Distribution of Digital Explicit Inversive Pseudorandom Numbers and Their Binary Threshold Sequence

Based on the cyclotomy classes of extension fields, a family of binary cyclotomic sequences are constructed and their pseudorandom measures (i.e., the well-distribution measure and the correlation measure of order k) are estimated using certain exponential sums. A lower bound on the linear complexity profile is also presented in terms of the correlation measure.

Linear complexity profile and correlation measure of order k are important pseudorandomness measures for sequences used in cryptography. We study both measures for a class of binary sequences called Legendre-Sidelnikov sequences. The proofs involve character sums.

A technique to study the correlation measures of binary sequences

We study measures used to assess Boolean functions, where the functions are understood to be derived from binary sequences. We prove relations between two complexity measures for these Boolean functions and the correlation measure of order k of the corresponding sequence. More precisely, we study sparsity and combinatorial complexity of the Boolean function. Moreover, for a sequence with 'typical' values of the correlation measure of order k we apply these relations to derive bounds on the complexity measures for the corresponding Boolean function. Finally, we apply our results to Sidelnikov sequences for which nice upper bounds on the correlation measure are known.

Linear complexity profile of m-ary pseudorandom sequences with small correlation measure

Following the idea of Xing et al., we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence. With the help of the tool of exponential sums on elliptic curves, we study their periods, linear complexities, linear complexity profiles, distributions of r-patterns, periodic correlation, partial period distributions, and aperiodic correlation in detail. The results show that they have nice randomness.

We essentially improve lower bounds on the linear complexity of a sequence representing the residues of the discrete logarithm in a finite field modulo a divisor of the order of the multiplicative group. More generally, we present the result as a bound on the linear complexity profile. The proof is based on character sum bounds.

We study several pseudorandom properties of the Liouville function and the Möbius function of polynomials over a finite field. More precisely, we obtain bounds on their balancedness as well as their well-distribution measure, correlation measure, and linear complexity profile.

On an inequality between pseudorandom measures of lattices

Recently, Dartyge and Sarkozy defined the measures, i.e., the well- distribution measure and the correlation measure of order k, of pseudo-randomness of subsets of the set {1, 2, . . . , N}, and they presented several examples for subsets with strong pseudo-random properties when N is a prime number. In this article, we present a construction of pseudo-random subsets for N = pq and give some partial results on the pseudo-random measures.

An algorithm for shifted continued fraction expansions in parallel linear time

The authors explore the conditions of parameters in a Continuously Decimalized Maximum length (CDM) sequence to realize its uniform distribution bit patterns and evaluate the specific characteristics of CDM-sequences with the parameters.

Inversive methods provide an attractive alternative to linear methods for pseudorandom number and vector generation. We present the first nontrivial results on the distribution of sequences of digital inversive pseudorandom numbers and inversive pseudorandom vectors in parts of the period. The proofs are based on a new bound for certain exponential sums over finite fields.

We obtain a lower bound on the linear complexity profile of the power generator of pseudo-random numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the Blum-Blum-Shub (1986) generator. In particular, these results imply that lattice reduction attacks on such generators are not feasible.

On the correlation of binary sequences