# Ternary Syndrome Decoding with Large Weight

## TL;DR

This paper investigates the ternary Syndrome Decoding problem with large weight, revealing its increased difficulty over binary cases and updating cryptosystem parameters accordingly.

## Contribution

It provides the first thorough algorithmic analysis of large-weight ternary Syndrome Decoding and demonstrates its hardness compared to binary decoding.

## Key findings

- Large-weight ternary Syndrome Decoding is significantly harder than binary decoding.
- Updated parameters for the Wave signature scheme based on new algorithmic insights.
- Potential for improved security in code-based cryptosystems using ternary codes.

## Abstract

The Syndrome Decoding problem is at the core of many code-based cryptosystems. In this paper, we study ternary Syndrome Decoding in large weight. This problem has been introduced in the Wave signature scheme but has never been thoroughly studied. We perform an algorithmic study of this problem which results in an update of the Wave parameters. On a more fundamental level, we show that ternary Syndrome Decoding with large weight is a really harder problem than the binary Syndrome Decoding problem, which could have several applications for the design of code-based cryptosystems.