Short Cycles in Repeated Exponentiation Modulo a Prime

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system

Periodic Structure of the Exponential Pseudorandom Number Generator

We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$

Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods

Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficiently large primes and gave estimates for the number of such pairs when g and h are primitive roots modulo p. In 2000, Campbell showed that the answer to Brizolis was "yes" for all primes. The first author has extended this question to questions about counting fixed points, two-cycles, and collisions of the discrete exponential map. In this paper, we use p-adic methods, primarily Hensel's lemma and p-adic interpolation, to count fixed points, two cycles, collisions, and solutions to related equations modulo powers of a prime p.Comment: 14 pages, no figure

Software and hardware implementation of the RSA public key cipher

Cryptographic systems and their use in communications are presented. The advantages obtained by the use of a public key cipher and the importance of this in a commercial environment are stressed. Two two main public key ciphers are considered. The RSA public key cipher is introduced and various methods for implementing this cipher on a standard, nondedicated, 8 bit microprocessor are investigated. The performance of the different algorithms are evaluated and compared. Various ways of increasing the performance are considered. The limitations imposed by the performance on the practical use of the cipher are discussed. The importance of the key to the security of the cipher is assessed. Different forms of attack are mentioned and a procedure for generating keys, which minimise the probability of a sucessful attack is presented. This procedure is implemented on a minicomputer. Use of the method on personal computers or microprocessors is examined. Methods for performing multiplication in hardware, with particular emphasis on the use of these methods in modular multiplication, are detailed. An algorithm for performing part of the encryption function in hardware and the hardware necessary for it is described. Different methods for implementing the hardware are discussed and one is choosen. A description of the hardware unit is given. The design and development of an application specific integrated circuit (ASIC) to perform key elements of the encryption function is described. The various stages of the design process are detailed. The results expected from this device and its integration into the overall encryption scheme are presented