Generalized Fermat-Mersenne number theoretic transform

A generalization of the Fermat and Mersenne number transform is suggested. The transforms are defined over finite fields and rings. This paper establishes the conditions necessary for these numbers to be prime. The length of the transforms is a highly composite number. An algorithm for finding primitive roots of unity is also discussed. The proposed transforms are characterized by respectable combinations of transform length, dynamic range and computational efficiency and can be used for fast convolution of integer sequences. >

[1]  L. Dickson.  History of the Theory of Numbers , 1924, Nature.

[2]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[3]  Raphael M. Robinson,et al.  A report on primes of the form ⋅2ⁿ+1 and on factors of Fermat numbers , 1958 .

[4]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[5]  R. Singleton.  An algorithm for computing the mixed radix fast Fourier transform , 1969 .

[6]  Charles M. Rader,et al.  Discrete Convolutions via Mersenne Transrorms , 1972, IEEE Transactions on Computers.

[7]  C. Burrus,et al.  Fast Convolution using fermat number transforms with applications to digital filtering , 1974 .

[8]  S. Golomb.  Properties of the sequences 3⋅2ⁿ+1 , 1976 .

[9]  S. Winograd.  On computing the Discrete Fourier Transform. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[10]  L. Leibowitz.  A simplified binary arithmetic for the Fermat number transform , 1976 .

[11]  Trieu-Kien Truong,et al.  Fast number-theoretic transforms for digital filtering , 1976 .

[12]  Henri J. Nussbaumer.  Digital filtering using complex Mersenne transforms , 1976 .

[13]  J. Pollard.  Implementation of number-theoretic transforms , 1976 .

[14]  H. Nussbaumer.  Digital filtering using pseudo fermat number transforms , 1977 .

[15]  C. Jesshope,et al.  The solution of elliptic partial differential equations using number theoretic transforms with application to narrow or limited computer hardware , 1977 .

[16]  I. S. Reed,et al.  Integer Convolutions over the Finite Field $GF( {3 \cdot 2^n + 1} )$ , 1977 .

[17]  Trieu-Kien Truong,et al.  The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[18]  Anastasios N. Venetsanopoulos,et al.  The generalized discrete Fourier transform in rings of algebraic integers , 1980 .

[19]  J. Martens,et al.  Convolutions of long integer sequences by means of number theoretic transforms over residue class polynomial rings , 1983 .

[20]  Jean-Bernard Martens.  Number theoretic transforms for the calculation of convolutions , 1983 .

[21]  Jean-Bernard Martens,et al.  Convolution using a conjugate symmetry property for number theoretic transforms over rings of regular integers , 1983 .

[22]  Jean-Bernard Martens,et al.  Two-dimensional convolutions by means of number theoretic transforms over residue class polynomial rings , 1984 .

[23]  J. Martens.  Recursive cyclotomic factorization--A new algorithm for calculating the discrete Fourier transform , 1984 .

[24]  David Y. Y. Yun,et al.  Binary paradigm and systolic array implementation for residue arithmetic , 1985, 1985 IEEE 7th Symposium on Computer Arithmetic (ARITH).

[25]  Jae Lee,et al.  Realization of adaptive digital filters using the Fermat number transform , 1985, IEEE Trans. Acoust. Speech Signal Process..

[26]  Said Boussakta,et al.  Fast multidimensional discrete Hartley transform using Fermat number transform , 1988 .

[27]  Gabriele Steidl,et al.  Number-theoretic Transforms in Rings of Cyclotomic Integers , 1988, J. Inf. Process. Cybern..

[28]  Weiping Li,et al.  FIR filtering by the modified Fermat number transform , 1990, IEEE Trans. Acoust. Speech Signal Process..

[29]  Number theoretic fast algorithms for bilinear and other generalized transformations , 1990 .

[30]  Samuel C. Lee,et al.  A new approach to solve the sequence-length constraint problem in circular convolution using number theoretic transform , 1991, IEEE Trans. Signal Process..