Some Mathematical Remarks On The Polynomial Selection In Nfs

In this work, we consider the proportion of friable (free of large prime factors) values of a binary form F(X-1, X-2) is an element of Z[X-1, X-2]. In the particular case of quadratic forms, we give an asymptotic equivalent for this proportion which depends on F. This is related to Murphy's a function, which is known in the cryptologic community, but which has not been studied before from a mathematical point of view. This has consequences on the first step, called polynomial selection, of the Number Field Sieve, the fastest algorithm of integer factorization.