Proth Search Page for Proth--the Windows program by Yves Gallot Yves Gallot wrote an excellent Win95/98/NT4.0 program which makes it easy for anyone to find record size or otherwise interesting primes, but this creates a problem: without a coordinated effort, many of us were be searching the same ranges of numbers for primes! Some spent hundreds of hours checking ranges that were already known ...

Carmichael Number ...

Lucas Sequences in Cryptography This is a short note on the practical usefulness of Lucas sequences in applied cryptography. A Lucas sequence is a sequence of integers characterized by two parameters, P and Q. In practice Q is always 1 and the sequence is taken modulo a large integer. Calculating an element of a Lucas sequence is very similar to exponentiation. It may be helpful to think of P as ...

Least primitive root of prime numbers Least prime primitive root of prime numbers Least base necessary to prove the primality of a number Introduction Results References Links Contact Introduction Let p be a prime number. Fermat's little theorem states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between 1 and p-1. A primitive root of p is a number r such that any ...

Lucas's Primality Test With Factored N-1 Fermat's Little Theorem assures us that if N is a prime then b^(N-1) = 1 (mod N) (1) for every integer b coprime to N. In contrast, if N is composite it is quite rare for the above congruence to be satisfied for ANY b. This fact enables us to test for compositeness quite easily, simply by checking to see if b^(N-1) is congruent to 1 (mod N) for some ...

Generalized Carmichael Numbers Ashish Kumar Singh (97076) and Surendra Kumar Pathak (97354), April, 2001 Supervisor: Prof. Manindra Agrawal Fermat's little theorem has been the basis of many primality testing algorithms. However the major problem with these test has been the existence of Carmichael Numbers. In this report we have tried to extend the Fermat's test to bigger fields and we ...