Let G be an arbitrary cyclic group with generator g and order jGj with known factorization. G could be the subgroup generated by g within a larger group H. Based on an assumption about the existence of smooth numbers in short intervals, we prove that breaking the Diie-Hellman protocol for G and base g is equivalent to computing discrete logarithms in G to the base g when a certain side informat...

Let G be an arbitrary cyclic group with generator g and order jGj with known factorization. G could be the subgroup generated by g within a larger group H. Based on an assumption about the existence of smooth numbers in short intervals, we prove that breaking the Diie-Hellman protocol for G and base g is equivalent to computing discrete logarithms in G to the base g when a certain side informat...

Remember that we are using natural logarithms, where the base is e ≈ 2.71828. Logarithms may have other bases, for instance the decimal logarithm of base 10. (The base 10 logarithm is used in the definition of the Richter scale, for instance, measuring the intensity of earthquakes as Richter = log(intensity). This is why an earthquake of magnitude 9 is 100 times more powerful than an earthquake...

Let G be a group of prime order q with generator g. We study hardcore subsets H ⊂ G of the discrete logarithm (DL) logg in the model of generic algorithms. In this model we count group operations such as multiplication and division, while computations with non-group data are for free. It is known from Nechaev [Math. Notes 55 (1994) 165] and Shoup [Lecture Notes in Comp. Sci., Vol. 1233, Springe...

Cryptographic applications play a significant role in secure and private communication of information. Secrecy and security in areas such as military intelligence are of paramount importance. Many of the modern day cryptographic algorithms are deemed to be secure, based on the intractability of the discrete logarithm problem over the elliptic curves defined in Galois fields. The purpose of this...

We investigate the asymptotic behavior of the partition function pΛ(n) defined by ∑∞ n=0 pΛ(n)x n = ∏∞ m=1(1−x), where Λ(n) denotes the von Mangoldt function. Improving a result of Richmond, we show that log pΛ(n) = 2 √ ζ(2)n+O( √ n exp{−c(logn)(log2 n)(log3 n)−1/3}), where c is a positive constant and logk denotes the k times iterated logarithm. We also show that the error term can be improved...

In this report we provide a detailed description of four signcryption schemes all of which have been proven secure. The first two of the schemes are based on discrete logarithm on finite fields and the other two schemes are based on elliptic curves. We also discuss issues on the implementation of signcryption schemes.

This report discusses the elliptic curve discrete logarithm problem and the known methods to solve it. We consider the implications of these methods for choosing the domain parameters in elliptic curve based cryptographic schemes. We also study special classes of elliptic curves. In particular, we discuss the security of Koblitz curves.

Barnes and Sloane recently described a "general construction" for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in U with density A satisfying log2 A ~ — m logf m, as m -> oo, where logf m is the smallest value of k for which the fc-th it...