Generation of a message m of order (n) for a test of the integrity of the generation of a pair of cryptographic keys within the multiplicative group of integers modulo n=p.Math.q, including: key pair generation including, to generate p and q: a random selection of candidate integers; and a primality test; a first search of the multiplicative group of integers modulo p for a generator a; a second search of the multiplicative group of integers modulo q for a generator b; a third search for a number y, as message m, verifying: 1n1, where =a mod p and =b mod q, the first or second search being performed during the primality test.

This disclosure relates to methods for generating a prime number, which can be implemented in an electronic device. An example method can include calculating a candidate prime number using a formula Pr=2P.Math.R+1, where P is a prime number and R is an integer. The method can also include applying the Pocklington primality test to the candidate prime number and rejecting the candidate prime number if it fails the Pocklington test. The integer can be generated from an invertible number belonging to a set of invertible elements modulo the product of numbers belonging to a group of small prime numbers greater than 2, where the candidate prime number is not divisible by any number of the group. The prime number P having a number of bits equal to within one bit, to half or a third of the number of bits of the candidate prime number.

The invention relates to a method for generating a prime number, implemented in an electronic device, the method including steps of generating a prime number from another prime number using the formula Pr=2P.Math.R+1, where P is a prime number having a number of bits lower than that of the candidate prime number, and R is an integer, and applying the Pocklington primality test to the candidate prime number, the candidate prime number being proven if it passes the Pocklington test. According to the invention, the size in number of bits of the candidate prime number is equal to three times the size of the prime number, to within one unit, the generated candidate prime number being retained as candidate prime number only if the quotient of the integer division of the integer by the prime number is odd.

A new mathematical technique called the T-sequence is a primality testing method. A similar approach can be applied to perform fast factoring for numerous special cases, a method that can, in all likelihood, be extended to the general case, making possible a general and fast factoring algorithm. The same T-sequence can be used to construct a prime number formula and a good random number generator.