On pseudorandom properties of some Dirichlet characters (original) (raw)
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Unif. Distrib. Theory, 2009
The pseudorandom properties of a two dimensional "binary plattice" related to the Legendre symbol are studied. 1 Introduction Pseudorandom binary sequences have many important applications. In particular, they are used as a key stream in the classical stream cipher called the Vernam cipher. The standard approach to the theory of pseudorandomness of binary sequences is based on complexity theory. However, this approach has certain limitations and weak points. Thus recently Mauduit and Sárközy [9] (see also the survey paper [12]) initiated a new, constructive approach to the theory of pseudorandomness. They defined and studied new measures of pseudorandomness. These measures provide a quantitative characterization of pseudorandomness of a given binary sequence. In the last 10 years numerous binary sequences have been tested for pseudorandomness.
Measures of pseudorandomness for binary sequences constructed using finite fields
Discrete Mathematics, 2009
We extend the results of Goubin, Mauduit and Sárközy on the well-distribution measure and the correlation measure of order k of the sequence of Legendre sequences with polynomial argument in several ways. We analyze sequences of quadratic characters of finite fields of prime power order and consider in each case two, in general, different definitions of well-distribution measure and correlation measure of order k, respectively.
Multiplicative character sums of Fermat quotients and pseudorandom sequences
Periodica Mathematica Hungarica, 2012
We prove a bound on sums of products of multiplicative characters of shifted Fermat quotients modulo p. From this bound we derive results on the pseudorandomness of sequences of modular discrete logarithms of Fermat quotients modulo p: bounds on the well-distribution measure, the correlation measure of order , and the linear complexity.
Measures of Pseudorandomness for Finite Sequences: Minimal Values
Combinatorics, Probability & Computing, 2006
Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences EN ∈ {−1, 1} N in order to measure their 'level of randomness'. Two of these parameters are the normality measure N (EN ) and the correlation measure C k (EN ) of order k, which focus on different combinatorial aspects of EN . In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.
Pseudorandom binary sequences: quality measures and number-theoretic constructions
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2023
In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order , expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.
Construction of large families of pseudorandom binary sequences
Journal of Number Theory, 2004
In a series of papers Mauduit and S ark ozy (partly with coauthors) studied nite pseudorandom binary sequences. They showed that the Legendre symbol forms a \good" pseudorandom sequence, and they also tested other sequences for pseudorandomness, however, no large family of \good" pseudorandom sequences has been found yet. In this paper, a large family of this type is constructed. Again, the construction is related to the Legendre symbol. Moreover, by using elliptic curves large families of binary sequences are constructed. It is expected that these sequences form \good" pseudorandom binary sequences.
Measures of Pseudorandomness: Arithmetic Autocorrelation and Correlation Measure
2017
We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation, and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials.
Trace representation of pseudorandom binary sequences derived from Euler quotients
Applicable Algebra in Engineering, Communication and Computing, 2015
We give the trace representation of a family of binary sequences derived from Euler quotients by determining the corresponding defining polynomials. Trace representation can help us producing the sequences efficiently and analyzing their cryptographic properties, such as linear complexity.
Uniform distribution theory, 2017
In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.