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## secp256k1 arithmetic

### Signatures

```solidity
function ecAddAffine(uint256 pointX1, uint256 pointY1, uint256 pointX2, uint256 pointY2)
    external
    pure
    returns (uint256 resultX, uint256 resultY);

function ecAddProjective(
    uint256 pointX1,
    uint256 pointY1,
    uint256 pointZ1,
    uint256 pointX2,
    uint256 pointY2,
    uint256 pointZ2
) external pure returns (uint256 resultX, uint256 resultY, uint256 resultZ);

function ecMulAffine(uint256 pointX, uint256 pointY, uint256 scalar)
    external
    pure
    returns (uint256 resultX, uint256 resultY);

function ecMulProjective(uint256 pointX, uint256 pointY, uint256 pointZ, uint256 scalar)
    external
    pure
    returns (uint256 resultX, uint256 resultY, uint256 resultZ);
```

### Description

These cheatcodes add and multiply points on the secp256k1 curve. Except for the point-at-infinity encodings described below, coordinates must be valid field elements and represent points on the curve.

* `ecAddAffine` and `ecAddProjective` add two points.
* `ecMulAffine` and `ecMulProjective` multiply a point by a scalar. The scalar is reduced modulo the secp256k1 group order.

#### Coordinate representations

The affine point at infinity is represented as `(0, 0)`.

Projective inputs use homogeneous coordinates: `(X, Y, Z)` represents the affine point `(X / Z, Y / Z)` over the secp256k1 base field. These are not Jacobian coordinates, which use `(X / Z², Y / Z³)`.

For projective inputs, the point at infinity is `(0, y, 0)` for any non-zero `y`. Projective results are normalized: finite points return `(x, y, 1)`, and the point at infinity returns `(0, 1, 0)`.

:::note
Use the affine cheatcodes unless you already have a point in homogeneous projective coordinates.
:::