On the complexity of a family based on irreducible polynomials

Ahlswede, Khachatrian, Mauduit and Sárközyy introduced the f-complexity measure ("f" for family) in order to study pseudorandom properties of large families of binary sequences. So far several families have been studied by this measure. In the present paper I considerably improve on my earlier result in [7], where the f-complexity measure of a family based on the Legendre symbol and polynomials over Fp is studied. This paper also extends the earlier results to a family restricted on irreducible polynomials

Pszeudovéletlenség, elliptikus görbék és egész számok sorozatai = Pseudorandomness, elliptic curves and sequences of integers

A projekt időtartama alatt 19 cikket írtam, ezek közül 11 dolgozatom jelent meg, és 4 dolgozatomat fogadtak el már közlésre neves hazai vagy külföldi folyóiratokban. További 4 cikket nyújtottam be közlésre. Az alábbiakban szeretném főbb eredményeimet röviden összefoglalni. Mostanában a pszeudovéletlen sorozatok mellett a többdimenziós pszeudovéletlen objektumok is bekerültek a kutatás főirányvonalába. Kutatásaimat főképp ebben az irányban folytattam, részben egyedül, részben társszerzőkkel. On new measures of pseudorandomness of binary lattices című cikkemben új pszeudovéletlen mértékeket vezettem be pszeudovéletlen rácsok vizsgálatára. On the correlation of subsequences című cikkemben konstruáltam egy sorozatot, melynek a rövid részsorozatai is erős pszeudovéletlen tulajdonságokkal rendelkeznek. Christian Mauduittal és Sárközy Andrással közösen vizsgáltuk az egy és többdimenziós elmélet közötti kapcsolatot. Majd megadtunk több kétdimenziós konstrukciót erős pszeudovéletlen tulajdonságokkal rendelkező rácsokra. Ezeknek a rácsoknak több alkalmazása is van a kriptográfiában, páldául képek, térképek titkosítása során alkalmazzák őket. További dolgozatokban összehasonlítottuk a pszeudovéletlenség különböző mértékeit. Tanulmányoztuk pszeudovéletlen rácsok nagy családjainak a pszeudovéletlen mértékeit. Ruzsa Imrével közösen a négyzetszámok sorozatai között vizsgáltunk 3-tagú számtani sorozat mentes halmazt. | During the project I wrote 19 papers, 11 of them have been appeared, other 4 have been accepted for publication in strong leading Hungarian and foreign journals. Further 4 papers have been submitted for publication. Below, I summarize my main results. Recently, besides the pseudorandom sequences, the multi-dimensional pseudorandom-objects are in the center of the research. I continue my research in this direction. I work partly alone and partly with coauthors. In my paper ""On pseudorandomness of binary lattices"", I introduced new measures of pseudorandomness of binary lattices. In ""Correlation of subsequences"" I constructed a sequence such that its all subsequences have strong pseudorandom properties. With Christian Mauduit and András Sárközy we studied the connection between the one and multidimensional theory. Later we constructed several constructions of binary lattices with strong pseudorandom properties. These lattices have many applications in cryptography, for example they can be used in encryption of maps and images. In more papers we studied the connection between different pseudorandom measures. We also studied pseudorandom measures of large families of binary lattices. With Imre Ruzsa we gave a large subset of squares which contains no 3-term arithmetic progression

On reducible and primitive subsets of F_p, II

In Part I of this paper we introduced and studied the notion of reducibility and primitivity of subsets of F_p: a set A is said to be reducible if it can be represented in the form A = B + C with |B|, |C| > 1. Here we introduce and study strong form of primitivity and reducibility: a set A is said to be k-primitive if changing at most k elements of it we always get a primitive set, and it is said to be k - reducible if it has a representation in the form A = B_1 + B_2 + ... + B_k with |B_1|, |B_2|, ..., |B_k| > 1

On the cross-combined measure of families of binary lattices and sequences

The cross-combined measure (which is a natural extension of cross-correlation measure) is introduced and important constructions of large families of binary lattices with optimal or nearly optimal cross-combined measures are presented. These results are also strongly related to the one-dimensional case: An easy method is showed obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of families of lattices. The important feature of this result is that so far there exists only one type of constructions of very large families of binary sequences with small cross-correlation measure, and this only type of constructions was based on one-variable irreducible polynomials. Since it is very complicated to construct one-variable irreducible polynomials over $\mathbb F_p$, it became necessary to show other types of constructions where the generation of sequences is much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided. (Since, contrary to one-variable polynomials, using Sch\"oneman-Eisenstein criteria it is possible to generate two-variable irreducible polynomials over $\mathbb F_p$ fast.

On finite pseudorandom binary lattices

Pseudorandom binary sequences play a crucial role in cryptography. The classical approach to pseudorandomness of binary sequences is based on computational complexity. This approach has certain weak points thus in the last two decades years a new, more constructive and quantitative approach has been developed. Since multidimensional analogs of binary sequences (called binary lattices) also have important applications thus it is a natural idea to extend this new approach to the multidimensional case. This extension started with a paper published in 2006, and since that about 25 papers have been written on this subject. Here our goal is to present a survey of all these papers