Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj denotes the j-th iterated logarithm, so that log1 n = logn and logj n = log(logj−1 n) for each j ≥ 2.) The best upper bound known is Vinogradov’s |ζ(1 + it)| (log t). Littlewood had shown that |ζ(1+it...

In this paper, we propose a new signature scheme based on factoring and discrete logarithm. This scheme is based on two hard problems and provides higher level security as compare to a single hard problem. Most of the designated signature schemes are based on a single hard problem. Although these schemes secure but in a future if an enemy manages to solve this problem, are then he can recover a...

In this paper we present that some statistical properties of points on elliptic curve can be used to form new equivalence classes. This can have an impact on solving discrete logarithm (ECDLP) owing to the reduction of the number of points among which a logarithm is searched to points of particular features. It should lead to an improvement of the Pollard-rho algorithm.

In this paper we give an introduction to elliptic curve public key cryptosystems. We explain how the discrete logarithm in an elliptic curve group can be used to construct cryptosystems. We also focus on practical aspects such as implementation, standardization and intellectual property.

Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over nite elds. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are suuciently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not ...

This paper will examine the role of elliptic curves in the field of cryptography. The applicability of an analogous discrete logarithm problem to elliptic curve groups provides a basis for the security of elliptic curves. Two cryptographic protocols which implement elliptic curves are examined as well as two popular methods to solve the elliptic curve discrete logarithm problem. Finally, a comp...

The generalized secret sharing scheme is a method used to divide secret into a set of participants such that only the qualified subsets of participants can reconstruct the secret. In this paper, we first propose a new generalized secret sharing scheme with cheater detection. The security of the scheme is based on discrete logarithm problem. Based on this scheme, we present a group-oriented gene...

Frey and Rück gave a method to transform the discrete logarithm problem in the divisor class group of a curve over Fq into a discrete logarithm problem in some finite field extension Fqk . The discrete logarithm problem can therefore be solved using index calculus algorithms as long as k is small. In the elliptic curve case it was shown by Menezes, Okamoto and Vanstone that for supersingular cu...

Cryptographic protocols often make use of the inherent hardness of the classical discrete logarithm problem, which is to solve gx ≡ y (mod p) for x. The hardness of this problem has been exploited in the Diffie-Hellman key exchange, as well as in cryptosystems such as ElGamal. There is a similar discrete logarithm problem on elliptic curves: solve kB = P for k. Therefore, Diffie-Hellman and ElG...

In 1989, Koblitz proposed using the Jacobian of a hyperelliptic curve defined over a finite field to implement discrete logarithm cryptographic protocols. The discrete logarithm problem of the Jacobian is called hyperelliptic curve discrete logarithm problem (HCDLP). For a hyperelliptic curve of genus g over the finite field Fq, the group order of the Jacobian is ( ) g O q which is larger than ...