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

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.

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.

We study the distribution of s-dimensional points of digital explicit inversive pseudorandom numbers with arbitrary lags. We prove a discrepancy bound and derive results on the pseudorandomness of the binary threshold sequence derived from digital explicit inversive pseudorandom numbers in terms of bounds on the correlation measure of order k and the linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations of Mauduit, Niederreiter and Sárközy between discrepancy and correlation measure of order k and of Brandstätter and the third author between correlation measure of order k and linear complexity profile, respectively.

This paper studies the error linear complexity profiles of binary sequences with period 2n. We give a precise categorization of those sequences having 2 distinct critical points in their profiles, as well as an enumeration of these sequences. We also give an upper bound on the maximum number of distinct critical points that the profile of a sequence can have, along with several constructions for sequences having many distinct critical points.

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.

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.

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.

An asymptotically fast algorithm for solving the generalized rational interpolation problem is presented. This problem has been studied as part of system theory and is related to the solution of the classical and Welch-Berlekamp (1983) key equations which arise in Reed-Solomon decoding. The algorithm can also be used to compute the linear complexity profile of a binary sequence of length m using only O(m(log m)/sup 2/ log log m) bit operations.

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.

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.

On the joint linear complexity profile of explicit inversive multisequences

We study the linear complexity of sequences over the prime field \({\mathbb{F}_d}\) introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation.KeywordsSidel’nikov sequenceLinear complexityLinear complexity profileAperiodic autocorrelation

Pseudorandom number generators are required to generate pseudorandom numbers which have not only good statistical properties but also unpredictability in cryptography. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence (this sequence is referred to as the generalized NTU sequence), and showed the period and periodic autocorrelation. In this paper, we investigate the linear complexity of the generalized NTU sequences. Under some conditions, we can ensure that generalized NTU sequences have large linear complexity from the results on linear complexity of Sidel'nikov sequences.

Recently, multisequences have gained increasing interest for applications in cryptography and quasi-Monte Carlo methods. We study the (generalized) joint linear complexity of a class of nonlinear pseudorandom multisequences introduced by the first two authors as well as the linear complexity of its coordinate sequences. We prove lower bounds which are much stronger than in the case of single sequences since the multidimensional case brings in new and favourable effects.