Fast Constructive Recognition of a Black Box Group Isomorphic toSnorAnusingGoldbach’s Conjecture

The well-known Goldbach Conjecture (GC) states that any sufficiently large even number can be represented as a sum of two odd primes. Although not yet demonstrated, it has been checked for integers up to 1014. Using two stronger versions of the conjecture, we offer a simple and fast method for recognition of a gray box group G known to be isomorphic to Sn(or An) with knownn ≥ 20, i.e. for construction1of an isomorphism from GtoSn (or An). Correctness and rigorous worst case complexity estimates rely heavily on the conjectures, and yield times of O([ρ + ν + μ ] nlog2n) or O([ ρ + ν + μ ] nlogn / loglogn) depending on which of the stronger versions of the GC is assumed to hold. Here,ρ is the complexity of generating a uniform random element of G, ν is the complexity of finding the order of a group element in G, and μ is the time necessary for group multiplication in G. Rigorous lower bound and probabilistic approach to the time complexity of the algorithm are discussed in the Appendix.