Scrambled Linear Pseudorandom Number Generators

F 2 -linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer from linear artifacts that show as failures in linearity-related statistical tests such as the binary-rank and the linear-complexity test. In this article, we give two new contributions. First, we introduce two new F 2 -linear transformations that have been handcrafted to have good statistical properties and at the same time to be programmable very efficiently on superscalar processors, or even directly in hardware. Then, we describe some scramblers , that is, nonlinear functions applied to the state array that reduce or delete the linear artifacts, and propose combinations of linear transformations and scramblers that give extremely fast pseudorandom number generators of high quality. A novelty in our approach is that we use ideas from the theory of filtered linear-feedback shift registers to prove some properties of our scramblers, rather than relying purely on heuristics. In the end, we provide simple, extremely fast generators that use a few hundred bits of memory, have provable properties, and pass strong statistical tests.

Generalized Galois-Fibonacci Matrix Generators Pseudo-Random Sequences

The article discusses various options for constructing binary generators of pseudo-random numbers (PRN) based on the so-called generalized Galois and Fibonacci matrices. The terms "Galois matrix" and "Fibonacci matrix" are borrowed from the theory of cryptography, in which the linear feedback shift registers (LFSR) generators of the PRN according to the Galois and Fibonacci schemes are widely used. The matrix generators generate identical PRN sequences as the LFSR generators. The transition from classical to generalized matrix PRN generators (PRNG) is accompanied by expanding the variety of generators, leading to a significant increase in their cryptographic resistance. This effect is achieved both due to the rise in the number of elements forming matrices and because generalized matrices are synthesized based on primitive generating polynomials and polynomials that are not necessarily primitive. Classical LFSR generators of PRN (and their matrix equivalents) have a significant drawback: they are susceptible to Berlekamp-Messi (BM) attacks. Generalized matrix PRNG is free from BM attack. The last property is a consequence of such a feature of the BM algorithm. This algorithm for cracking classical LFSR generators of PRN solves the problem of calculating the only unknown – a primitive polynomial generating the generator. For variants of generalized matrix PRNG, it becomes necessary to determine two unknown parameters: both an irreducible polynomial and a forming element that produces a generalized matrix. This problem turns out to be unsolvable for the BM algorithm since it is designed to calculate only one unknown parameter. The research results are generalized for solving PRNG problems over a Galois field of odd characteristics.

Rational complexity of binary sequences, F$\mathbb {Q}$SRs, and pseudo-ultrametric continued fractions in $\mathbb {R}$

We introduce rational complexity, a new complexity measure for binary sequences. The sequence s ∈ Bω is considered as binary expansion of a real fraction $s \equiv {\sum }_{k\in \mathbb {N}}s_{k}2^{-k}\in [0,1] \subset \mathbb {R}$ s ≡ ∑ k ∈ ℕ s k 2 − k ∈ [ 0 , 1 ] ⊂ ℝ . We compute its continued fraction expansion (CFE) by the Binary CFE Algorithm, a bitwise approximation of s by binary search in the encoding space of partial denominators, obtaining rational approximations r of s with r → s. We introduce Feedback in$\mathbb {Q}$ ℚ Shift Registers (F$\mathbb {Q}$ ℚ SRs) as the analogue of Linear Feedback Shift Registers (LFSRs) for the linear complexity L, and Feedback with Carry Shift Registers (FCSRs) for the 2-adic complexity A. We show that there is a substantial subset of prefixes with “typical” linear and 2-adic complexities, around n/2, but low rational complexity. Thus the three complexities sort out different sequences as non-random.

Gaussian Pseudorandom Number Generator Using Linear Feedback Shift Registers in Extended Fields

A new proposal to generate pseudorandom numbers with Gaussian distribution is presented. The generator is a generalization to the extended field GF(2n) of the one using cyclic rotations of linear feedback shift registers (LFSRs) originally defined in GF(2). The rotations applied to LFSRs in the binary case are no longer needed in the extended field due to the implicit rotations found in the binary equivalent model of LFSRs in GF(2n). The new proposal is aligned with the current trend in cryptography of using extended fields as a way to speed up the bitrate of the pseudorandom generators. This proposal allows the use of LFSRs in cryptography to be taken further, from the generation of the classical uniformly distributed sequences to other areas, such as quantum key distribution schemes, in which sequences with Gaussian distribution are needed. The paper contains the statistical analysis of the numbers produced and a comparison with other Gaussian generators.