The Hidden Subgroup Problem and Post-quantum Group-based Cryptography

TL;DR

This paper explores the Hidden Subgroup Problem's role in post-quantum cryptography, reviewing its complexity, solution methods, and the current state of group-based cryptosystems, highlighting open challenges.

Contribution

It provides a comprehensive review of HSP in the context of post-quantum cryptography, including complexity analysis and the status of algorithms for infinite groups.

Findings

Efficient algorithms for infinite group HSP are not yet known.

The relationship between HSP and other computational problems is clarified.

Current platforms for group-based cryptosystems are overviewed.

Abstract

In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in such infinite group platforms are not yet known.