Faster Initial Splitting for Small Characteristic Composite Extension Degree Fields

Abstract

Abstract Let p be a small prime and n = n 1 n 2 > 1 be a composite integer. For the function field sieve algorithm applied to F p n , Guillevic (2019) had proposed an algorithm for initial splitting of the target in the individual logarithm phase. This algorithm generates polynomials and tests them for B-smoothness for some appropriate value of B. The amortised cost of generating each polynomial is O ( n 2 2 ) multiplications over F p n 1 . In this work, we propose a new algorithm for performing the initial splitting which also generates and tests polynomials for B-smoothness. The advantage over Guillevic splitting is that in the new algorithm, the cost of generating a polynomial is O ( n log p \u2061 ( 1 / π ) ) multiplications in F p , where π is the relevant smoothness probability.