Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

TL;DR

This paper introduces efficient methods for constructing and sampling fast forward permutations that are indistinguishable from random permutations, significantly reducing the complexity of certain permutation-based queries.

Contribution

It presents a novel approach to sampling the cycle structure of permutations and constructs fast forward permutations that remain indistinguishable from random ones even with advanced queries.

Findings

Number of queries needed to distinguish permutations is Theta(N) without special queries.

With P^m(x) queries, the complexity reduces to Theta(1).

Introduces an efficient sampling method for permutation cycle structures.

Abstract

A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$ is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation on {1,..,N} is Theta(N) if one does not use queries of the form P^m(x), but is only Theta(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form P^m(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.