From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

Invariant pattern recognition: A review☆

In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

Generalized FFTS - A Survey of Some Recent Results

It is known that the discrete Fourier transform (DFT) used in digital signal processing can be characterized in the framework of the representation theory of algebras, namely, as the decomposition matrix for the regular module ${\mathbb{C}}[Z_n] = {\mathbb{C}}[x]/(x^n - 1)$. This characterization provides deep insight into the DFT and can be used to derive and understand the structure of its fast algorithms. In this paper we present an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of Chebyshev polynomials. Then we derive most of their known algorithms by pure algebraic means. We identify the mathematical principle behind each algorithm and give insight into its structure. Our results show that the connection between algebra and digital signal processing is stronger than previously understood.

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The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms

Permutations are ubiquitous in many real-world problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the "low-frequency" terms of a Fourier decomposition to represent distributions over permutations compactly. We present Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations. Low order Fourier-based approximations, however, may lead to functions that do not correspond to valid distributions. To address this problem, we present a quadratic program defined directly in the Fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camera-based multi-person tracking scenario.

/pdf/fourier-theoretic-probabilistic-inference-over-permutations-2r2474id88.pdf

Fourier Theoretic Probabilistic Inference over Permutations

Fast Fourier transforms

Recently, it has been proved that a Fourier transform for the symmetric group S, based on Young's seminormal form can be evaluated in less than 0.5(n3 + n2)n! arithmetic operations. We look at this algorithm in more detail and show that it allows an efficient software implementation using appropriate data structures. We also describe a similarly efficient algorithm for the inverse Fourier transform. We compare the time and memory requirements of our program to those of other existing implementations.

/pdf/fast-fourier-transforms-for-symmetric-groups-theory-and-5d5dypikvc.pdf

Fast Fourier transforms for symmetric groups : theory and implementation

The linear complexityL
K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L
K(G):= min{L
K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L
K(G)≤8.5|G|log|G| for any supersolvable groupG andL
K(G)≤1.5|G|log|G| for any 2-groupG.

Existence and efficient construction of fast Fourier transforms on supersolvable groups

Applying an interpolation algorithm invented by D. V. Chudnovsky and G. V. Chudnovsky to the Fermat-curvex3 + y3 = 1 overF4, we present a bilinear algorithm for the multiplication in theF4-algebraF256 which requires 8 essential multiplications.

An optimal algorithm for multiplication in F256/F4

For $2 \leqq c \leqq \infty $, the c-linear complexity $L_c (A)$ of a complex matrix A is defined as the minimal number of additions, subtractions, and multiplications by complex constants of absol...

Ulrich Baum

Papers

Fast Fourier transforms

Fast Fourier transforms for symmetric groups : theory and implementation

Existence and efficient construction of fast Fourier transforms on supersolvable groups

An optimal algorithm for multiplication in F256/F4

Some lower and upper complexity bounds for generalized Fourier transforms and their inverses