Paper 2019/306

Faster Initial Splitting for Small Characteristic Composite Extension Degree Fields

Madhurima Mukhopadhyay and Palash Sarkar

Abstract

Let $p$ be a small prime and $n=n_1n_2>1$ be a composite integer. For the function field sieve algorithm applied to $\mathbb{F}_{p^n}$, 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 $\mathbb{F}_{p^{n_1}}$. 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 $O(n\log(1/\pi))$ multiplications in $\mathbb{F}_p$, where $\pi$ is the relevant smoothness probability.