An approach to constructing a sequence of pseudorandom numbers based on decompositions of polyadic numbers

Abstract

The aim of the work is to construct pseudorandom number generators based on expansions of almost polyadic numbers in powers of a given number. A polyadic number is usually called a
series of the form Σ︀∞ 𝑛=0 𝑎𝑛𝑛!, where 0 ⩽ 𝑎𝑛 ⩽ 𝑛, 𝑎𝑛 is an integer. Series of this type, converging in all fields of 𝑝–adic numbers, except for a finite number of them, having rational coefficients, are called almost polyadic numbers.
We shall assume that 𝜆𝑖 = 𝑎𝑖 𝑏𝑖 , 𝑖 = 1, . . . ,𝑚, where 𝑎𝑖, 𝑏𝑖 are positive integers, N.O.D. (𝑎𝑖, 𝑏𝑖) = 1, 𝑖 = 1, . . . ,𝑚 and 𝜆𝑖 − 𝜆𝑗 ̸∈ Z for 𝑖 ̸= 𝑗. It can be shown that under these conditions the series
Σ︀∞ 𝑛=0 (𝜆𝑖)𝑛(𝑏𝑖)𝑛𝑍𝑛, 𝑖 = 1, . . . ,𝑚 are algebraically independent over the field of rational functions of 𝑧 [1].
This implies the infinite algebraic independence of polyadic numbers Σ︀∞ 𝑛=0 (𝜆𝑖)𝑛(𝑏𝑖)𝑛, 𝑖 = 1, . . . ,𝑚 [2].
It can be assumed that the expansion digits of the partial sums
Σ︀𝑁 𝑛=0 (𝜆𝑖)𝑛(𝑏𝑖)𝑛, 𝑖 = 1, . . . ,𝑚 of the series under consideration have good statistical properties. The article describes the results
of the experiments conducted.