It is widely known that if p and q are relatively prime positive integers then (a) the set of linear combinations of p and q with nonnegative integer coefficients includes all integers greater than pq − p − q, (b) exactly half the integers between 0 and pq − p − q belong to this set and (c) an integer m belongs to this set if and only if pq−p−q−m does not. A multidimensional version of statemen...

Blum integers (BL), which has extensively been used in the domain of cryptography, are integers with form p1q2 , where p and q are different primes both ≡ 3 mod 4 and k1 and k2 are odd integers. These integers can be divided two types: 1) M = pq, 2) M = p1q2 , where at least one of k1 and k2 is greater than 1. In [3], Bruce Schneier has already proposed an open problem: it is unknown whether th...

Let pq be a rational number. Numeration in base p q is defined by a function that evaluates each finite word over Ap = {0, 1, . . . , p− 1} to a rational number in some set Np q . In particular, Np q contains all integers and the literature on base pq usually focuses on the set of words that are evaluated to integers; it is a rather chaotic language which is not context-free. On the contrary, w...

We discuss the problem of factoring N = pq and survey some approaches. We then present a specialized factoring algorithm that runs in time Õ(q0.31), which is comparable to the runtime Õ(p) of the factoring algorithm for integers of the form N = pq presented in [1]. We then survey the factoring algorithm of [1] and discuss the number of advice bits needed for it to run in polynomial time. Furthe...

We consider two data-structuring problems which involve performing priority queue (pq) operations on a set of integers in the range 0 . . 2 − 1 on a unit-cost RAM with word size w bits. A monotone min-PQ has the property that the minimum value stored in the pq is a non-decreasing function of time. We give a monotone minpq that, starting with an empty set, processes a sequence of n insert and de...

We prove that the following are equivalent. We denote by Pq the vector space of functions from a finite field Fq to itself, which can be represented as the space Pq := Fq[x]/(x q − x) of polynomial functions. We denote by On ⊂ Pq the set of polynomials that are either the zero polynomial, or have at most n distinct roots in Fq. Given two subspaces Y,Z of Pq, we denote by 〈Y,Z〉 their span. • Let...

In this paper, we study some RSA-based semantically secure encryption schemes (IND-CPA) in the standard model. We first derive the exactly tight one-wayness of Rabin-Paillier encryption scheme which assumes that factoring Blum integers is hard. We next propose the first IND-CPA scheme whose one-wayness is equivalent to factoring general n = pq (not factoring Blum integers). Our reductions of on...

We propose a novel probabilistic public-key encryption, based on the RSA cryptosystem. We prove that in contrast to the (standardmodel) RSA cryptosystem each user can choose his own encryption exponent from a more extensive set of positive integers than it can be done by the creator of the concrete RSA cryptosystem who chooses and distributes encryption keys among all users. Moreover, we show t...

In 1995, Kuwakado, Koyama and Tsuruoka presented a new RSA-type scheme based on singular cubic curves y2 ≡ x3+bx2 (mod N) where N = pq is an RSA modulus. Then, in 2002, Elkamchouchi, Elshenawy and Shaban introduced an extension of the RSA scheme to the field of Gaussian integers using a modulus N = PQ where P and Q are Gaussian primes such that p = |P| and q = |Q| are ordinary primes. Later, in...

We consider binary abelian codes of length pq where p and q are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or bounds of these weights. We give examples showing that these bounds are attained in some cases.