[Paper Review] Polynomial-time isomorphism testing of groups of most finite orders

Heiko Dietrich, James B. Wilson|arXiv (Cornell University)|Jun 22, 2018

Coding theory and cryptography30 references4 citations

TL;DR

This paper presents polynomial-time isomorphism testing algorithms for abelian and coprime meta-cyclic groups of black-box type, using a type-theoretic model to ensure input correctness. It proves that for a dense set of group orders, isomorphism can be tested in time polynomial in log n, significantly improving over prior exponential-time methods for black-box groups.

ABSTRACT

[PLEASE SEE COMMENT] We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for abelian groups of black-box type of order $n$ can be done in time polynomial in $\log n$. We also prove that for a dense set of orders $n$ with given prime factors, one can test isomorphism for coprime meta-cyclic groups of black-box type of order $n$ in time polynomial in $\log n$. Prior methods for these two classes of groups have running times exponential in $\log n$.

Motivation & Objective

• To develop efficient isomorphism testing algorithms for finite abelian and meta-cyclic groups in the black-box group model, where group operations are accessed via oracles.
• To overcome the limitations of prior exponential-time algorithms by identifying a dense set of group orders for which isomorphism testing becomes polynomial-time.
• To address the foundational issue of malformed inputs in black-box group algorithms by adopting a type-theoretic model that guarantees inputs are valid groups.
• To reconcile the practical use of black-box group algorithms with formal complexity theory by shifting from promise problems to decision problems via type safety.

Proposed method

• Leverages a type-theoretic model where all inputs are terms of a known group type, ensuring that every input is a valid group by construction, thus eliminating malformed inputs.
• Applies number-theoretic density results and group-theoretic constraints to define sets D and D̂ of integers for which group structures are constrained enough to allow efficient isomorphism testing.
• Uses the structure of groups of order n ∈ D (with small prime powers and large square-free parts) to reduce isomorphism testing to solving discrete logarithm and factorization problems in controlled settings.
• Employs Las Vegas randomized algorithms for meta-cyclic groups, with correctness guaranteed by the type system and efficiency ensured by the density and structure of the allowed orders.
• Introduces constructive membership testing that works on arbitrary inputs, not just those within a larger ambient group, enabling absolute membership checks without relying on promise problems.
• Relies on prior results from Erdős-Pálfi and Karagiorgos-Poulakis to characterize the density of orders with favorable group-theoretic properties, enabling the construction of efficient algorithms.

Experimental results

Research questions

• RQ1 Can isomorphism testing for abelian black-box groups be made polynomial-time for a dense set of group orders?
• RQ2 For which classes of non-abelian groups does isomorphism testing become feasible in polynomial time under the black-box model?
• RQ3 How can the foundational issue of malformed inputs in black-box group algorithms be resolved within a formal computational model?
• RQ4 To what extent can type theory be used to transform promise problems into decision problems in group-theoretic computation?
• RQ5 What structural constraints on group orders enable efficient isomorphism testing despite the general hardness of the problem?

Key findings

• For a dense set of integers D, isomorphism testing of abelian black-box groups of order n ∈ D runs in time polynomial in log n, significantly improving over prior exponential-time methods.
• The set D consists of integers n = p₁^e₁⋯pₖ^eₖb, where primes pᵢ ≤ log log n, pᵢ^eᵢ ≤ log n, and b is square-free with all prime divisors > log log n, and |D ∩ {1,…,10⁸}| / 10⁸ ≈ 0.848.
• For a dense subset D̂ ⊆ D, a polynomial-time Las Vegas algorithm tests isomorphism of coprime meta-cyclic black-box groups of order n ∈ D̂, with |D̂ ∩ {1,…,10⁸}| / 10⁸ ≈ 0.721.
• The type-theoretic model ensures that all inputs are valid groups, eliminating the need for promise problems and enabling correct, deterministic reasoning in proofs.
• Constructive membership testing is achieved as a by-product, allowing membership checks on arbitrary elements without assuming they lie in a larger ambient group.
• The results demonstrate that for most group orders, the isomorphism problem becomes tractable under the black-box model, despite the general problem being hard for non-deterministic Turing machines.