Abstract: In studying how to communicate over a public channel with an active adversary, Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor dramatically strengthens the notion of a strong extractor. A strong extractor takes two inputs, a weakly-random $x$ and a uniformly random seed $y$, and outputs a string which appears uniform, even given $y$. For a non-malleable extractor $ m$, the output $ m(x,y)$ should appear uniform given $y$ as well as $ m(x,\adv(y))$, where $\adv$ is an arbitrary function with $\adv(y) eq y$. We show that an extractor introduced by Chor and Gold reich is non-malleable when the entropy rate is above half. It outputs a linear number of bits when the entropy rate is $1/2 + \alpha$, for any $\alpha>0$. Previously, no nontrivial parameters were known for any non-malleable extractor. To achieve a polynomial running time when outputting many bits, we rely on a widely-believed conjecture about the distribution of prime numbers in arithmetic progressions. Our analysis involves a character sum estimate, which may be of independent interest. Using our non-malleable extractor, we obtain protocols for ``privacy amplification & quot;: key agreement between two parties who share a weakly-random secret. Our protocols work in the presence of an active adversary with unlimited computational power, and have asymptotically optimal entropy loss. When the secret has entropy rate greater than $1/2$, the protocol follows from a result of Dodis and Wichs, and takes two rounds. When the secret has entropy rate $\delta$ for any constant~$\delta>0$, our new protocol takes a constant (polynomial in $1/\delta$) number of rounds. Our protocols run in polynomial time under the above well-known conjecture about primes.

Abstract: Let q≥2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the re-partition of the sequence (sq(P(n)))n∈ in arithmetic progressions when P∈[XS is such that P()⊂. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence ( sq(P(n)))n∈ for ∈ , provided that q is a large enough prime number co-prime with the leading coefficient of P.

Abstract: In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal representations of GLm(\( \mathbb{A} \)E) × GLm′ (\( \mathbb{A} \)F) over cyclic algebraic number fields E and F which are invariant under the Galois action, by exploiting a result proved by Arthur and Clozel, and prove a prime number theorem for this L-function.

Abstract: A Wieferich prime is a prime p such that 2 p−1 ≡ 1 (mod p 2 ). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new search algorithm for Wieferich primes using double-precision Montgomery arithmetic and a memoryless sieve, which runs significantly faster than previously published algorithms, allowing us to report that there are no other Wieferich primes p < 6.7 × 10 15 . Furthermore, our method allowed for the efficent collectionof statistical data on Fermat quotients, leading to a strong empirical confirmation of a conjecture of Crandall, Dilcher, and Pomerance. Our methods proved flexible enough to search for new solutions of a p−1 ≡ 1 (mod p 2 ) for other small values of a, and to extend the search for Fibonacci-Wieferich primes. We conclude, among other things, that there are no Fibonacci-Wieferich primes less than p < 9.7 × 10 14 .

Abstract: Let p be a prime number. For a positive integer n and a p -adic number ξ , let λ n ( ξ ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that qξ p , qξ 2 p ,..., qξ n p are all less than q − λ − 1 . Here, x p denotes the infimum of | x − n | p as n runs through the integers. We study the set of values taken by the function λ n

Abstract: Let p be an odd prime number. We define a family of quaternary sequences of period 2p using generalized cyclotomic classes over the residue class ring modulo 2p. We compute exact values of the linear complexity, which are larger than half of the period. Such sequences are 'good' enough from the viewpoint of linear complexity.

Abstract: Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$.

Abstract: Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N {0}. Let Cpn {ζ | ζpn 1} be the cyclic group of order p and let

Abstract: Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density $\delta_{q;a_1,...,a_r}$ of the set of real numbers $x$ such that $\pi(x;q,a_1)>\pi(x;q,a_2)>...>\pi(x;q,a_r),$ as $q\to\infty$; conditionally on the assumption of the Generalized Riemann Hypothesis GRH and the Grand Simplicity Hypothesis GSH. Several applications concerning these prime number races are then deduced. Indeed, comparing with a recent work of D. Fiorilli and G. Martin for the case $r=2$, we show that these densities behave differently when $r\geq 3$. Another consequence of our results is the fact that, unlike two-way races, biases do appear in races involving three of more squares (or non-squares) to large moduli. Furthermore, we establish a conjecture of M. Rubinstein and P. Sarnak (on biased races) in certain cases where the $a_i$ are assumed to be fixed and $q$ is large. We also prove that a conjecture of A. Feuerverger and G. Martin concerning "bias factors" (which follows from the work of Rubinstein and Sarnak for $r=2$) does not hold when $r\geq 3$. Finally, we use a variant of our method to derive Fiorilli and Martin asymptotic formula for the densities in two-way races.

Abstract: Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let \lambda_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute \lambda_A(h) and estimate its growth rate.

Abstract: In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus 2 and study various related issues, such as the probability of cyclicity and the probability of primality of the number of points on the curve itself. Finally, we discuss the asymptotic behavior as the genus tends to infinity.

Abstract: In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d>2, n>1, (n,d) eq (2,4), and \gcd(q,d)=\gcd(q,d-1)=1. This allows us to give a complete criterion in the case where q=p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p (d-1)^n then X is isomorphic to the Klein hypersurface, n=2 or n+2 is prime, and p=\Phi_{n+2}(1-d) where \Phi_{n+2} is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.

Abstract: We consider the divisibility of the class numbers of imaginary quadratic fields , where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 7 mod 8.

Abstract: For a number field k and a prime number p, let k∞ be the cyclotomic Zp-extension of k with finite layers kn. We study the finiteness of the Galois group X∞ over k∞ of the maximal abelian unramified p-extension of k∞ when it is assumed to be cyclic. We then focus our attention to the case where p = 2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of kn. As a consequence, we determine the complete list of real quadratic number fields for which X∞ is cyclic non trivial. We then apply these results to the study of Greenberg’s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.

Abstract: In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algorithm studied by Brandt and Damgaard and its many variants implemented in numerous software packages, which reduce the number of random bits used at the price of a distribution easily distinguished from uniform. Our new method is also no more computationally expensive than the ones in current use, and opens up interesting options for prime number generation in constrained environments.

Abstract: A phenomenon of interdependency between the structure of positive integers and the form of their prime factors is discovered. This paper was prepared for publication by E. A. Karatsuba and M. E. Changa, based on A. A. Karatsuba's drafts and notes from 2007-2008. Details of calculations are due to Changa. Bibliography: 10 titles.

Abstract: RSA is a typical algorithm of public key cryptography algorithm, analyzing the reason of large prime numbers which is the factor of restricting thisalgorithm's safety, and giving the method of determining the large prime numbers. Design fitness function, crossover and mutation strategies which can be used in genetic algorithm. Finally design the algorithm of producing large prime numbers.

Abstract: Recently, pairing-based cryptographic application sch-emes have attracted much attentions. In order to make the schemes more efficient, not only pairing algorithm but also arithmetic operations in extension field need to be efficient. For this purpose, the authors have proposed a series of cyclic vector multiplication algorithms (CVMAs) corresponding to the adopted bases such as type-I optimal normal basis (ONB). Note here that every basis adapted for the conventional CVMAs are just special classes of Gauss period normal bases (GNBs). In general, GNB is characterized with a certain positive integer h in addition to characteristic p and extension degree m, namely type- GNB in extension field Fpm. The parameter h needs to satisfy some conditions and such a positive integer h infinitely exists. From the viewpoint of the calculation cost of CVMA, it is preferred to be small. Thus, the minimal one denoted by hmin will be adapted. This paper focuses on two remaining problems: 1) CVMA has not been expanded for general GNBs yet and 2) the minimal hmin sometimes becomes large and it causes an inefficient case. First, this paper expands CVMA for general GNBs. It will improve some critical cases with large hmin reported in the conventional works. After that, this paper shows a theorem that, for a fixed prime number r, other prime numbers modulo r uniformly distribute between 1 to r-1. Then, based on this theorem, the existence probability of type- GNB in Fpm and also the expected value of hmin are explicitly given.

Abstract: The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe's fast algorithm for computing $R_1,R_2,...,R_n$.

Abstract: This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized O*(pk) time algorithm for testing p-monomials in an n-variate polynomial of degree k represented by an arithmetic circuit, while a deterministic O*((6.4p)k) time algorithm is devised when the circuit is a formula, here p is a given prime number. Second, we present a deterministic O*(2k) time algorithm for testing multilinear monomials in ΠmΣ2Πt×ΠkΣ3 polynomials, while a randomized O*(1.5k) algorithm is given for these polynomials. Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.

Abstract: In this article, we show that if G is a simply connected Chevalley group of either classical type of rank bigger than 1 or type E6, and q > 9 is a power of a prime number p > 5, then G = G(F_q((1/t))), up to an automorphism, has a unique lattice of minimum covolume, which is G(F_q[t]).

Abstract: In this paper, we study the diophantine equation 2 x +p y = z 2 where where p is a prime number and x;y and z are non-negative integers.

Abstract: In this work, we applied the classical numerical method of the secant in the p-adic case to calculate the cubic root of a p-adic number $a\in\mathbb{Q}_{p}^{\ast }$ where $p$ is a prime number, and this through the calculation of the approximate solution of the equation $x^{3}-a=0$. We also determined the rate of convergence of this method and evaluated the number of iterations obtained in each step of the approximation. Computing both the cubic root and other roots of a p-adic number is useful both for their theoretical values as for their theoretical applications in the field of theoretical computer science and cryptography.