WORKING WITH CHRISTIAN MAUDUIT (In memoriam Christian Mauduit) - - PowerPoint PPT Presentation

WORKING WITH CHRISTIAN MAUDUIT (In memoriam Christian Mauduit)

András Sárközy

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• 1. THE BEGINNING AND THE CONTINUATION

I met Christian first at a CIRM meeting. Then he invited me for a one month visit to the Institut de Mathématiques de Luminy in Marseille. This first visit of mine in 1994 was very fruitful and enjoyable. We ended up with a joint paper, and decided to continue the cooperation. Since then I have visited Marseille 24 times for 1, 2 or 3 month visits (for altogether 39 months) and Christian also visited me several times in Budapest. We have written 48 joint papers which make 26% of his papers and 18% of mine (in 31 of these papers we also had further coauthors). According to the citation statistics of the Mathematical Reviews the citation list of both of us is led by the same joint paper of ours, and the same 5 papers on the list (each of them written in pseudorandom binary sequences) out of the first 7 in my case and 6 in his case are our joint papers.

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All our joint papers are written in number theory, and their subjects belong to one of the following 4 fields: digit properties, pseudorandom (to be abbreviated as PR) binary sequences, PR binary lattices, pseudorandomness in other situations. Now I will present a short survey of our papers grouping them according to their subjects and focusing on the most important ones.

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• 2. DIGIT PROPERTIES

Our first two papers with Christian (written in 1994 and 1995, resp.) were written on digit properties, more precisely, we studied the arithmetic structure of sets characterized by digit

• properties. This subject was proposed by Christian since he had been working earlier on problems of

this flavour, but I was also interested in it since the digit properties can be controlled very well by using the generating function principle, and I was wondering: how well can one utilize this fact to study arithmetic properties? The first paper in this direction was written by Gelfond.

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We continued Gelfond’s work by studying the distribution of the values Sg(a + b) in residue classes where Sg(n) denotes the sum of base g digits of n and a ∈ A, b ∈ B for two “dense” sets A, B of positive integers. We also studied the typical and extremal values of the function ω(n) (= number of distinct prime factors of n) over the integers n belonging to the set Ur,m(N) = {n : n ≤ N, Sg(n) ≡ r (mod m)}. In our second (much more difficult) paper we studied similar arithmetic properties of the integers n some of whose digits are fixed, i.e., which belong to the set Vk(N) = {n : n ≤ N, Sg(n) = k} (the difficulty is that this set is much thinner than the set Ur,m(N)).

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In two triple papers with P. Erdős we studied similar arithmetic properties as Gelfond in his paper and two of us in our two earlier papers, but we considered even thinner sets characterized by digit properties: namely, we considered integers with “missing digits”, i.e., integers n such that in their base g representation a fixed digit does not occur. In a triple paper with Konyagin we also studied the number of prime factors of integers characterized by digit properties. In a triple paper with C. Dartyge we considered finite fields Fq of order q = pr with r ≥ 2, and the analog of the base p representation of integers with “missing digits”. We showed that under certain conditions there are squares x2, more generally polynomial values f (x), and generators g such that x2, f (x) and f (g), resp., are of missing digits.

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In two recent rather difficult and complicated triple papers with J. Rivat we returned to our early papers with Christian on the sum of digits function. In one of these new papers we proved a conjecture appearing in one of our early papers, while in the other new paper we sharpened some results proved in an other early paper of ours.

• 3. PSEUDORANDOM BINARY SEQUENCES

We have written 17 papers on this subject whose study was proposed by me. As a student Szemerédi and I settled a combinatorial extremum problem of Erdős, and to show that the bound given for the extremal value can be achieved we used the Legendre symbol sequence and utilized (in a non-trivial way) its random-type structure.

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I thought already then that this property can be also utilized elsewhere. Later I read about the Vernam cipher which is still a frequently applied encrypting algorithm and it is based on the use of PR binary sequences. I also heard a talk on H. Niederreiter’s work on PR sequences of real numbers from the interval [0, 1) (to be used in numerical analysis). Based on these facts I suggested to Christian to try to develop a Niederreiter-type quantitative and constructive theory of PR binary

• sequences. He liked the idea, so that we started to work in this direction.

First we did a lot of reading and made some inquiries, and it took us 3 months to end up with

• ur fist paper of this type; this is probably our best paper and it is certainly the most frequently

cited one.

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In this paper we considered finite binary sequences of type (1) EN = (e1, e2, . . . , eN) ∈ {−1, +1}N. First we introduced the measures of pseudorandomness. The most important ones are the following: The well-distribution measure of the sequence EN in (1) is defined as W (EN) = max

a,b,t

• t−1
• j=0

ea+jb

• ,

where the maximum is taken over all positive integers a, b, t such that 1 ≤ a ≤ a + (t − 1)b ≤ N, while the correlation measure of order k of EN is defined as Ck(EN) = max

M,D

• M
• n=1

en+d1en+d2 . . . en+dk

• ,

where the maximum is taken over all D = (d1, d2, . . . , dk) and M such that d1, d2, . . . , dk and M are non-negative integers with 0 ≤ d1 < d2 < · · · < dk ≤ N − M.

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We also defined the combined (well-distribution-correlation) PR-measure of order k: Qk(EN) = max

a,b,t,D

• t
• j=0

ea+jb+d1ea+jb+d2 . . . ea+jb+dk

• where a, b, t, (0 ≤)d1 < d2 < · · · < dk are positive integers, D = (d1, d2, . . . , dk) and we consider

all the sums such that all the subscripts a + jb + di belong to {1, 2, . . . , N}. Then the sequence EN is considered as a “good” PR sequence if the measures W (EN) and Ck(EN) (at least for “small” k) are “small”. (This terminology was justified later.) We also proved in this paper (by using Weil’s theorem) that the Legendre symbol sequence (2) Ep−1 = 1 p

• ,

2 p

• , . . . ,

p − 1 p

• is “good”.

The paper described above was the first part of a series of 7 papers. In part II we applied the PR measures introduced in part I to study the PR properties of certain important special sequences:

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We studied the Champernowne, Rudin–Shapiro and Thue–Morse sequences, and we also presented a further “good” PR sequence: we generalized the Legendre symbol sequence construction (2) so that we replaced the n-th element

• n

p

• by
• g(n)

p

• where g(x) is a permutation

polynomial over Fp satisfying a certain mild condition. Parts III and IV of the series were 5 author papers written jointly with J. Cassaigne, S. Ferenczi and J. Rivat on the PR properties of the Liouville function (defined as λ(n) = (−1)Ω(n) where Ω(n) denotes the number of prime factors of n counted with multiplicity). This is a difficult question whose importance is based on the fact that this function is closely related to the Möbius function playing a central role in multiplicative number theory.

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In parts V and VI we considered a problem of Erdős. Let k be a positive integer, α an irrational number such that the partial quotients in its continued fraction expansion are bounded. Then we studied the PR properties of {nkα}, more precisely, of its distribution between the intervals [0, 1/2) and [1/2, 1). In part VII of the series written jointly with J. Cassaigne we studied the measures of pseudorandomness defined earlier. We proved in this important paper that for a random binary sequence EN ⊂ {−1, +1}N with large probability we have N1/2 ≪ W (EN) ≪ (N log N)1/2 and N1/2 ≪ Ck(EN) ≪ (kN log N)1/2, so that, indeed, for a “good” PR sequence EN W (EN) and Ck(EN) cannot be much larger than N1/2. We also studied the minimum values of the PR measures.

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Another important part of this paper is the comparison of correlations of different order. We showed that if k | ℓ, N → ∞ and Cℓ(EN) is “small”, then Ck(EN) also must be small, while if k ∤ ℓ, then Ck(EN) and Cℓ(EN) are independent in the sense that either of them can be small, while the

• ther one is large (so that the PR measures C2, C3, C5, . . . , Cp, . . . are independent).

In another paper with Christian we studied the connection between the measures W and Ck. Jointly with J. Rivat we studied the PR properties of {nc} for n = 1, 2, . . . where c is not integer and c > 1, i.e., we considered its distribution between the intervals [0, 1/2), [1/2, 1) for n = 1, 2, . . . , N. If one wants to apply PR binary sequences in cryptography then one has to construct large families of them (and possibly many constructions of this type). We have presented the following constructions:

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(i) Jointly with Louis Goubin we studied the following generalization of our original Legendre symbol construction: for a prime p and f (x) ∈ Fp[x] define the sequence Ep = (e1, e2, . . . , ep) by en =   

• f (n)

p

• for p ∤ f (n),

+1 for p | f (n) for n = 1, 2, . . . , p. In this paper with Goubin we presented certain sufficient conditions ensuring that this construction should work (in particular, we proved that it always produces a “good” sequence if f (x) has no multiple zero in the algebraic closure of Fp and 2 is a primitive root modulo p), and we also gave an example showing that assuming f (x) = (g(x))2 is not enough, since it may occur that this holds, however, a correlation type quantity is “large”.

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(ii) We presented a construction based on the fact that if p is a prime and f (x) ∈ Fp[x] satisfies certain mild conditions, then the multiplicative inverse of f (n) is distributed modulo p in a random way between the intervals

• 0, p

2

• ,

p

2, 1

• .

(iii) In a triple paper with J. Rivat we presented a construction using additive characters which also involves polynomials over Fp and it is simple and easy to handle, and the correlations of small

• rder are small, but the correlations of large order can be large; this illustrates the importance of

estimating correlations of large order as well. (iv) In a triple paper with K. Gyarmati we showed that if a “good” PR sequence is given which satisfies certain conditions, then one can construct “many” further “good” sequences starting out of it, and we presented an algorithm for doing this “blowing up a single sequence” procedure.

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Note that in almost all of our constructions and in the related proofs Weil’s theorem is a crucial tool, but lengthy computations, harmonic analysis and combinatorial tools are also needed. In a triple paper with H. Niederreiter we studied the connection between the pseudorandomness

• f binary sequences and of [0, 1) sequences, and we showed that there is a strong connection

between these fields. In applications of PR binary sequences in cryptography it is not enough to construct large families of PR sequences, it is more important to know that the family constructed has a “rich”, “complex” structure. To study this requirement, in a quadruple paper with R. Ahlswede and L. Khachatrian we introduced and studied the notion of family complexity, and we constructed a large family also having high family complexity.

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In another paper we studied the connection between family complexity and the so-called VC-dimension (answering a question of Csiszár and Gács). In a triple paper with K. Gyarmati we also introduced and studied the notion of cross-correlation measure for investigating the structure of families of binary sequences.

• 4. A PHOTO, A (SLIGHTLY) RELATED PROBLEM

AND MY CONNECTION WITH CHRISTIAN I have many photos of Christian made on different occasions but I have found just one of them showing him as a university professor:

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This photo was taken in 2003 and it shows Christian as my laudator when I got my “doctor honoris causa” title in Marseille. He did it very well, enjoyed it very much; clearly, he was a natural born actor. After all, his mother was Greek and don’t forget that theatre culture originated mostly in Greece; think of Aristophanes, etc. Indeed, he had a Greek temper: he lived with double intensity, enjoyed life very much, and he was also very fast and sharp in mathematics. I have a very different temper, I am much slower (in mathematics as well). However, this did not cause any problems in our connection, just the

• pposite: we complemented each other nicely.

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I was sitting, thinking, analyzing, finally ending up with a direction to go, then recalling some related ideas from earlier papers, while Christian came up with a series of bright and original ideas

• f mostly combinatorial nature or coming from his uniform distribution background; these nice

ideas of his also inspired me to find further new ideas, etc. In fact, his partly Greek origin made a further link between us: my wife was also Greek (she was born in Greece and moved to Hungary at the age of 9), thus my children also had a Greek mother like Christian, besides their age was much closer to Christian’s age than to mine. Once he invited my children for a one week visit to Marseille; then they became friends, and also remained in touch later. Thus the difference in our tempers and backgrounds did not bother us, it just made our cooperation more entertaining. It was much more important that our top priorities were similar: we both loved mathematics, in particular, number theory, our political views were also similar, we loved humor, nature, etc.

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The honorary doctor ceremony also led to a mathematical problem on pseudorandomness. Namely, after the honorary degree ceremony there was a party, where I had a nice chat with a Dutch Nobel prize winner physicist called Veltman who also got the honorary doctorate then. He told me a story on pseudorandomness (of sequences of real numbers) which led to the conclusion that it is not enough to study PR sequences of a fixed length, but one also must study PR sequences such that their short subsets formed by consecutive elements are also PR, and also the closely related problem of infinite sets such that cutting them anywhere the initial elements form a “good” PR sequence. To construct sequences of this type seems to be one of the most important open problems to settle in the theory of pseudorandomness. Some small steps in this direction have been made but there is still a long way to go. (Among others we have two related triple papers with C. Dartyge and K. Gyarmati, and now J. Rivat has also joined us to work on the continuation of those papers.)

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• 5. PSEUDORANDOM BINARY LATTICES

We have also studied multidimensional generalization of the theory of pseudorandomness of binary sequences in 18 papers; here we will confine ourselves to the two-dimensional case. Let N be a positive integer, consider a square whose side length is N, divide it into N2 unit squares, and write a −1 or +1 in each of these unit squares. Then the scheme formed by these −1’s and +1’s is called a (two-dimensional) binary lattice. Binary lattices of PR nature also have applications, e.g., in encripting bit maps, watermarking, steganography, etc. Thus we extended our work to also study pseudorandomness of binary lattices. In some cases we could generalize our

• ne-dimensional definitions and results to the multidimensional case (e.g., we defined the

two-dimensional PR measure of order k by generalizing the definition of the one-dimensional combined measure Qk(EN)).

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In some other cases we could achieve only partial success. (E.g., some of the most important

• ne-dimensional constructions can be extended to the two-dimensional case, but while in one

dimension the PR measures can be estimated optimally well by using Weil’s theorem, in two dimensions we run into difficulties since Deligne’s theorem is the two-dimensional analog of Weil’s theorem, and because of a rather inconvenient condition in this theorem one cannot use it. Thus instead one has to apply more elementary tools giving non-trivial estimates which are only midway to the optimal.) Let me mention just one special problem where we ran into serious difficulties. We wanted to define the notion of linear complexity in two dimensions. With a lot of work both of us ended up with a definition but first time in our 25 years of joint work we disagreed.

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I presented two different (algebraic, resp. analytical) approaches leading to the same definition which was clearly the generalization of the one-dimensional one, but very complicated and difficult to use. Christian used an other, elementary-geometric approach leading to a more transparent definition which, however, was not the extension of the one-dimensional definition. After a little dispute we made a compromise: in our paper we presented both definitions and we gave the two complexity definitions different names. Very recently we decided to continue the study of two-dimensional linear complexity jointly with further coauthors. However, some unexpected technical difficulties came up, then Christian died, and I decided not to write on this subject without him and to abandon this project.

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• 6. GENERALIZATIONS AND EXTENSIONS OF PSEUDORANDOMNESS

OF BINARY SEQUENCES IN VARIOUS DIRECTIONS The study of pseudorandomness of binary sequences has been extended in various directions. In particular, Christian and I have written 3 joint papers on pseudorandomness of k symbol sequences. In our first paper we introduced and studied the measures of pseudorandomness in this case (which are rather different from the one-dimensional ones). We also presented an example for a “good” PR k symbol sequence (by using additive characters), and we proved that, indeed, it is “good”. In two triple papers with R. Ahlswede we constructed large families of “good” PR k symbol

• sequences. Moreover, we extended the definition of family complexity to k symbol sequences, and

we showed that a certain large family of “good” PR sequences also possesses large family complexity.

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(Both Christian and I also have many papers on different extensions and generalizations of pseudorandomness of binary sequences written jointly not with us but with other coauthors: e.g., pseudorandomness of vectors, subsets, trees, graphs, arithmetic functions, sequences from finite fields were studied.)

• 7. THE END

Christian called me about 3 months ago to ask me whether I would visit Marseille in this academic year. I answered him that because of health problems I will not make long visits in the future; I will do, perhaps, a few short ones, but for sure, I will visit Marseille for his birthday

• conference. Moreover, I told him that he is welcome in Hungary at any time; perhaps, we might

continue the work on the multidimensional linear complexity (jointly with others).

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Then two or three weeks later Joël Rivat called me and said that Christian had died. I thought I had misunderstood him (although he has a good and clear English) so that I asked him to repeat what he said. Then he confirmed that, indeed, Christian had died. I was shocked. He was 19 years younger than I was, and he was still in a very good shape. I wanted to use his birthday conference to tell him how much I enjoyed to work with him, and how grateful I am for all the fun that we had together. Now I will not have more opportunity to tell him these things, but I am sure he knew them anyway.

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THANK YOU FOR YOUR ATTENTION.