A fast generalized DFT for finite groups of lie type
Authors: 
Chloe Ching-Yun Hsu and Chris UmansAuthors Info & Claims
Chloe Ching-Yun Hsu and Chris UmansAuthors Info & ClaimsJanuary 2018
Pages 1047 - 1059
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 [EQUATION] exponent for general finite groups G, which beats the longstanding previous best upper bound (assuming ω = 2) of 3/2.
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- SIGACT: ACM Special Interest Group on Algorithms and Computation Theory
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Published: 07 January 2018
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