In this paper we propose an algorithm for computing large primes p and q such that q divides p4 + p3 + p2 + p+ 1 or p4 − p3 + p2 − p+ 1. Such primes are key parameters for the GiulianiGong Public Key System.

We develop a method for generating shared, secret, safe primes applicable to use in threshold RSA signature schemes such as the one developed by Shoup. We would like a scheme usable in practical settings, so our protocol is robust and efficient in asynchronous, hostile environments. We show that the techniques used for robustness need special care when they must be efficient. Specifically, we s...

The primary goal of our project is to implement the RSA cryptosystem in Maple 8. Our implementation includes finding primes suitable for RSA, translating text messages into blocks of large numbers and back, as well as encrypting and decrypting the numerical blocks. We found that our program could quickly encrypt long messages using primes with at least 512 bits and was capable of encrypting mes...

The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation [4]; for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors [2, 3]. ...

A prime number sieve is an algorithm that nds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O(p n=(log log n) 2) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l > 1...

An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the rst prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: { it employs only (n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for xed n; { the transition from n to n + 1 takes on...

In the [16] has been proven that the multiplicative group Z/pZ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. M...

Gennaro, Krawczyk and Rabin gave the first undeniable signature scheme based on RSA signatures. However, their solution required the use of RSA moduli which are a product of safe primes. This paper gives techniques which allow RSA-based undeniable signatures for gen-

A BBS-like generator is discussed for which the quadratic function F : x 7→ x − 2 (mod n), where n = p · q is the product of two distinct primes, is used. The maximal cycle length of the orbits produced by iterating F is obtained and the particular important cases in which p, q are both 1and 2-safe are analyzed in deeper detail.