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Numerical and Non-Asymptotic Analysis of Elias’s and Peres’s Extractors with Finite Input Sequences^{†}

^{1}

Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama 240-8501, Japan

^{2}

Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan

^{*}

Author to whom correspondence should be addressed.

^{†}

This paper is an extended version of our paper published in 51st Annual Conference on Information Sciences and Systems (CISS), Baltimore, MD, USA, 22–24 March 2017.

Received: 29 July 2018 / Revised: 13 September 2018 / Accepted: 19 September 2018 / Published: 23 September 2018

(This article belongs to the Section Information Theory, Probability and Statistics)

Many cryptographic systems require random numbers, and the use of weak random numbers leads 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) $h\left(p\right)$ if the block size tends to infinity, where $h(\xb7)$ is the binary entropy function and p is the probability that each bit of input sequences occurs. Peres’s extractor achieves the optimal rate $h\left(p\right)$ if the length of the input and the number of iterations tend to infinity. Previous research related to both extractors has made no reference to 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 extractor in terms of the maximum redundancy (or the rates) if we do not pay attention to the 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. To do so, 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 extractor for given finite input sequences under a very similar 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.