Polynomial evaluation over finite fields: new algorithms and complexity bounds

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques, when the degree of the polynomial is large enough compared to the field characteristic. Specifically, if n is the degree of the polynomiaI, the asymptotic complexity is shown to be $${O(\sqrt{n})}$$, versus O(n) of classical algorithms. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighted.

[1]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  J. Rosenthal,et al.  The Gate Complexity of Syndrome Decoding of Hamming Codes , 2004 .

[4]  Joachim Rosenthal,et al.  On the decoding complexity of cyclic codes up to the BCH bound , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[5]  Larry J. Stockmeyer,et al.  On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials , 1973, SIAM J. Comput..

[6]  Dilip V. Sarwate,et al.  Semi-Fast Fourier Transforms over GF(2m). , 1978, IEEE Transactions on Computers.

[7]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[8]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[9]  V. Pan.  METHODS OF COMPUTING VALUES OF POLYNOMIALS , 1966 .

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

[11]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[12]  S Winograd,et al.  On the number of multiplications required to compute certain functions. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

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

[14]  Michele Elia,et al.  On the Inherent Space Complexity of Fast Parallel Multipliers for GF(2/supm/) , 2002, IEEE Trans. Computers.

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

[16]  Allan Borodin,et al.  The computational complexity of algebraic and numeric problems , 1975, Elsevier computer science library.