[Paper Review] Efficient Isomorphism Testing for a Class of Group Extensions

François Le Gall|arXiv (Cornell University)|Dec 12, 2008

Cryptography and Data Security25 references18 citations

TL;DR

This paper presents a deterministic, almost linear-time algorithm for testing isomorphism between groups in a specific class of non-abelian groups—extensions of an abelian group A by a cyclic group Zm, where |A| and m are coprime. The method leverages standard group decompositions and efficiently tests conjugacy of automorphisms induced by group generators, achieving a time complexity of (√n + γ)^{1+o(1)} for groups of size n and parameter γ.

ABSTRACT

A finite group of order n can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order n can be stored using O(n²) words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order n that uses o(n²) space but can still answer a multiplication query in constant time. We design a constant query-time data structure that can store any finite group using O(n) words where n is the order of the group. Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of Ω(n) on the number of words to store a group of order n. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).

Motivation & Objective

• To address the long-standing open problem of efficiently testing isomorphism for non-abelian groups, particularly in classes beyond abelian groups.
• To focus on a natural class of non-abelian groups: extensions of an abelian group A by a cyclic group Zm with gcd(|A|, m) = 1.
• To develop a deterministic algorithm that runs in almost linear time for black-box groups in this class.
• To provide a complete and efficient solution for isomorphism testing in this class, which includes many non-abelian groups and generalizes abelian group isomorphism.

Proposed method

• Introduce a standard decomposition of groups in class S as an invariant, ensuring isomorphic groups have similar decompositions.
• Use black-box group operations to compute standard decompositions efficiently in O(n^{1/2+o(1)}) time.
• Reduce the isomorphism problem to testing conjugacy of automorphisms of the abelian normal subgroup induced by the cyclic generator.
• Compute matrix representations of these automorphisms with respect to group bases, exploiting block-diagonal structure over Sylow subgroups.
• Test conjugacy of these matrices block-wise over Sylow p-subgroups using a specialized algorithm from Section 4.
• Construct an explicit isomorphism from the conjugacy data when found, using the isomorphism condition from Proposition 5.1.

Experimental results

Research questions

• RQ1 Can isomorphism between non-abelian groups in the class S be tested in almost linear time using black-box group operations?
• RQ2 Is there a structural invariant—such as a standard decomposition—that allows efficient reduction of the isomorphism problem to conjugacy testing in automorphism groups?
• RQ3 Can the conjugacy of automorphisms induced by group generators be tested efficiently in the context of group extensions with coprime orders?
• RQ4 Does the complexity of isomorphism testing in this class match or improve upon known bounds for abelian groups?
• RQ5 Can the algorithm be made deterministic and efficient without relying on exponential-time cohomology or automorphism group computations?

Key findings

• The paper presents a deterministic algorithm for group isomorphism testing in the class S with time complexity bounded by (√n + γ)^{1+o(1)}, where n is the size of the smaller input group and γ is the minimum of the orders of the cyclic generators.
• The algorithm runs in almost linear time, achieving O(n^{1/2+o(1)}) time when γ = O(n^{1/2}), matching the best known bound for abelian groups in the black-box model.
• The method relies on computing standard decompositions of groups in S, which are invariant under isomorphism and can be computed efficiently in O(n^{1/2+o(1)}) time.
• The core of the algorithm reduces isomorphism testing to checking conjugacy of automorphisms of the abelian normal subgroup, which is solved via matrix conjugacy over Sylow subgroups.
• The algorithm efficiently computes an explicit isomorphism when one exists, using a block-diagonal matrix conjugacy solution from Theorem 4.2.
• The approach avoids exponential-time computations in automorphism or cohomology groups, providing a rigorous and efficient alternative to existing heuristic or impractical methods.