The paper proposes justification of using Mersenne primes in the following cryptoschemes: commutative and publickey encryption algorithms and zero-knowledge protocol. The cryptoschemes are based on computational difficulty of finding discrete logarithm in the finite fields GF (2), where s is a sufficiently large prime such that 2s−1 is also a prime, for example s = 1279, s = 2203, and s = 4253.

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set’s underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the min...

In this paper, we report efficient implementations of the linear sieve and the cubic sieve methods for computing discrete logarithms over prime fields. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two times faster than the linear sieve method even in cases of small prime fields of the size about 150 bits. We also prov...

We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the elliptic curve discrete logarithm problem restricted to curves over the fields Fqi i can be solved in subexponential expected time (qi i ) . We also show that there exists a sequen...

We present a (random) polynomial-time algorithm to generate a randomGaussian integer with the uniform distribution among those with norm at most N , along with its prime factorization. The method generalizes to finding a random ideal in the ring of integers of a quadratic number field together with its prime ideal factorization. We also discuss the analogous problem for higher degree number fie...

For any prime p the density of elements in Fp having order, respectively index, congruent to a(mod d) is being considered. These densities on average are determined, where the average is taken over all finite fields of prime order. Some connections between the two densties are established. It is also shown how to compute these densities with high numerical accuracy.

This paper presents the results of numerical experiments to determine the probability, over concrete fixed finite fields, of prime-order elliptic curves having a prime-order twist. These curves are called “elliptic twins” by [7], and are useful for a variety of cryptographic applications. Most notable is that such curves are secure against an “insecure twist” attack. This attack was introduced ...