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=22m+1 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)loglog(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{}(2|X|) field operations.Finally, we give a Las Vegas algorithm that, given h〈X〉=Sz(q) for some h∈GL(4,q), finds some g such that g〈X〉=Sz(q). The running time is O{}(log(q)loglog(q)+|X|) field operations.Implementations of the algorithms are available for the computer system Magma.