Search problems in groups and branching processes

TL;DR

This paper investigates the complexity of random Dehn search problems in finitely presented groups, demonstrating that most instances are computationally easy and that cryptographic keys can be efficiently broken.

Contribution

It introduces a probabilistic analysis using Crump-Mode-Jagers processes to show the typical difficulty of these problems and implications for cryptography.

Findings

Most random instances are easy to solve.

Cryptographic keys in Wagner-Wagner protocol can be broken efficiently.

Analysis applies probabilistic models to group-theoretic problems.

Abstract

In this paper we study complexity of randomly generated instances of Dehn search problems in finitely presented groups. We use Crump-Mode-Jagers processes to show that most of the random instances are easy. Our analysis shows that for any choice of a finitely presented platform group in Wagner-Wagner public key encryption protocol the majority of random keys can be broken by a polynomial time algorithm.