1 research outputs found
Pattern occurrence in the dyadic expansion of square root of two and an analysis of pseudorandom number generators
Recently, designs of pseudorandom number generators (PRNGs) using
integer-valued variants of logistic maps and their applications to some
cryptographic schemes have been studied, due mostly to their ease of
implementation and performance. However, it has been noted that this ease is
reduced for some choices of the PRNGs accuracy parameters. In this article, we
show that the distribution of such undesirable accuracy parameters is closely
related to the occurrence of some patterns in the dyadic expansion of the
square root of 2. We prove that for an arbitrary infinite binary word, the
asymptotic occurrence rate of these patterns is bounded in terms of the
asymptotic occurrence rate of zeroes. We also present examples of infinite
binary words that tightly achieve the bounds. As a consequence, a classical
conjecture on asymptotic evenness of occurrence of zeroes and ones in the
dyadic expansion of the square root of 2 implies that the asymptotic rate of
the undesirable accuracy parameters for the PRNGs is at least 1/6.Comment: 21 pages, extended abstract presented in FPSAC 200