A <I>&beta;</I>-ary to binary conversion for random number generation using a <I>&beta;</I> encoder

Abstract

Abstract:A β encoder is an analog-to-digital (A/D) converter, proposed by Daubechies et al. in 2002, that outputs a truncated sequence of β expansion of an input value x. It is known that the conventional pulse code modulation (PCM) that outputs the binary expansion of x is sensitive to the offset of the threshold voltage, while a β encoder is robust to such an offset. We propose an algorithm that calculates the binary expansion of an interval that is identified by an output sequence from a β encoder. Such a method is referred to as a β-ary to binary converter. We generate sequences of random numbers, using a hardware β encoder followed by the β-ary to binary converter. The randomness of the generated binary random numbers is verified by the National Institute of Standards and Technology (NIST) statistical test suite.

Pseudo-random number generation with $β$-encoders

“…It is verified in [JM16] that the resulting output sequences {a n } pass the NIST statistical test suite from [RSN + 01], which shows that this method performs well as a pseudo-random number generator. A natural question asked in [JM16] is the following: If we use u = (u n ) n≥1 to denote the consecutive (random) threshold values u n , what is the number k(m, u, x) of bits {b n } from the β-encoder that are necessary to obtain m digits in base 2 of the number x via this process? In [JM16] the lower bound k(m, u, x) ≥ m log 2 log β was found.…”

“…In recent years β-encoders were also considered as sources for random number generation, see [JMKA13,SJO15,JM16,KJ16]. If x is chosen uniformly at random in [0, 1], then the digits {a n (x)} n≥1 from (1) form a sequence of binary independent identically distributed random variables with P(a n = 0) = P(a n = 1) = 1 2 .…”

“…On the other hand, it is known that successive bits {b n } in the output of a β-encoder are correlated and therefore not immediately applicable as pseudo-random numbers. Under the assumption that the amplification factor β does not fluctuate, Jitsumatsu and Matsumura proposed in [JM16] a coding scheme which 'removes' the dependence between the bits and converts the output bits {b n } of the β-encoder into the binary digits {a n } in base 2 of the number it represents. It is verified in [JM16] that the resulting output sequences {a n } pass the NIST statistical test suite from [RSN + 01], which shows that this method performs well as a pseudo-random number generator.…”

“…Under the assumption that the amplification factor β does not fluctuate, Jitsumatsu and Matsumura proposed in [JM16] a coding scheme which 'removes' the dependence between the bits and converts the output bits {b n } of the β-encoder into the binary digits {a n } in base 2 of the number it represents. It is verified in [JM16] that the resulting output sequences {a n } pass the NIST statistical test suite from [RSN + 01], which shows that this method performs well as a pseudo-random number generator. A natural question asked in [JM16] is the following: If we use u = (u n ) n≥1 to denote the consecutive (random) threshold values u n , what is the number k(m, u, x) of bits {b n } from the β-encoder that are necessary to obtain m digits in base 2 of the number x via this process?…”

“…A natural question asked in [JM16] is the following: If we use u = (u n ) n≥1 to denote the consecutive (random) threshold values u n , what is the number k(m, u, x) of bits {b n } from the β-encoder that are necessary to obtain m digits in base 2 of the number x via this process? In [JM16] the lower bound k(m, u, x) ≥ m log 2 log β was found. 1 The authors of [JM16] remarked that a theoretical analysis of the expected value of k(m, u, x) is relevant as an indication of the efficiency of the proposed pseudo-random number generator.…”