Search

In this paper, we proposed a topologically conjugate map, equivalent to the well known logistic map. This constructed map is defined on the integer domain [0, 2 n ) with a view to be used as a random number generator (RNG) based on an integer domain as is the required in classical cryptography. The maps were found to have a one to one correspondence between points in their respective defining intervals defined on an n-bits precision. The dynamics of the proposed map similar with that of the logistic map, in terms of the Lyapunov exponents with the control parameter. This similarity between the curves indicates topological conjugacy between the maps. With a view to be applied in cryptography as a Pseudo-Random number generator (PRNG), the complexity of the constructed map as a source of randomness is determined using both the permutation entropy (PE) and the Lempel-Ziv (LZ-76) complexity measures, and the results are compared with numerical simulations.

An elliptic curve E defined over a finite field K, E(K) is the set of solutions to the general Weierstrass polynomial E: y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 where the coefficients a1, a2, a3, a4, a6 є K. There exist a well defined addition of points on each curve such that the points form an abelian group under the addition operation. This group is either cyclic or isomorphic to the product of two cyclic groups. These set of solutions that form the group lie in the closure of the field K over which the curve is defined. If we allow the set to lie only in a particular extension of K, the addition operation is well defined there too. Therefore we can associate a group to every extension K' of the field K denoted by E(K'). Will the structure of the group defined over the base field K, be affected if the same group is made to lie in the extension K' of K?

In this article, we have investigated synchronization phenomenon between two discrete chaotic systems. A general scheme for synchronization between two discrete maps with adaptive coupling has been studied analytically. The scheme can be successfully implemented for generalized synchronization between two chaotic maps. Conditional Lyapunov exponents (CLE) and Transverse Lyapunov exponents (TLE) can quantifies the robustness of synchronization. A secure communication scheme based on synchronization between two Logistic maps is also demonstrated. Numerical results show the effectiveness of our proposed scheme.

The generation of pseudo-random numbers (bits) plays a critical role in a large number of applications such as statistical mechanics, numerical simulations, gaming industry, communication or cryptography. The choice of secret keys for cryptographic primitives largely depends on the quality of random numbers used. These random numbers are fundamental tools in the generation of secret keys and initialization variables of encryption for cryptographic application, masking protocols, or for internet gambling. Chaotic Pseudorandom numbers were found to be very efficient in this aspect. The relevance of chaotic pseudorandom sequences in ensuring security in cryptosystems is considered, at the same time reviewing statistical tests required to make such sequences cryptographically secure. This paper intends to review the development of chaotic pseudorandom number generators through the years and the statistical tests they are required to pass as a measure of their randomness.