A Simple Unpredictable Pseudo-Random Number Generator

Abstract

Two closely-related pseudo-random sequence generators are presented: The ${1 / P}$generator, with input P a prime, outputs the quotient digits obtained on dividing 1 by P. The $x^2 \bmod N$generator with inputs N, $x_0 $ (where $N = P \cdot Q$ is a product of distinct primes, each congruent to 3 mod 4, and $x_0 $ is a quadratic residue $\bmod N$), outputs $b_0 b_1 b_2 \cdots $ where $b_i = {\operatorname{parity}}(x_i )$ and $x_{i + 1} = x_i^2 \bmod N$. From short seeds each generator efficiently produces long well-distributed sequences. Moreover, both generators have computationally hard problems at their core. The first generator’s sequences, however, are completely predictable (from any small segment of $2|P| + 1$ consecutive digits one can infer the “seed,” P, and continue the sequence backwards and forwards), whereas the second, under a certain intractability assumption, is unpredictable in a precise sense. The second generator has additional interesting properties: from knowledge of $x_0 $ and N but notP or Q, one can generate the sequence forwards, but, under the above-mentioned intractability assumption, one can not generate the sequence backwards. From the additional knowledge of P and Q, one can generate the sequence backwards; one can even “jump” about from any point in the sequence to any other. Because of these properties, the $x^2 \bmod N$generator promises many interesting applications, e.g., to public-key cryptography. To use these generators in practice, an analysis is needed of various properties of these sequences such as their periods. This analysis is begun here.