Recognising the small Ree groups in their natural representations

TL;DR

This paper introduces polynomial-time Las Vegas algorithms for recognizing and testing membership in small Ree groups in their natural 7-dimensional representations, with implementations available in MAGMA.

Contribution

It provides the first efficient algorithms for constructive recognition and membership testing of Ree groups in their natural representations.

Findings

Algorithms are polynomial time given a discrete logarithm oracle.

Constructive recognition and membership testing are practically implementable.

Algorithms are available in MAGMA for real-world use.

Abstract

We present Las Vegas algorithms for constructive recognition and constructive membership testing of the Ree groups 2G_2(q) = Ree(q), where q = 3^{2m + 1} for some m > 0, in their natural representations of degree 7. The input is a generating set X. The constructive recognition algorithm is polynomial time given a discrete logarithm oracle. The constructive membership testing consists of a pre-processing step, that only needs to be executed once for a given X, and a main step. The latter is polynomial time, and the former is polynomial time given a discrete logarithm oracle. Implementations of the algorithms are available for the computer algebra system MAGMA.