Recognising the Suzuki groups in their natural representations

Under the assumption of a certain conjecture, for which there exists strong experimental evidence, we produce an efficient algorithm for constructive membership testing in the Suzuki groups Sz(q), where q = 2 for some m > 0, in their natural representations of degree 4. It is a Las Vegas algorithm with running time O(log(q)) field operations, and a preprocessing step with running time O(log(q) log log(q)) field operations. The latter step needs an oracle for the discrete logarithm problem in Fq. We also produce a recognition algorithm for Sz(q) = 〈X〉. This is a Las Vegas algorithm with running time O(|X|) field operations. Finally, we give a Las Vegas algorithm that, given 〈X〉 = Sz(q) for some h ∈ GL(4, q), finds some g such that 〈X〉 = Sz(q). The running time is O(log(q) log log(q) + |X|) field operations. Implementations of the algorithms are available for the computer system Magma.