Generalizing Syndrome Decoding problem to the totally Non-negative Grassmannian

TL;DR

This paper extends syndrome decoding to codes derived from the totally non-negative Grassmannian, connecting algebraic geometry with coding theory, and analyzes decoding complexity and failure probabilities.

Contribution

It introduces a novel generalization of syndrome decoding for Grassmannian-based codes, linking boundary measurement maps to Tanner graph constructions and deriving new analytical bounds.

Findings

Complexity increases with size of Plücker coordinates.

Decoding failure probability analyzed and compared with LDPC codes.

New bounds on parameters of Grassmannian-based codes derived.

Abstract

The syndrome decoding problem has been proposed as a computational hardness assumption for code based cryptosystem that are safe against quantum computing. The problem has been reduced to finding the codeword with the smallest non-zero columns that would satisfy a linear check equation. Variants of Information set decoding algorithms has been developed as cryptanalytic tools to solve the problem. In this paper, we study and generalize the solution to codes associated with the totally non-negative Grassmannian in the Grassmann metric. This is achieved by reducing it to an instance of finding a subset of the plucker coordinates with the smallest number of columns. Subsequently, the theory of the totally non negative Grassmann is extended to connect the concept of boundary measurement map to Tanner graph like code construction while deriving new analytical bounds on its parameters. The derived bounds shows that the complexity scales up on the size of the plucker coordinates. Finally, experimental results on decoding failure probability and complexity based on row operations are presented and compared to Low Density parity check codes in the Hamming metric.