Lattice Sieving in Three Dimensions for Discrete Log in Medium Characteristic

Gary McGuire, Oisín Robinson
2020 Journal of Mathematical Cryptology  
AbstractLattice sieving in two dimensions has proven to be an indispensable practical aid in integer factorization and discrete log computations involving the number field sieve. The main contribution of this article is to show that a different method of lattice sieving in three dimensions will provide a significant speedup in medium characteristic. Our method is to use the successive minima and shortest vectors of the lattice instead of transition vectors to iterate through lattice points. We
more » ... howcase the new method by a record computation in a 133-bit subgroup of ${{\mathbb{F}}_{{{p}^{6}}}}$, with p6 having 423 bits. Our overall timing is nearly 3 times faster than the previous record of a 132-bit subgroup in a 422-bit field. The approach generalizes to dimensions 4 or more, overcoming one key obstruction to the implementation of the tower number field sieve.
doi:10.1515/jmc-2020-0008 fatcat:t5rj4mip6za4xiztfjrjmfenjm