A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

Fast Software Implementation of Serial Test and Approximate Entropy Test of Binary Sequence

In many cryptographic applications, random numbers and pseudorandom numbers are required. Many cryptographic protocols require using random or pseudorandom numbers at various points, e.g., for auxiliary data in digital signatures or challenges in authentication protocols. In NIST SP800-22, the focus is on the need for randomness for encryption purposes and describes how to apply a set of statistical randomness tests. These tests can be used to evaluate the data generated by cryptographic algorithms. This paper will study the fast software implementation of the serial test and the approximate entropy test and propose two types of fast implementations of these tests. The first method is to follow the basic steps of these tests and replace bit operations with byte operations. Through this method, compared with the implementation of Fast NIST STS, the efficiency of the serial test and approximate entropy test is increased by 2.164 and 2.100 times, respectively. The second method is based on the first method, combining the statistical characteristics of subsequences of different lengths and further combining the two detections with different detection parameters. In this way, compared to the individual implementation of these tests, the efficiency has been significantly improved. Compared with the implementation of Fast NIST STS, the efficiency of this paper is increased by 4.078 times.

A Hybrid Inversive Congruential Pseudorandom Number Generator with High Period

Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence unusable for various stochastic simulations. An extension which showed promise for such applications is a generator obtained by using a first-order recurrence based upon the inversive modulo a prime or a prime power, called inversive congruential generator (ICG). A lot of work has been dedicated to investigate the periods (under some conditions of the parameters), the lattice test passing, discrepancy and other statistical properties of such a generator. Here, we propose a new method, which we call hybrid inversive congruential generator (HICG), based upon a second order recurrence using the inversive modulo $M$, a power of 2. We investigate the period of this pseudorandom numbers generator (PRNG) and give necessary and sufficient conditions for our PRNG to have periods $M$ (thereby doubling the period of the classical ICG) and $M/2$ (matching the one of the ICG). Moreover, we show that the lattice test complexity for a binary sequence associated to (a full period) HICG is precisely M/2.

CELLULAR AUTOMATA ALGORITHMS FOR PSEUDORANDOM NUMBERS GENERATION

Work describes four permutation algorithms of square matrices based on cyclic rows and columns shifts. This choice of discrete transformation algorithms is justified by the convenience of the cellular automaton (CA) formulation. Output matrices can be considered as pseudo-random sequences of numbers. As a result of numerical calculation, empirical formulas are obtained for the permutation period and the function of the period of a single CA-cell on the order of the matrix n. As a parameter of CA dynamics, we analyze two "mixing metrics" on permutations of the matrix (compared to the initial matrix).

Development of a Scalable Coding for the Encryption of Images using Diagonal Min-Max Block Truncation Code

This paper proposes a scalability coding on encrypted images, especially resolution scalability on lossy compression images. In the compression stage, the input gray level image is compressed using Diagonal Min-Max Block Truncation Coding technique. The compressed input image is encrypted using pseudorandom numbers masked by modulo-256, and then the encoded bit streams are transmitted. The pseudorandom numbers generated will be the encrypted key and the same is shared to the receiver. In the receiver side, the encoded bit stream is decrypted by using the shared encrypted key, which gives the compressed pixel value. Then the original image is reconstructed by using Diagonal Min-Max Block Truncation Coding Technique.