14 research outputs found
A new algorithm for fast generalized DFTs
We give an new arithmetic algorithm to compute the generalized Discrete
Fourier Transform (DFT) over finite groups . The new algorithm uses
operations to compute the generalized DFT over
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 is the exponent of matrix multiplication, so the exponent
is optimal if . 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 )
for families of linear groups of fixed dimension, and indeed the previous
best-known algorithm for had exponent despite being the focus
of significant effort. We unconditionally achieve exponent at most for
this group, and exponent one if . Our algorithm also yields an
improved exponent for computing the generalized DFT over general finite groups
, which beats the longstanding previous best upper bound, for any .
In particular, assuming , we achieve exponent , while the
previous best was
Fast Fourier Transforms for the Rook Monoid
We define the notion of the Fourier transform for the rook monoid (also
called the symmetric inverse semigroup) and provide two efficient
divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing
it. This paper marks the first extension of group FFTs to non-group semigroups
Fast generalized DFTs for all finite groups
For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication
Fast generalized DFTs for all finite groups
For any finite group , we give an arithmetic algorithm to compute
generalized Discrete Fourier Transforms (DFTs) with respect to , using
operations, for any . Here,
is the exponent of matrix multiplication
Branching Diagrams for Group Inclusions Induced by Field Inclusions
A Fourier transform for a finite group G is an isomorphism from the complex group algebra CG to a direct product of complex matrix algebras, which are determined beforehand by the structure of G. Given such an isomorphism, naive application of that isomorphism to an arbitrary element of CG takes time proportional to |G|2. A fast Fourier transform for some (family of) groups is an algorithm which computes the Fourier transform of a group G of the family in less than O(|G|2) time, generally O(|G| log |G|) or O(|G|(log |G|)2). I describe the construction of a fast Fourier transform for the special linear groups SL(q) with q = 2n
Fast generalized DFTs for all finite groups
For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication
Inverse semigroup spectral analysis for partially ranked data
Motivated by the notion of symmetric group spectral analysis developed by
Diaconis, we introduce the notion of spectral analysis on the rook monoid (also
called the symmetric inverse semigroup), characterize its output in terms of
symmetric group spectral analysis, and provide an application to the
statistical analysis of partially ranked (voting) data. We also discuss
generalizations to arbitrary finite inverse semigroups. This paper marks the
first non-group semigroup development of spectral analysis.Comment: v3: Significant changes in terms of organization and presentation.
Accepted for publication in Appl. Comput. Harmon. Anal. 25 pages, 5 tables.
v2: Some reformatting, minor typos correcte
Decimation-in-Frequency Fast Fourier Transforms for the Symmetric Group
In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research
Cryptographic schemes, key exchange, public key.
General cryptographic schemes are presented where keys can be one-time or ephemeral. Processes for key exchange are derived. Public key cryptographic schemes based on the new systems are established. Authentication and signature schemes are easy to implement.
The schemes may be integrated with error-correcting coding schemes
so that encryption/coding and decryption/decoding may
be done simultaneously