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A survey about Hidden Subgroup Problem from a mathematical and cryptographic perspective

Simone Dutto, Pietro Mercuri, Nadir Murru, Lorenzo Romano

TL;DR

The paper surveys the Hidden Subgroup Problem as a unifying lens for post-quantum cryptography, highlighting that abelian HSP admits efficient quantum solutions via Kitaev’s framework and Fourier sampling, while non-abelian HSP lacks a general efficient algorithm. It systematically reviews key non-abelian groups—dihedral, symmetric, and semidirect products—and the specialized techniques (Fourier sampling, black-box methods) used to tackle HSP in these contexts, including links to lattice problems and graph isomorphism. It also details the problem reductions from cryptographic primitives (order finding, factorization, discrete log, code equivalence) to HSP, and discusses broader approaches, such as reductions, hidden-shift formulations, and black-box models. Overall, the work clarifies the mathematical machinery underpinning HSP and its cryptographic relevance, while outlining the current limits and directions for achieving quantum-resilient cryptosystems.

Abstract

We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, where Kitaev's algorithm yields an efficient quantum solution to the HSP, recalling how classical problems (such as order finding, integer factorization, and discrete logarithm) can be formulated as abelian HSP instances. We then examine the current state of the art for non-abelian HSP, where no general efficient quantum solution is known, focusing on some relevant groups including dihedral group (connected to the shortest vector problem), symmetric groups (connected to the graph isomorphism problem), and semidirect product constructions (connected, in a special case, to the code equivalence problem). We also describe the main techniques for addressing the HSP in non-abelian cases, namely Fourier sampling and the black-box approach. Throughout the paper, we highlight the mathematical notions required and exploited in this context, providing a cryptography-oriented perspective.

A survey about Hidden Subgroup Problem from a mathematical and cryptographic perspective

TL;DR

The paper surveys the Hidden Subgroup Problem as a unifying lens for post-quantum cryptography, highlighting that abelian HSP admits efficient quantum solutions via Kitaev’s framework and Fourier sampling, while non-abelian HSP lacks a general efficient algorithm. It systematically reviews key non-abelian groups—dihedral, symmetric, and semidirect products—and the specialized techniques (Fourier sampling, black-box methods) used to tackle HSP in these contexts, including links to lattice problems and graph isomorphism. It also details the problem reductions from cryptographic primitives (order finding, factorization, discrete log, code equivalence) to HSP, and discusses broader approaches, such as reductions, hidden-shift formulations, and black-box models. Overall, the work clarifies the mathematical machinery underpinning HSP and its cryptographic relevance, while outlining the current limits and directions for achieving quantum-resilient cryptosystems.

Abstract

We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, where Kitaev's algorithm yields an efficient quantum solution to the HSP, recalling how classical problems (such as order finding, integer factorization, and discrete logarithm) can be formulated as abelian HSP instances. We then examine the current state of the art for non-abelian HSP, where no general efficient quantum solution is known, focusing on some relevant groups including dihedral group (connected to the shortest vector problem), symmetric groups (connected to the graph isomorphism problem), and semidirect product constructions (connected, in a special case, to the code equivalence problem). We also describe the main techniques for addressing the HSP in non-abelian cases, namely Fourier sampling and the black-box approach. Throughout the paper, we highlight the mathematical notions required and exploited in this context, providing a cryptography-oriented perspective.

Paper Structure

This paper contains 27 sections, 18 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Group theory
  4. Representation theory
  5. Polynomial-time reductions
  6. Quantum computing
  7. Hidden Subgroup Problem
  8. Abelian HSP
  9. Simon's problem
  10. Order finding and integer factorization
  11. Discrete logarithm
  12. Discrete root
  13. DLP-like approach.
  14. Mixed approach.
  15. Exponent approach.
  16. ...and 12 more sections

Key Result

Theorem 2.7

Every finite abelian group $G$ can be written as where $N_1,N_2,\ldots,N_\ell$ are powers of (not necessarily distinct) primes.

Theorems & Definitions (76)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: Fundamental Theorem of Finite Abelian Groups
  • Definition 2.8
  • Lemma 2.9: Schur's lemma
  • Corollary 2.10
  • ...and 66 more