Statistical Analysis of the Chaos-Driven Elliptic Curve Pseudo-random Number Generators

Abstract

In this paper, after a short survey describing several known constructions recommended for generating sequences of pseudo-random numbers based on elliptic curves over finite fields of prime order, we propose a method of generating such sequences of points with algorithms driven by a chaotic map. Our construction improves randomness of the sequence generated since it combines good statistical properties of an ECPRNG (Elliptic Curve Pseudo-Random Number Generator) and a CPRNG (Chaotic Pseudo-Random Number Generator). Theoretical analysis shows that periods of the proposed constructions are longer than in the case of the ECPRNG without modulation by a chaotic map. In the second part of the paper we present numerical analysis of the proposed construction to obtain optimal parameters of the generator. We also use some tests from the NIST’s SP 800-22 Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications to analyze statistical properties of the proposed constructions for different values of parameters.