Length-Based Attacks for Certain Group Based Encryption Rewriting Systems

TL;DR

This paper presents a length-based probabilistic attack on certain group-based public key cryptosystems, exploiting canonical representatives and length functions to compromise the conjugacy problem.

Contribution

It introduces a novel length attack method targeting group-based cryptosystems with known polynomial word problem solutions but hard conjugacy problems.

Findings

The attack successfully compromises braid group cryptosystems.

Canonical representatives enable effective length-based cryptanalysis.

The method highlights vulnerabilities in certain group-based encryption schemes.

Abstract

In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.