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References [1] Ahlswede , R. , Mauduit , C. , Sárközy , A. 2006 Large families of pseudorandom sequences of k symbols and their complexity, Part I General Theory of

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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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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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Using the discrete logarithm in [7] and [9] a large family of pseudorandom binary sequences was constructed. Here we extend this construction. An interesting feature of this extension is that in certain special cases we get sequences involving points on elliptic curves.

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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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In an earlier paper [3] Cassaigne et al studied the pseudorandom properties of the Liouville function. In this paper some of their results are generalized and sharpened considerably.

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

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In this paper a large family of pseudorandom binary lattices is constructed by using the multiplicative characters of finite fields. This construction generalizes several one-dimensional constructions to arbitrary dimensions.

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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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