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Multiplicative number theory 74 MAUDUIT, C. and SÁRKÖZY, A., On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the

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Summary By using the multiplicative inverse modulo p, a large family of finite binary sequences is constructed with strong pseudorandom properties. The crucial tool in the proofs is an (additive) character sum estimate of Eichenauer--Hermann and Niederreiter.

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Periodica Mathematica Hungarica
Authors: Rudolf Ahlswede, Levon Khachatrian, C. Mauduit, and A. Sárközy

Abstract  

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated.

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Abstract  

Recently a constructive theory of pseudorandomness of binary sequences has been developed and many constructions for binary sequences with strong pseudorandom properties have been given. In the applications one usually needs large families of binary sequences of this type. In this paper we adapt the notions of collision and avalanche effect to study these pseudorandom properties of families of binary sequences. We test two of the most important constructions for these pseudorandom properties, and it turns out that one of the two constructions is ideal from this point of view as well, while the other construction does not possess these pseudorandom properties.

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Summary  

Recently, Goubin, Mauduit, Rivat and Srkzy have given three constructions for large families of binary sequences. In each of these constructions the sequence is defined by modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} congruences where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} is a prime number. In this paper the three constructions are extended to the case when the modulus is of the form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $q$ \end{document} are two distinct primes not far apart (note that the well-known Blum-Blum-Shub and RSA constructions for pseudorandom sequences are also of this type). It is shown that these modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions also have certain strong pseudorandom properties but, e.g., the (``long range'') correlation of order \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $4$ \end{document} is large (similar phenomenon may occur in other modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions as well).

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Abstract  

In a series of papers Mauduit and Sárközy (partly with coauthors) studied finite pseudorandom binary sequences and they constructed sequences with strong pseudorandom properties. In these constructions fields with prime order were used. In this paper a new construction is presented, which is based on finite fields of order 2k.

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Abstract  

In a series of papers Mauduit and Sárközy introduced measures of pseudorandomness and they constructed large families of sequences with strong pseudorandom properties. In later papers the structure of families of binary sequences was also studied. In these constructions fields with prime order were used. Throughout this paper the structure of a family of binary sequences based on GF(2k) will be studied.

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Abstract  

In an earlier paper we studied collisions and avalanche effect in two of the most important constructions given for large families of binary sequences possessing strong pseudorandom properties. It turned out that one of the two constructions (which is based on the use of the Legendre symbol) is ideal from this point of view, while the other construction (which is based on the size of the modulo p residue of f(n) for some polynomial f(x) ∈
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_p$$ \end{document}
[x]) is not satisfactory since there are “many” collisions in it. Here it is shown that this weakness of the second construction can be corrected: one can take a subfamily of the given family which is just slightly smaller and collision free.
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Abstract  

In the applications it may occur that our initial pseudorandom binary sequence turns out to be not long enough, thus we have to take the concatenation or merging of it with other pseudorandom binary sequences. Here our goal is study when we can form the concatenation of several pseudorandom binary sequences belonging to a given family? We introduce and study new measures which can be used for answering this question.

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Abstract  

The pseudorandom properties of finite binary sequences have been studied recently intensively. In the papers written on this subject the two distinct elements of the sequences are chosen equally with probability 1/2. In this paper the authors extend the work to the more general case when the two elements are chosen with probability p, resp. 1-p.

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