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**11 - 17**of**17**### THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY

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"... Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: wor ..."

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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining

### On the Foundamental Theorem in Arithmetic Progession of Primes

"... Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [1-3]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved that the ..."

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Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [1-3]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved that the primes contain arbitrarily long arithmetic progressions which is absolutely false[4-9].They do not understand the arithmetic progression of primes [4-15].

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"... Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [1-3]. P i�1 � P1 ..."

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Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [1-3]. P i�1 � P1

### A SIMPLE REGULARIZATION OF GRAPHS

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"... Abstract. The well-known regularity lemma of E. Szemerédi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It is done just by taking a constant-bounded number of ..."

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Abstract. The well-known regularity lemma of E. Szemerédi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It is done just by taking a constant-bounded number of random vertex samplings only one time (thus, iteration-free). Since it is independent from the definition of quasi-randomness, it can be generalized very naturally to hypergraph regularization. In this expository note, we show only a graph case of the paper [5] on hypergraphs, but may help the reader to access [5]. 1.

### FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS

"... Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part ..."

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Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourier-analytic properties of singular measures. It became clear over the years that the theory would have to involve sophisticated geometric and combinatorial input. Two particularly important turning points were Fefferman’s work in the 1970s invoking the ”Kakeya problem ” in this context, and Bourgain’s application of Gowers’s additive number theory techniques to the Kakeya problem almost 30 years later. All this led harmonic analysts to explore areas previously foreign to them, such as combinatorial geometry, graph theory, and additive number theory. Although the Kakeya and restriction problems remain stubbornly open, the

### Cube Testers and Key Recovery Attacks On

"... Abstract. CRYPTO 2008 saw the introduction of the hash function MD6 and of cube attacks, a type of algebraic attack applicable to cryptographic functions having a low-degree algebraic normal form over GF(2). This paper applies cube attacks to reduced round MD6, finding the full 128-bit key of a 14-r ..."

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Abstract. CRYPTO 2008 saw the introduction of the hash function MD6 and of cube attacks, a type of algebraic attack applicable to cryptographic functions having a low-degree algebraic normal form over GF(2). This paper applies cube attacks to reduced round MD6, finding the full 128-bit key of a 14-round MD6 with complexity 2 22 (which takes less than a minute on a single PC). This is the best key recovery attack announced so far for MD6. We then introduce a new class of attacks called cube testers, based on efficient property-testing algorithms, and apply them to MD6 and to the stream cipher Trivium. Unlike the standard cube attacks, cube testers detect nonrandom behavior rather than performing key extraction, but they can also attack cryptographic schemes described by nonrandom polynomials of relatively high degree. Applied to MD6, cube testers detect nonrandomness over 18 rounds in 2 17 complexity; applied to a slightly modified version of the MD6 compression function, they can distinguish 66 rounds from random in 2 24 complexity. Cube testers give distinguishers on Trivium reduced to 790 rounds from random with 2 30 complexity and detect nonrandomness over 885 rounds in 2 27, improving on the original 767-round cube attack.