Let n be an RSA modulus, that is, n = pq, where p, q are two large primes. We deene the discrete logarithm problem for Lucas sequences and show that solving the discrete logarithm problem for Lucas sequences modulo n gives a polynomial factorisation algorithm. Let d = jq?pj. We calculate m and Vd(m; 1) mod n in polynomial time and then solve for d in O(~ d 1 2 +) time where ~ d is an upper esti...

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomial p lie in the convex hull of the roots of p, called the Lucas polygon of p. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots of p′ lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon of p. Based on a simple proof...

Robert Lucas' ([1972b] 1981a) article on the neutrality of money represented first effective challenge to Samuelson’s neoclassical synthesis methodological separation between static microeconomic optimisation and macroeconomic dynamics. Lucas rejected disequilibrium price dynamics, as expressed by Walrasian tâtonnement auctioneer mechanisms. Lucas’ new treatment equilibrium an expectational con...

From a special sequence of squares of k-Fibonacci numbers, the kLucas sequences are obtained in a natural form. Then, we will study the properties of the k-Lucas numbers and will prove these properties will be related with the k-Fibonaci numbers. In this paper we examine some of the interesting properties of the k-Lucas numbers themselves as well as looking at its close relationship with the k-...

We introduce the notions of unsigned and signed generalized Lucas sequences and prove certain polynomial recurrence relations on their characteristic polynomials. We also characterize when these characteristic polynomials are irreducible polynomials over a finite field. Moreover, we obtain the explicit expressions of the remainders of Dickson polynomials of the first kind divided by the charact...

We describe public key cryptosystems and analyse the RSA cryptosystem, pointing out a weakness (already known) of the RSA system. We define Lucas functions and derive some of their properties. Then we introduce a public key system based on Lucas functions instead of exponentiation. The computational requirements of the new system are only a little greater than those for the RSA system, and we p...

The aim of this paper is to present an original modelling tool for biogeographical systems. This software named LUCAS (LUcifer Cellular Automata Simulator) has been developed within the European programme LUCIFER (Land Use Change Interactions with Fire in MeditERrranean landscapes). LUCAS constitutes general software, based on coupling local dynamics described by a set of nonlinear ordinary dif...

Let f be a binary string and d C 1. Then the generalized Lucas cube Qd(JÐf ) is introduced as the graph obtained from the d-cube Qd by removing all vertices that have a circulation containing f as a substring. The question for which f and d, the generalized Lucas cube Qd(JÐf ) is an isometric subgraph of the d-cube Qd is solved for all binary strings of length at most five. Several isometricall...