On the Distribution of the Power Generator over a Residue Ring for Parts of the Period

This paper studies the distribution of the power generator of pseudorandom numbers over a residue ring for parts of the period. These results compliment some recently obtained distribution bounds of the power generator modulo an arbitrary number for the entire period. Also, the arbitrary modulus case may have some cryptography related applications and could be of interest in other settings which require quality pseudorandom numbers.This paper studies the distribution of the power generator of pseudorandom numbers over a residue ring for parts of the period. These results compliment some recently obtained distribution bounds of the power generator modulo an arbitrary number for the entire period. Also, the arbitrary modulus case may have some cryptography related applications and could be of interest in other settings which require quality pseudorandom numbers

On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators

In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degreegrowth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.Comment: Mathematics of Computation (to appear

Portable random number generators

Computers are deterministic devices, and a computer-generated random number is a contradiction in terms. As a result, computer-generated pseudorandom numbers are fraught with peril for the unwary. We summarize much that is known about the most well-known pseudorandom number generators: congruential generators. We also provide machine-independent programs to implement the generators in any language that has 32-bit signed integers-for example C, C++, and FORTRAN. Based on an extensive search, we provide parameter values better than those previously available.Programming (Mathematics) ; Computers

Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation

Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG). In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.Comment: To appear in Mathematics of Computatio

Fractional jumps: complete characterisation and an explicit infinite family

In this paper we provide a complete characterisation of transitive fractional jumps by showing that they can only arise from transitive projective automorphisms. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

On Pseudo-Random Number Generators Using Elliptic Curves and Chaotic Systems

Elliptic Curve Cryptography (ECC) is a relatively recent branch of cryptography which is based on the arithmetic on elliptic curves and security of the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Elliptic curve cryptographic schemes are public-key mechanisms that provide encryption, digital signature and key exchange capabilities. Elliptic curve algorithms are also applied to generation of sequences of pseudo-random numbers. Another recent branch of cryptography is chaotic dynamical systems where security is based on high sensitivity of iterations of maps to initial conditions and parameters. In the present work, we give a short survey describing state-of-the-art of several suggested constructions for generating sequences of pseudorandom number generators based on elliptic curves (ECPRNG) over finite fields of prime order. In the second part of the paper we propose a method of generating sequences of pseudorandom points on elliptic curves over finite fields which is driven by a chaotic map. Such a construction improves randomness of the sequence generated since it combines good statistical properties of an ECPRNG and a CPRNG (Chaotic Pseudo- Random Number Generator). The algorithm proposed in this work is of interest for both classical and elliptic curve cryptography