[Paper Review] Black box groups isomorphic to ${ m PGL}(2,2^e)$

William M. Kantor, Martin Kassabov|arXiv (Cornell University)|Sep 15, 2013

Finite Group Theory Research6 references3 citations

TL;DR

This paper presents a deterministic polynomial-time algorithm that constructs an explicit isomorphism between a black box group isomorphic to ${\rm PGL}(2,2^e)$ and the abstract group ${\rm PGL}(2,2^e)$, using involution-producing formulas and field recovery techniques. The key contribution is a black box recognition algorithm running in $O(\mu e^3 \log e)$ time, with efficient straight-line program generation for group elements.

ABSTRACT

A deterministic polynomial-time algorithm constructs an isomorphism between ${ m PGL}(2,2^e)$ and a black box group to which it is isomorphic.

Motivation & Objective

• To develop a deterministic polynomial-time algorithm for recognizing black box groups isomorphic to ${\rm PGL}(2,2^e)$.
• To construct an effective isomorphism between such a black box group and the abstract ${\rm PGL}(2,2^e)$ group.
• To overcome the longstanding obstacle of lacking deterministic involution generation in black box groups of even characteristic.
• To enable efficient computation of straight-line programs for arbitrary group elements using new generators.

Proposed method

• Leverages a novel involution-producing formula (Proposition 4) that deterministically generates involutions in a black box group isomorphic to ${\rm PGL}(2,2^e)$.
• Uses the involution to construct a Borel subgroup and recover a field $\mathfrak{F} \cong \mathbb{F}_{2^e}$ inside the black box group.
• Applies Proposition 9 to recover matrix entries using rational expressions in square roots of field elements derived from group operations.
• Employs straight-line programs (SLPs) to compute images and preimages of group elements efficiently.
• Uses minimal polynomial computation to define the field $\mathbb{F}_{2^e} = \mathbb{F}_2[s]$ for the abstract group representation.
• Relies on group-theoretic properties such as conjugation actions and trace identities to reconstruct matrix entries from group elements.

Experimental results

Research questions

• RQ1 Can a deterministic polynomial-time algorithm construct an isomorphism between a black box group and ${\rm PGL}(2,2^e)$?
• RQ2 Is it possible to generate involutions in a black box group isomorphic to ${\rm PGL}(2,2^e)$ without relying on discrete logarithms or probabilistic methods?
• RQ3 Can the field $\mathbb{F}_{2^e}$ be effectively recovered from group operations within the black box?
• RQ4 What is the computational complexity of generating straight-line programs for arbitrary elements in such groups?
• RQ5 Can matrix entries be reconstructed from group elements using only group operations and field arithmetic?

Key findings

• The algorithm constructs an isomorphism in $O(\mu e^3 \log e)$ time, where $\mu$ is the time per group operation.
• A 3-element generating set for the black box group and the abstract ${\rm PGL}(2,2^e)$ is produced, with an effective isomorphism between them.
• Straight-line programs of length $O(e)$ can be computed in $O(\mu e^3)$ time for both directions: from ${\rm PGL}(2,2^e)$ to $G$ and vice versa.
• The field $\mathbb{F}_{2^e}$ is recovered entirely within the black box group using group operations and involution-based constructions.
• Matrix entries of any group element can be reconstructed using rational expressions involving square roots of field elements derived from specific group products.
• The algorithm avoids discrete logarithm oracles, relying only on group operations and field arithmetic, making it applicable in purely black box settings.