Preprint Article Version 1 This version is not peer-reviewed

Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequence

Version 1 : Received: 29 July 2018 / Approved: 2 August 2018 / Online: 2 August 2018 (07:58:37 CEST)

A peer-reviewed article of this Preprint also exists.

Prasitsupparote, A.; Konno, N.; Shikata, J. Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequences. Entropy 2018, 20, 729. Prasitsupparote, A.; Konno, N.; Shikata, J. Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequences. Entropy 2018, 20, 729.

Journal reference: Entropy 2018, 20, 729
DOI: 10.3390/e20100729

Abstract

Many cryptographic systems require random numbers, and weak random numbers lead to insecure systems. In the modern world, there are several techniques for generating random numbers, of which the most fundamental and important methods are deterministic extractors proposed by von Neumann, Elias, and Peres. Elias’s extractor achieves the optimal rate (i.e., information theoretic upper bound) h(p) if the block size tends to infinity, where h(·) is the binary entropy function and p is probability that each bit of input sequences occurs. Peres’s extractor achieves the optimal rate h(p) if the length of input and the number of iterations tend to infinity. The previous researches related to both extractors did not mention practical aspects including running time and memory-size with finite input sequences. In this paper, based on some heuristics, we derive a lower bound on the maximum redundancy of Peres’s extractor, and we show that Elias’s extractor is better than Peres’s one in terms of the maximum redundancy (or the rates) if we do not pay attention to time complexity or space complexity. In addition, we perform numerical and non-asymptotic analysis of both extractors with a finite input sequence with any biased probability under the same environments. For doing it, we implemented both extractors on a general PC and simple environments. Our empirical results show that Peres’s extractor is much better than Elias’s one for given finite input sequences under the almost same running time. As a consequence, Peres’s extractor would be more suitable to generate uniformly random sequences in practice in applications such as cryptographic systems.

Subject Areas

true random number generation; von Neumann’s extractor; Peres’s extractor; Elias’s extractor

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