On the efficiency of algorithms for polynomial factoring

Algorithms for factoring polynomials over finite fields are discussed. A construction is shown which reduces the final step of Berlekamp's algorithm to the problem of finding the roots of a polynomial in a finite field Zp. It is shown that if the characteristic of the field is of the form p = L 21 + 1, where I L, then the roots of a polynomial of degree n may be found in O(n log p + n log2 p) steps. As a result, a modification of Berlekamp's method can be performed in O(n + n log p + n log p) steps. If n is very large then an alternative method finds the factors of the polynomial in O(n2log n + n lognlogp). Some consequences and empirical evidence are discussed.

[1]  A. Adrian Albert,et al.  Fundamental Concepts of Higher Algebra , 1958 .

[2]  Ronald L. Graham,et al.  Problem #7 , 1974, SIGS.

[3]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[4]  B. F. Caviness.  On Canonical Forms and Simplification , 1970, JACM.

[5]  George E. Collins,et al.  Computing multiplicative inverses in () , 1969 .

[6]  David R. Musser,et al.  SAC-1 solution of problem #7 , 1974, SIGS.

[7]  J. D. Lipson,et al.  Chinese remainder and interpolation algorithms , 1971, SYMSAC '71.

[8]  R. Moenck.  Studies in fast algebraic algorithms. , 1973 .

[9]  Paul S. Wang,et al.  Factoring multivariate polynomials over the integers , 1973, SIGS.

[10]  David Y. Y. Yun,et al.  On algorithms for solving systems of polynomial equations , 1973, SIGS.

[11]  A. Albert,et al.  Fundamental Concepts of Higher Algebra , 1958 .

[12]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1971, SYMSAC '71.

[13]  V. Strassen.  Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten , 1973 .

[14]  G. E. Collins.  Computing time analyses for some arithmetic and algebraic algorithms , 1969 .

[15]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[16]  George E. Collins,et al.  The SAC-1 system: An introduction and survey , 1971, SYMSAC '71.

[17]  D. Musser.  Algorithms for polynomial factorization. , 1971 .

[18]  G. E. Collins.  Computing Multiplicative Inverses in GF(p) , 1969 .

[19]  Joel Moses,et al.  Solution to problem #7 using MACSYMA , 1974, SIGS.

[20]  W. Burnside,et al.  Theory of equations , 1886 .

[21]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

[22]  R. Risch,et al.  Symbolic integration of elementary functions , 1969 .

[23]  George E. Collins,et al.  The Calculation of Multivariate Polynomial Resultants , 1971, JACM.

[24]  Richard Y. Kain,et al.  Symbolic factoring of polynomials in several variables , 1966, CACM.

[25]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[26]  Solomon W. Golomb,et al.  ON THE FACTORIZATION OF TRINOMIALS OVER GF 2 , 1959 .