Cyclotomic FFTs With Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm

In this paper, we first propose a novel common subexpression elimination (CSE) algorithm for matrix-vector multiplications over characteristic-2 fields. As opposed to previously proposed CSE algorithms, which usually focus on complexity savings due to recurrences of subexpressions, our CSE algorithm achieves two types of complexity reductions, differential savings and recurrence savings, by taking advantage of the cancellation property of characteristic-2 fields. Using our CSE algorithm, we reduce the additive complexities of cyclotomic fast Fourier transforms (CFFTs). Using a weighted sum of the numbers of multiplications and additions as a metric, our CFFTs achieve smaller total complexities than previously proposed CFFTs and other FFTs, requiring both fewer multiplications and fewer additions in many cases.

[1]  Martin Vetterli,et al.  Computing m DFT's over GF(q) with one DFT over GF(qm) , 1993, IEEE Trans. Inf. Theory.

[2]  J.L. Massey,et al.  Theory and practice of error control codes , 1986, Proceedings of the IEEE.

[3]  R. Hartley.  Subexpression sharing in filters using canonic signed digit multipliers , 1996 .

[4]  Allan O. Steinhardt,et al.  Fast algorithms for digital signal processing , 1986, Proceedings of the IEEE.

[5]  Richard E. Blahut.  Transform Techniques for Error Control Codes , 1979, IBM J. Res. Dev..

[6]  Yao Wang,et al.  A fast algorithm for the Fourier transform over finite fields and its VLSI implementation , 1988, IEEE J. Sel. Areas Commun..

[7]  Athar Mahboob,et al.  Lookup table based multiplication technique for GF(2m) with cryptographic significance , 2005 .

[8]  Meghanad D. Wagh,et al.  A new structured design method for convolutions over finite fields, Part I , 1983, IEEE Trans. Inf. Theory.

[9]  J. Cooley,et al.  New algorithms for digital convolution , 1977 .

[10]  Keshab K. Parhi,et al.  Implementation approaches for the Advanced Encryption Standard algorithm , 2002 .

[11]  Elena Costa,et al.  On computing the syndrome polynomial in Reed-Solomon decoder , 2004, Eur. Trans. Telecommun..

[12]  Peter Trifonov,et al.  Matrix-Vector Multiplication via Erasure Decoding , 2007 .

[13]  Chip-Hong Chang,et al.  Information Theoretic Approach to Complexity Reduction of FIR Filter Design , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Shmuel Winograd.  Some bilinear forms whose multiplicative complexity depends on the field of constants , 2005, Mathematical systems theory.

[15]  Tsung-Ching Lin,et al.  A Fast Algorithm for the Syndrome Calculation in Algebraic Decoding of Reed–Solomon Codes , 2007, IEEE Transactions on Communications.

[16]  Miodrag Potkonjak,et al.  Multiple constant multiplications: efficient and versatile framework and algorithms for exploring common subexpression elimination , 1996, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[17]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[18]  L. J. Wang,et al.  Fast, prime factor, discrete Fourier transform , 2004 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  Stephen B. Wicker,et al.  Reed-Solomon Codes and Their Applications , 1999 .

[21]  A. Dempster,et al.  Use of minimum-adder multiplier blocks in FIR digital filters , 1995 .

[22]  Trieu-Kien Truong,et al.  Fast, prime factor, discrete Fourier transform algorithms over GF(2m) for 8 leq m leq 10 , 2006, Inf. Sci..

[23]  Berk Sunar,et al.  Mastrovito Multiplier for All Trinomials , 1999, IEEE Trans. Computers.

[24]  A. Dempster,et al.  Common subexpression elimination algorithm for low-cost multiplierless implementation of matrix multipliers , 2004 .

[25]  K. Steiglitz,et al.  Some complexity issues in digital signal processing , 1984 .

[26]  L. Wanhammar,et al.  Design of high-speed multiplierless filters using a nonrecursive signed common subexpression algorithm , 2002 .

[27]  T. C. Cheng,et al.  Fast Transform for Decoding Both Errors and Erasures of Reed – Solomon Codes Over GF ( 2 m ) for 8 m 10 , 2006 .

[28]  Trieu-Kien Truong,et al.  Fast transform for decoding both errors and erasures of Reed-Solomon codes over GF(2/sup m/) for 8/spl les/m/spl les/10 , 2006, IEEE Transactions on Communications.

[29]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[30]  Peter D. Welch,et al.  The Fast Fourier Transform and Its Applications , 1969 .

[31]  Peter Trifonov,et al.  A Method for Fast Computation of the Fourier Transform over a Finite Field , 2003, Probl. Inf. Transm..

[32]  Paul M. Embree,et al.  C++ algorithms for digital signal processing , 1998 .

[33]  Oscar Gustafsson,et al.  Complexity Reduction of Constant Matrix Computations over the Binary Field , 2007, WAIFI.

[34]  Patrick Schaumont,et al.  A new algorithm for elimination of common subexpressions , 1999, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[35]  Sergei V. Fedorenko.  A method for computation of the discrete Fourier transform over a finite field , 2006, Probl. Inf. Transm..

[36]  S. Wicker.  Error Control Systems for Digital Communication and Storage , 1994 .

[37]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[38]  R. Blahut.  Theory and practice of error control codes , 1983 .

[39]  A. Prasad Vinod,et al.  A New Common Subexpression Elimination Algorithm for Realizing Low-Complexity Higher Order Digital Filters , 2008, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[40]  T. C. Cheng,et al.  Fast Transform for Decoding Both Errors and Erasures of Reed-Solomon Codes Over GF for , 2006 .