Quasi-optimal Permutation Ranking and Applications to PERK

A ranking function for permutations maps every permutation of length n to a unique integer between 0 and \(n!-1\) . For permutations of size that are of interest in cryptographic applications, evaluating such a function requires multiple-precision arithmetic. This work introduces a quasi-optimal ranking technique that allows us to rank a permutation efficiently without needing a multiple-precision arithmetic library. We present experiments that show the computational advantage of our method compared to the standard lexicographic optimal permutation ranking. As an application of our result, we show how this technique improves the signature sizes and the efficiency of PERK digital signature scheme.