# EXODUS - EXchange Of Digital Uniformed Signatures

## Overview

This module optimizes a hot spot in the verification formula used for missing‑signer recovery. The target expression is:

`k*P + l*Q + d*M`

which requires three scalar multiplications and two point additions. Naive elliptic‑curve routines make this very gas‑expensive; this work reduces cost while maintaining correctness.

For full background and protocol details see the [project wiki](https://git.ghostchain.io/ghostchain/ghost-exodus-draft/wiki/Description).

## Goals

* Reduce gas for the targeted combination of scalar multiplications and additions.
* Keep implementation compact and auditable for on‑chain use.
* Maintain correctness and safety for cryptographic operations.

## Design choices

* Use Projective coordinates (not Jacobian) to cut down on the number of `mulmod`/`addmod` operations where possible while retaining simple formulas for point addition and doubling.
* Perform the final conversion to affine coordinates with an optimized `Extended Euclidean Algorithm` implemented in inline assembly to reduce gas compared with high‑level inversion routines.
* Benchmark against the Jacobian implementation from the [witnet elliptic‑curve‑solidity project](https://github.com/witnet/elliptic-curve-solidity) as a reference.

## Rationale

* Projective coordinates: fewer modular multiplications in the common path, making point operations cheaper on average.
* Assembly `Extended Euclidean Algorithm` for inversion: this algorithm in optimized inline assembly typically yields lower gas for single inversions compared with repeated `mulmod` exponentiation or other higher‑level approaches.
* Comparing to a well‑maintained Jacobian implementation gives a meaningful baseline for gas and correctness.

## Gas Usage

`Jacobian` is the original implementation used as a reference implementation, while `Projective` is optimized one.

```
╭----------------------------------------+-----------------+-------+--------+-------+---------╮
| src/MathTester.sol:MathTester Contract |                 |       |        |       |         |
+=============================================================================================+
| Deployment Cost                        | Deployment Size |       |        |       |         |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| 1065323                                | 4715            |       |        |       |         |
|----------------------------------------+-----------------+-------+--------+-------+---------|
|                                        |                 |       |        |       |         |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| Function Name                          | Min             | Avg   | Median | Max   | # Calls |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| addJacobian                            | 1768            | 1768  | 1768   | 1768  | 44      |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| addProjective                          | 992             | 992   | 992    | 992   | 44      |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| doubleJacobian                         | 777             | 777   | 777    | 777   | 45      |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| doubleProjective                       | 532             | 532   | 532    | 532   | 45      |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| toAffineJacobian                       | 30564           | 36206 | 36300  | 40841 | 89      |
|----------------------------------------+-----------------+-------+--------+-------+---------|
| toAffineProjective                     | 11838           | 13792 | 13796  | 15155 | 89      |
╰----------------------------------------+-----------------+-------+--------+-------+---------╯
```

## Contributing

All contributions are welcome — whether it's code, documentation, tests, performance benchmarks, or review. Please submit commits, issues, or pull requests; any help to improve correctness, security, or gas efficiency is greatly appreciated.