Pseudorandom number generation by $p$-adic ergodic transformations

Abstract

The paper study counter-dependent pseudorandom generators; the latter are generators such that their state transition function (and output function) is being modified dynamically while working: For such a generator the recurrence sequence of states satisfies a congruence $x_{i+1}\equiv f_i(x_i)\pmod{2^n}$, while its output sequence is of the form $z_{i}=F_i(u_i)$. The paper introduces techniques and constructions that enable one to compose generators that output uniformly distributed sequences of a maximum period length and with high linear and 2-adic spans. The corresponding stream chipher is provably strong against a known plaintext attack (up to a plausible conjecture). Both state transition function and output function could be key-dependent, so the only information available to a cryptanalyst is that these functions belong to some (exponentially large) class. These functions are compositions of standard machine instructions (such as addition, multiplication, bitwise logical operations, etc.) The compositions should satisfy rather loose conditions; so the corresponding generators are flexible enough and could be easily implemented as computer programs.