Uniformly distributed sequences of p-adic integers, II

TL;DR

This paper characterizes ergodic and measure-preserving functions on p-adic integers, providing explicit formulas especially for p=2, and applies these results to design pseudorandom generators with maximal period and analyze their linear complexity.

Contribution

It offers a detailed description of ergodic and measure-preserving functions on p-adic integers, including explicit formulas for p=2, and introduces nonlinear pseudorandom generators with optimal properties.

Findings

Explicit formulas for ergodic functions when p=2.

Construction of pseudorandom generators with maximal period.

Analysis of the linear complexity of generated sequences.

Abstract

The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In some cases (and especially for p=2) the descriptions are given by explicit formulae. Some of the results may be viewed as descriptions of ergodic isometric dynamical systems on p-adic unit disk. The study was motivated by the problem of pseudorandom number generation for computer simulation and cryptography. From this view the paper describes nonlinear congruential pseudorandom generators modulo M which produce stricly periodic uniformly distributed sequences modulo M with maximal possible period length (i.e., exactly M). Both the state change function and the output function of these generators could be, e.g., meromorphic functions (in particular, polynomials with rational, but not necessarily integer coefficients, or rational functions), or compositions of arithmetical operations (like addition, multiplication, exponentiation, raising to integer powers, including negative ones) with standard computer operations, such as bitwise logical operations (XOR, OR, AND, etc.). The linear complexity of the produced sequences is also studied.