A fast generalized DFT for finite groups of Lie type

Chloe Ching-Yun Hsu, Chris Umans

Abstract: We give an arithmetic algorithm using O(|G|ω/2+o(1)) operations to compute the generalized Discrete Fourier Transform (DFT) over group G for finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 2. Previously, “exponent one” algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known (even under the assumption ω = 2) for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for SL2(Fq) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group, and exponent one if ω = 2. We also show that ω = 2 implies a exponent for general finite groups G, which beats the longstanding previous best upper bound (assuming ω = 2) of 3/2.