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๐Ÿงฎ EC256

Introductionโ€‹

This library provides elliptic curve arithmetic over a 256-bit prime field (Weierstrass curve y^2 = x^3 + ax + b (mod p)).

Functionsโ€‹

To use the EC256 library, you need to import it.

import "@solarity/solidity-lib/libs/crypto/EC256.sol";

And optionally bind it to the type with the using statement.

using EC256 for *;

basepointโ€‹

function basepoint(
    EC256.Curve memory ec
) internal pure returns (EC256.APoint memory aPoint_);

Descriptionโ€‹

Returns the generator (base) point of the curve in affine form.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters

jbasepointโ€‹

function jbasepoint(
    EC256.Curve memory ec
) internal pure returns (EC256.JPoint memory jPoint_);

Descriptionโ€‹

Returns the generator (base) point of the curve in jacobian form.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters

toScalarโ€‹

function toScalar(
    EC256.Curve memory ec,
    uint256 u256_
) internal pure returns (uint256 scalar_);

Descriptionโ€‹

Reduces an arbitrary uint256 into the scalar field [0, n).

Returns the result of u256_ mod n.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
u256uint256The integer to reduce

isOnCurveโ€‹

function isOnCurve(
    EC256.Curve memory ec,
    EC256.APoint memory aPoint_
) internal pure returns (bool result_);

Descriptionโ€‹

Checks whether an affine point lies on the curve.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
aPointstruct EC256.APointThe affine point to test

isValidScalarโ€‹

function isValidScalar(
    EC256.Curve memory ec,
    uint256 scalar_
) internal pure returns (bool result_);

Descriptionโ€‹

Checks whether a scalar is in the valid range [0, n).

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
scalaruint256The scalar to test

toAffineโ€‹

function toAffine(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint_
) internal view returns (EC256.APoint memory aPoint_);

Descriptionโ€‹

Converts a point from Jacobian to affine coordinates.

Returns the equivalent affine point (x, y).

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPointstruct EC256.JPointThe Jacobian point (X, Y, Z)

toJacobianโ€‹

function toJacobian(
    EC256.APoint memory aPoint_
) internal pure returns (EC256.JPoint memory jPoint_);

Descriptionโ€‹

Converts an affine point to Jacobian coordinates.

Returns the point in Jacobian representation (x, y, 1).

Parameters:โ€‹
NameTypeDescription
aPointstruct EC256.APointThe affine point (x, y)

isJacobianInfinityโ€‹

function isJacobianInfinity(
    EC256.JPoint memory jPoint_
) internal pure returns (bool result_);

Descriptionโ€‹

Checks whether a Jacobian point is the point at infinity.

Parameters:โ€‹
NameTypeDescription
jPointstruct EC256.JPointThe Jacobian point to test

jinfinityโ€‹

function jinfinity() internal pure returns (EC256.JPoint memory jPoint_);

Descriptionโ€‹

Returns the Jacobian representation of the point at infinity.

Returns the point at infinity (0, 0, 0).

jEqualโ€‹

function jEqual(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint1_,
    EC256.JPoint memory jPoint2_
) internal view returns (bool result_);

Descriptionโ€‹

Compares two Jacobian points for equality in affine coordinates.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPoint1struct EC256.JPointThe first Jacobian point
jPoint2struct EC256.JPointThe second Jacobian point

jMultShamirโ€‹

function jMultShamir(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint_,
    uint256 scalar_
) internal pure returns (EC256.JPoint memory jPoint2_);

Descriptionโ€‹

Point multiplication: R = u*P using 4-bit windowed method.

Returns the Jacobian representation of result point R.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPointstruct EC256.JPointTThe Jacobian point P
scalaruint256The scalar u

jMultShamir2โ€‹

function jMultShamir2(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint1_,
    EC256.JPoint memory jPoint2_,
    uint256 scalar1_,
    uint256 scalar2_
) internal pure returns (EC256.JPoint memory jPoint3_);

Descriptionโ€‹

Simultaneous double-scalar multiplication: R = u1P1 + u2P2 via Straussโ€“Shamir.

Returns the Jacobian representation of result point R.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPoint1struct EC256.JPointThe first Jacobian point P1
jPoint2struct EC256.JPointThe second Jacobian point P2
scalar1uint256The first scalar u1
scalar2uint256The second scalar u2

jAddPointโ€‹

function jAddPoint(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint1_,
    EC256.JPoint memory jPoint2_
) internal pure returns (EC256.JPoint memory jPoint3_);

Descriptionโ€‹

Adds two Jacobian points: R = P1 + P2.

Returns the Jacobian representation of result point R.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPoint1struct EC256.JPointThe first Jacobian point P1
jPoint2struct EC256.JPointThe second Jacobian point P2

jDoublePointโ€‹

function jDoublePoint(
    EC256.Curve memory ec,
    EC256.JPoint memory jPoint1_
) internal pure returns (EC256.JPoint memory jPoint2_);

Descriptionโ€‹

Doubles a Jacobian point: R = 2*P.

Returns the Jacobian representation of result point R.

Parameters:โ€‹
NameTypeDescription
ecstruct EC256.CurveThe curve parameters
jPointstruct EC256.JPointThe Jacobian point P to double

Exampleโ€‹

EC256.Curve public secp256k1CurveParams =
    EC256.Curve({
        a: 0x0000000000000000000000000000000000000000000000000000000000000000,
        b: 0x0000000000000000000000000000000000000000000000000000000000000007,
        gx: 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
        gy: 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8,
        p: 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,
        n: 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
    });

function affineInfinity() external view returns (EC256.APoint memory) {
    return secp256k1CurveParams.toAffine(EC256.jinfinity());
}

function basepoint() external view returns (EC256.APoint memory) {
    return secp256k1CurveParams.basepoint();
}

function checkBasepointAddition() external view returns (bool) {
    EC256.JPoint memory G_ = secp256k1CurveParams.jbasepoint();

    EC256.JPoint memory doubledG_ = secp256k1CurveParams.jDoublePoint(G_);
    EC256.JPoint memory scalarMultipliedG_ = secp256k1CurveParams.jMultShamir(G_, 2);

    return secp256k1CurveParams.jEqual(scalarMultipliedG_, doubledG_);
}