Elliptic Curve Operations in Solidity

Elliptic cureve point multiplication performed in projective coordinates.

In the specific contract the curve is alt-bn128 but it works for any curve.

To run points addition simply run:

function add(uint256 x0, uint256 y0,
             uint256 x1, uint256 y1) public pure
    returns(uint256 x2, uint256 y2)

To run double a point run:

function twice(uint256 x0, uint256 y0) public pure
    returns(uint256 x1, uint256 y1)

To multiply a point in some power of two scalar run:

function multiplyPowerBase2(uint256 x0, uint256 y0, 
                            int exp) public pure
    returns(uint256 x1, uint256 y1)

To multiply a point in any scalar run:

function multiplyScalar(uint256 x0, uint256 y0,
                        uint scalar) public pure
    returns(uint256 x1, uint256 y1)

Currently one addition or doubling a point is ~1600 gas.
There are precompiled contracts for the same operations in the curve bn128. It is more gas-wised to use the precompiled contracts for multiplying a base 2 power with exponent that is large (i.e., larger than ~30).
It is better to return to affine coordinates (if necessary) only at the end of the computation (since it requires division, thus costs more gas).

orbs-network/elliptic-curve-solidity