Pseudo-Random Number Generation using β

Pseudo-Random Number Generation using β-encoder CMOS circuit Yutaka Jitsumatsu1 , Kazuya Matsumura1 , Tohru Kohda1,2 , and Kazuyuki Aihara3 1 2 3

Dept. of Informatics, Kyushu University First Aihara Innovative Mathematical Modelling Project Institute of Industrial Science, University of Tokyo.

Abstract An analog-to-digital (A/D) converter based on pulse code modulation (PCM) is sensitive to the fluctuation of threshold value of a quantizer in the A/D converter. A β-encoder was proposed to improve the sensitivity of PCM-based A/D converters [1]. Dynamics of β-encoder is governed by β-expansion map, which is defined as { βx, for x < γν, Cβ,ν (x) = βx − 1, for x ≥ γν. Its associated quantizer is defined by Qν (x) = 0 for x < ν and Qν (x) = 1 for x ≥ ν, where ν is a unknown threshold value. β encoder has a self-correction property for fluctuations of β and ν. Implementation as well as mathematical analysis of β encoder have been researched extensively [2]. Such a situation motivates us to use a β-encoder as a robust pseudo-random number generator (PRNG). Advantage of PRNG based on β-encoder is that it can be implemented in an extremely small CMOS circuit with low power consumption as well as it can allow offsets of β and ν. In order to use sequences generated by such a PRNG as physical random numbers, statistical test must be examined. As a preliminary work, we gave a statistical test for PRNG using β encoders by numerical simulations [3]. In this work, we obtain several binary sequences from a β-encoder CMOS circuit and perform its statistical test. Simulation results shows that a set of binary sequences generated by a β-encoder passes most of the NIST statistical test suite [4].

References [1] I. Daubechies, R. DeVore, C. G¨ unt¨ urk, and V. Vaishampayan, “Beta expansions: a new approach to digitally corrected A/D conversion,” Proc. IEEE Int. Symp. Curc. Syst. 2002 (ISCAS2002), vol.2, pp. 784-787, 2002. [2] T. Kohda, Y. Horio, and K. Aihara, “β-expansion’s attractors observed in A/D converters,” IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical, pp. 90-91, Dec. 2011. [3] Y. Hirata, Y. Jitsumatsu, T. Kohda, and K. Aihara, “Pseudo-Random number generator using β-expansion attractors in A/D converters,” Proc. of the 30th Sympo. on Cryptography and Information Security (SCIS2013), Jan. 2013. [4] A. Rukhin, et.al, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST, Special Publication 800-22, May 2001.