16 research outputs found
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
The elliptic curve primality proving (ECPP) algorithm is one of the current
fastest practical algorithms for proving the primality of large numbers. Its
running time cannot be proven rigorously, but heuristic arguments show that it
should run in time O ((log N)^5) to prove the primality of N. An asymptotically
fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4).
The aim of this article is to describe this version in more details, leading to
actual implementations able to handle numbers with several thousands of decimal
digits
1. Kryptotag - Workshop über Kryptographie
Der Report enthält eine Sammlung aller Beiträge der Teilnehmer des 1. Kryptotages am 1. Dezember 2004 in Mannheim
A primality test for Kp^n+1 numbers
In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form N = Kpn +1. For these families, we present variations on the classical Pocklington's results and, in particular, a primality test whose computational complexity is Õ(log2 N) and, what is more important, that requires only one modular exponentiation modulo N similar to that of Fermat's test
A primality test for Kp^n+1 numbers
In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form N = Kpn +1. For these families, we present variations on the classical Pocklington's results and, in particular, a primality test whose computational complexity is Õ(log2 N) and, what is more important, that requires only one modular exponentiation modulo N similar to that of Fermat's test
Elements of high order in finite fields specified by binomials
Let be a field with elements, where is a power of a prime number . For any integer and such that the polynomial is irreducible in , we combine two different methods to explicitly construct elements of high order in the field . Namely, we find elements with multiplicative order of at least , which is better than previously obtained bound for such family of extension fields