64 lines
4.3 KiB
Markdown
64 lines
4.3 KiB
Markdown
# 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.
|