It is well known that if the higher half bits of a prime factor are known or the secret key is small enough then the RSA cryptosystem is broken (e.g. [Coppersmith, J. Cryptology, 1997] and [Boneh-Durfee, Eurocrypt’99]). Recently, Sarkar-MaitraSarkar [Cryptology ePrint Archiv, 2008/315] proposed attacks against RSA under the conditions that the higher bits of a prime factor is known and the secr...

Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prim...

Motivated by recent work of Drmota, Mauduit and Rivat, we discuss the possibility of counting the number of primes up to x whose sum of digits is also prime. We show that, although this is not possible unless we assume a hypothesis on the distribution of primes stronger than what is implied by the Riemann hypothesis, we can establish a Mertens-type result. That is, we obtain a formula for the n...

where / is an odd prime and x, y, z are rational integers prime to each other and none zero, we shall refer to the case where xyz is prime to / as case I ; if xyz = 0 (mod I) then we call this case II . I now give a sketch of a proof of a theorem which appears to be the principal result I have so far found concerning the first case of the last theorem. THEOREM 1. If (1) is possible in case I, t...

1. We say that g(n) is additive if g(mn)=g(m)+g(n) holds for every coprime pairs m, n of positive integers . If, moreover, g(p') =g(p)' for every prime power pa , then g(n) is called strongly additive . By p, pi , p2 , . . ., q, qr , q2 , . . . we denote prime numbers, c, Cl, c2i . . . are suitable positive constants . F(n) and x(n) denote the largest and the smallest prime factor of n . The sy...

Given an n-bit integer N , output YES if n is prime and NO otherwise. This is one of the most basic questions about numbers, with the following history. • By definition Prime ∈ coNP, because the prime decomposition is a short certificate for a number that is not prime. • [Pratt'75] showed that Prime ∈ NP. The Pratt certificate of a number N being prime, is by looking at all prime factor q of N ...

A variety of strong and electroweak interaction properties of the pion and the light scalar σ meson are computed in a relativistic quark model. Under the assumption that the resulting coupling of these mesons to the constituent quarks is identical, the σ meson mass is determined as Mσ = 385.4 MeV. We discuss in detail the gauging of the non-local meson-quark interaction and calculate the electr...