On pseudorandom properties of some Dirichlet characters (original) (raw)
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.
Publicationes Mathematicae Debrecen, 2011
In the last 15 years a new constructive theory of pseudorandomness of binary sequences has been developed. Later this theory was extended to n dimensions, i.e., to the study of pseudorandomness of binary lattices. In the applications it is not enough to consider single binary sequences, one also needs information on the structure of large families of binary sequences with strong pseudorandom properties. Thus the related notions of family complexity, collision and avalanche effect have been introduced. In this paper our goal is to extend these definitions to binary lattices, and we will present constructions of large families of binary lattices with strong pseudorandom properties such that these families also possess a nice structure.
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.
On the arithmetic autocorrelation of the Legendre sequence
Advances in Mathematics of Communications
The Legendre sequence possesses several desirable features of pseudorandomness in view of different applications such as a high linear complexity (profile) for cryptography and a small (aperiodic) autocorrelation for radar, gps, or sonar. Here we prove the first nontrivial bound on its arithmetic autocorrelation, another figure of merit introduced by Mandelbaum for error-correcting codes.
On the correlation of binary sequences, II
Discrete Mathematics, 2012
This paper concerns the study of the correlation measures of finite binary sequences, more particularily the dependence of correlation measures of even order and correlation measures of odd order. These results generalize previous results due to Gyarmati [7] and to Anantharam [3] and provide a partial answer to a conjecture due to Mauduit [12]. The last part of the paper concerns the generalization of this study to the case of finite binary n-dimensional lattices.
IEEE Transactions on Information Theory
It is known that Hall's sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period p. Here we study its correlation measure of order k and show that it is, up to a constant depending on k and some logarithmic factor, of order of magnitude p 1/2 , which is close to the expected value for a random sequence of length p. Moreover, we derive from this bound a lower bound on the N th maximum order complexity of order of magnitude log p, which is the expected order of magnitude for a random sequence of length p.
On discrete Fourier transform, ambiguity, and Hamming-autocorrelation of pseudorandom sequences
Designs, Codes and Cryptography, 2014
We estimate discrete Fourier transform, ambiguity, and Hamming-autocorrelation of m-ary sequences in terms of their (periodic) correlation measure of order 4. Roughly speaking, we show that every pseudorandom sequence, that is, any sequence with small correlation measure up to a sufficiently large order, cannot have a large discrete Fourier transform, ambiguity, or Hamming-autocorrelation.
A Formula of the Dirichlet Character Sum
In this paper, We use the Fourier series expansion of real variables function, We give a formula to calculate the Dirichlet character sum, and four special examples are given.
Arithmetic Crosscorrelation of Pseudorandom Binary Sequences of Coprime Periods
IEEE Transactions on Information Theory
The (classical) crosscorrelation is an important measure of pseudorandomness of two binary sequences for applications in communications. The arithmetic crosscorrelation is another figure of merit introduced by Goresky and Klapper generalizing Mandelbaum's arithmetic autocorrelation. First we observe that the arithmetic crosscorrelation is constant for two binary sequences of coprime periods which complements the analogous result for the classical crosscorrelation. Then we prove upper bounds for the constant arithmetic crosscorrelation of two Legendre sequences of different periods and of two binary m-sequences of coprime periods, respectively.
On the pseudorandomness of binary and quaternary sequences linked by the gray mapping
Periodica Mathematica Hungarica, 2010
Binary and quaternary sequences are the most important sequences in view of many practical applications. Any quaternary sequence can be decomposed into two binary sequences and any two binary sequences can be combined into a quaternary sequence using the Gray mapping. We analyze the relation between the measures of pseudorandomness for the two binary sequences and the measures for the corresponding quaternary sequences, which were both introduced by Mauduit and Sárközy. Our results show that each 'pseudorandom' quaternary sequence corresponds to two 'pseudorandom' binary sequences which are 'uncorrelated'.