Abstract: Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was realized in antiquity, when it was proved that in some sense they are the “building blocks” of the set of the positive integers N. Nowadays, the modern applications of prime numbers in areas such as physics, cryptography, and coding theory make any information that can be obtained about them of great value. In this paper we are interested in the distribution of prime numbers in M, where M ⊆ N is a given infinite set. This is a fundamental question, and it has been analyzed by some of the greatest mathematicians of all time, such as Euclid, Euler, Gauss, Dirichlet, Chebyshev, Riemann, Landau, Wiener, Hardy, Erdos, and many more. Their efforts gave birth to many new important mathematical ideas, e.g. analytic number theory, Riemann zeta function, and tauberian theorems. To make the problem concrete, let pn denote the nth prime in M, π(x) be the number of the primes in M not exceeding x, and δn = pn+1−pn. Here are some general “distribution” questions:

Abstract: Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

Abstract: Let G be a finite group. The prime graph of G is the graph whose vertex set is the set of all prime divisors of |G|, and two distinct primes p and q are joined by an edge if and only if G contains an element of order pq. A group M is called a CIT group or a C22 group if M is of even order and the centralizer of any involution is a 2-group. In this paper we determine finite groups G such that their prime graph is the same prime graph of M, where M is a CIT simple group. As a consequence of this result, we prove that if p>7 is a Mersenne prime or a Fermat prime, then PSL(2,p) is uniquely determined by its prime graph. Also we prove a few results by using the main theorem.

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: A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1 0$ depends only on $\alpha$. We also prove a similar result for primes $p=[\alpha n+\beta]$ such that $p\equiv a\pmod q$.

Abstract: Prime generating polynomial functions are known that can produce sequences of prime numbers (e.g. Euler polynomials). However, polynomials which produce consecutive prime numbers are much more difficult to obtain. In this paper, we propose approaches for both these problems. The first uses Cartesian Genetic Programming (CGP) to directly evolve integer based prime-prediction mathematical formulae. The second uses multi-chromosome CGP to evolve a digital circuit, which represents a polynomial. We evolved polynomials that can generate 43 primes in a row. We also found functions capable of producing the first 40 consecutive prime numbers, and a number of digital circuits capable of predicting up to 208 consecutive prime numbers, given consecutive input values. Many of the formulae have been previously unknown.

Abstract: In the RSA system, balanced modulus Ndenotes a product of two large prime numbers pand q, where q " , and are denoted as p E and q E . Thus p E and q E can be adopted to estimate p+ qmore accurately than by simply adopting $2\sqrt{N}$. In addition, we show that the Verheul and Tilborg's extension of the Wiener attack can be considered to be brute-guessing for the MSBs of p+ q. Comparing with their work, EPF can extend the Wiener attack to reduce the cost of exhaustive-searching for 2r+ 8 bits down to 2ri¾? 10 bits, where rdepends on Nand the private key d. The security boundary of private-exponent dcan be raised 9 bits again over Verheul and Tilborg's result.

Abstract: For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever the candidate base surface is not the 2-sphere, while for the 2-sphere exceptions do occur. A long-standing conjecture however asserts that when the candidate degree is a prime number, a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence: (1) The degrees giving realizable covers have asymptotically zero density in the naturals; (2) Each prime degree gives a realizable cover.

Abstract: We show that the set of prime factors of almost all integers are "Poisson distributed", and that this remains true (appropriately formulated) even when we restrict the number of prime factors of the integer. Our results have inspired analogous results about the distribution of cycle lengths of permutations.

Abstract: In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer–Siegel theorem. Using this we deduce an explicit version of the generalised Brauer–Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields. The classical Brauer–Siegel theorem is a well-known theorem which describes the behaviour of the quantity hR (the product of the class number and the regulator) in a family of number fields with growing genus under the conditions that the genus grows much faster than the degree and some additional properties like normality or the Generalised Riemann Hypothesis (GRH) to deal with the Siegel zeroes. These two hypotheses are of different nature: omitting the first one changes the final result, while the second one is a technical hypothesis. Tsfasman and Vluadut¸ [8] were able to remove the first hypothesis, which led to the so called generalised Brauer–Siegel theorem, and Zykin [10] was able to replace ”normality” by ”almost normality” in the second one using results of Stark and Louboutin. He also managed to generalise the Brauer–Siegel theorem to the case of smooth absolutely irreducible projective varieties over finite fields. As for the Mertens theorem, proven by Mertens in the case of Q, and much later generalised by Rosen [5] both in cases of number and function fields, it can be regarded as the Brauer–Siegel theorem in the finite steps of the family. An explicit Mertens theorem leads therefore to an explicit formulation of the generalised Brauer–Siegel theorem. We first recall the formulations of the (generalised) Brauer–Siegel theorem and Mertens theorem, then we prove their explicit versions for number fields and smooth projective absolutely irreducible varieties over finite fields, and finally we deduce the explicit generalised Brauer Siegel theorem.

Abstract: Let P denote the set of prime numbers, and let P.n/ denote the largest prime factor of an integer n > 1. We show that, for every real number 32=17 1s uch that for every integer a � 0, the set p ∈ P : p = P.q − a/ for some prime q with p � < q < c.�/ p �

Abstract: We develop a fast algorithm for the construction of good rank-1 lattice rules which are a quasi-Monte Carlo method for the approximation of multivariate integrals. A popular method to construct such rules is the component-by-component algorithm which is able to construct good lattice rules that achieve the optimal theoretical rate of convergence. The construction time of this algorithm is O(sn), or O(sn) when using O(n) memory, for an s-dimensional lattice rule with n points. We show how to construct good lattice rules in time O(sn log(n)), using O(n) memory, by means of a new algorithm, called the fast component-by-component algorithm. First this is shown for the base case when n is a prime number and the underlying function space is a weighted, shift-invariant and tensor-product reproducing kernel Hilbert space. Then we show that, by a minor increase in construction cost, also more generally weighted function spaces can be handled by the fast algorithm. In particular we show this for order-dependent weights. When n is not a prime number it turns out that fast construction is also possible, although the construction is more involved for numbers n which have a large number of unique prime factors. An additional advantage is obtained when choosing n to be a prime power, since then the rules are embedded for increasing powers of the prime. Using this embedding, we propose a new fast algorithm to construct lattice sequences which can be used point by point. Two natural extensions of the algorithm are the construction of polynomial lattice rules and so called copy rules. We show that also here the fast componentby-component algorithm can be applied. The quality of the constructed point sets is finally demonstrated on some finance and statistics examples.

Abstract: In this paper, we propose an efficient password-authenticated key exchange (PAKE) based on RSA, called RSA-EPAKE. Unlike SNAPI using a prime pubic key e greater than an RSA modulus n, RSA-EPAKE uses the public key e of a 96-bit prime, where e=2H(n, s)+1 for some s. By the Prime Number Theorem, it is easy to find such an s. But the probability that an adversary finds n and s with $\gcd(e, \phi(n)) eq 1$ is less than 2−80. Hence, in the same as SNAPI, RSA-EPAKE is also secure against e-residue attacks. The computational load on Alice (or Server) and Bob (or Client) in RSA-EPAKE is less than in the previous RSA-based PAKEs such as SNAPI, PEKEP ,CEKEP, and QR-EKE. In addition, the computational load on Bob in RSA-EPAKE is less than in PAKEs based on Diffie-Hellman key exchange (DHKE) with a 160-bit exponent. If we exclude perfect forward secrecy from consideration, the computational load on Alice is a little more than that in PAKEs based on DHKE with a 160-bit exponent. In this paper, we compare RSA-EPAKE with SNAPI, PEKEP, and CEKEP in computation and the number of rounds, and provide a formal security analysis of RSA-EPAKE under the RSA assumption in the random oracle model.

Abstract: In this article, we have proved the correctness of the PublicKey Cryptography and the Pepin’s Test for the Primality of Fermat Numbers (F (n) = 2 n + 1). It is a very important result in the IDEA Cryptography that F(4) is a prime number. At first, we prepared some useful theorems. Then, we proved the correctness of the Public-Key Cryptography. Next, we defined the Order’s function and proved some properties. This function is very important in the proof of the Pepin’s Test. Next, we proved some theorems about the Fermat Number. And finally, we proved the Pepin’s Test using some properties of the Order’s Function. And using the obtained result we have proved that F(1), F(2), F(3) and F(4) are prime number.

Abstract: Let p be a prime number, and let K be a finite extension of the field ℚ p of p-adic numbers. Let N be a fully ramified, elementary abelian extension of K. Under a mild hypothesis on the extension N/K, we show that every element of N with valuation congruent mod [N:K] to the largest lower ramification number of N/K generates a normal basis for N over K.

Abstract: We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.

Abstract: We generalize the classical definition of zeta-regularization of an infinite product. The extension enjoys the same properties as the classical definition, and yields new infinite products. With this generalization we compute the product over all prime numbers answering a question of Ch. Soule. The result is 4π2. This gives a new analytic proof, companion to Euler’s classical proof, that the set of prime numbers is infinite.

Abstract: In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to that of all the pro-p-groups with countably many generators.

Abstract: A portable electronic device (10) for exchanging encrypted data with other electronic devices includes a processor (22), a memory (16) operatively coupled to the processor, and a prime number generation circuit (12a) operatively coupled to the processor and memory. The prime number generation circuit includes logic that generates at least two prime numbers based on unique data stored in the electronic device, wherein said at least two prime numbers are always the same at least two prime numbers. The generated prime numbers then can be used to generate RSA public and private keys within the electronic device (10).

Abstract: We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{\alpha,\beta}$, where $\alpha>1$ is irrational and of finite type.

Abstract: Let p be an odd prime number, r a natural number, and H4pr the generalized quaternion group of order 4p r . Let k be a number field and Cl(k) its class group. Let Rm(k,H4pr) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to H4pr. In this article, we determine Rm(k,H4pr) and show that it is a subgroup of Cl(k). In particular, Rm(k,H4pr) is the full group Cl(k) if p does not divide the class number of k, or if either k contains a primitive p r th root of unity or p is unramified in k.

Abstract: The multiparty multiplication of two polynomially shared values over Zq with a public prime number q is an important module in distributed computations. The multiplication protocol of Gennaro, Rabin and Rabin (1998) is considered as the best protocol for this purpose. It requires a complexity of O(n2k log n + nk2) bit-operations per player, where k is the bit size of the prime q andn is the number of players. The present paper reduces this complexity to O(n2k + nk2) by using Newton's classical interpolation formula. The impact of the new method on distributed signatures is outlined.

Abstract: We prove that two sequences arising from two different domains are equal. The first one, {d(n)}"n"@?"N, comes from the following power expansion: (-ln(1-x)x)^m=(@?k=1+~x^kk+1)^[email protected]?n=0~B"n(m)d(n)x^n where B"n(X) is a primitive polynomial of Z[X]. The second sequence, {e(n)}"n"@?"N, is the factorial sequence of the set of prime numbers or, equivalently, e(n) is the denominator of the polynomials of degree @?n+1 that take integral values for all prime numbers.

Abstract: In this research, we proposed a parallel processing algorithm that runs on cluster architecture suitable for prime number generation. The proposed approach is meant to decrease computational cost and accelerate the prime number generation process. Several experimental results are shown to demonstrate the viability of our work.

Abstract: Algorithms are proposed for computing the basis of the solution set of a system of linear Diophantine homogeneous or inhomogeneous equations in the residue field modulo a prime number.

Abstract: We are studying representations obtained from ac- tions of Galois groups on torsors of paths on a projective line minus a finite number of points. Using these actions on torsors of paths, we construct geometrically representations of Galois groups which re- alize � -adically the associated graded Lie algebra of the fundamental group of the tannakian category of mixed Tate motives over SpecZ, Spec Z(i), Spec Z( 1 ), Spec OQ( √ −q) for any prime number q (q � in the last case) and over Spec O Q( √ −q) ( 1 ) for any prime number q congruent to 3 modulo 4 and also for q =2 .

Abstract: In this paper, we consider the problem of Mutually Unbiased Bases in prime dimension $d$. It is known to provide exactly $d+1$ mutually unbiased bases. We revisit this problem using a class of circulant $d \times d$ matrices. The constructive proof of a set of $d+1$ mutually unbiased bases follows, together with a set of properties of Gauss sums, and of bi-unimodular sequences.