Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes

TL;DR

This paper presents a polynomial-time attack on the McEliece cryptosystem when based on subcodes of algebraic geometry codes, exploiting their distinguishability via the Schur product and introducing the concept of t-closure.

Contribution

It introduces the t-closure of a code and extends cryptanalysis techniques to algebraic geometry codes beyond genus zero.

Findings

Successfully attacks the McEliece cryptosystem with AG codes

Shows distinguishability of AG codes from random codes using Schur product

Extends previous genus zero results to higher genera

Abstract

We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introducing and using a new notion, which we call the t-closure of a code.