Sentence examples for factorization of integers from inspiring English sources

However, Shor showed that factorization of integers and computation of discrete logarithm are done efficiently by using quantum computers [28].

What became clear along such lines is that in some of these extensions the Fundamental Theorem of Arithmetic asserting the unique factorization of all integers into powers of primes fails.

This cryptosystem is based on computationally hard problems, for example factorization of large integers and computation of discrete logarithm in large finite groups.

For example, Dedekind was the first to prove Fermat's two-square theorem using the unique factorization of Gaussian integers.

Factorization of large integers is believed to be a computationally very difficult problem, and the security of many modern cryptography systems is based upon its infeasibility.

In 1815, Carl Gauss used the Euclidean algorithm to demonstrate unique factorization of Gaussian integers, although his work was first published in 1832.

If one sets m ( 2 ) = 1 / 2, for notational convenience, then it appears that for any prime factorization of an integer greater than 2, m ( ∏ i = 1 k p i e i ) = ?

For example, the unique factorization of the Gaussian integers is convenient in deriving formulae for all Pythagorean triples and in proving Fermat's theorem on sums of two squares.

The identity between the Dirichlet series and the Euler product (taken over all prime numbers ) is an analytic version of the unique prime factorization in the ring of integers and reflects the importance of the zeta-function for number theory.

The security of this scheme was supposed to be based on the difficulty of integer factorization.

Hashimoto and Sakurai proposed a signature scheme (HS scheme), whose security is based on the difficulty of integer factorization.