ·      14.96  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  ecSub               ·      42634  ·      56236  ·      49717  ·         228  ·       3.05  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  invMod              ·      22153  ·      49255  ·      39627  ·          12  ·       2.43  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  isOnCurve           ·      23400  ·      24071  ·      23623  ·          16  ·       1.45  │
··················|······················|·············|·············|·············|··············|··············
|  EllipticCurve  ·  toAffine            ·      50145  ·      50850  ·      50498  ·           4  ·       3.10  │
·----------------------------------------|-------------|-------------|-------------|--------------|-------------·

Acknowledgements

Some functions of the contract are based on:

• Comparatively Study of ECC and Jacobian Elliptic Curve Cryptography by Anagha P. Zele and Avinash P. Wadhe
• Numerology by NuCypher
• solidity-arithmetic by Gnosis
• ecsol written by Jordi Baylina
• standard contracts written by Andreas Olofsson

License

elliptic-curve-solidity is published under the MIT license.