A new efficient factorization algorithm for polynomials over small finite fields
- 106 Downloads
We present a new deterministic factorization algorithm for polynomials over a finite prime fieldF p . As in other factorization algorithms for polynomials over finite fields such as the Berlekamp algorithm, the key step is the “linearization” of the factorization problem, i.e., the reduction of the problem to a system of linear equations. The theoretical justification for our algorithm is based on a study of the differential equationy(p−1)+y p =0 of orderp−1 in the rational function fieldFp(x). In the casep=2 the new algorithm is more efficient than the Berlekamp algorithm since there is no set-up cost for the coefficient matrix of the system of linear equations.
KeywordsFactorization of polynomials over finite fields Differential equations over rational function fields
Unable to display preview. Download preview PDF.
- 1.Berlekamp, E. R.: Factoring polynomials over finite fields. Bell System Tech. J.46, 1853–1859 (1967)Google Scholar
- 2.Camion, P.: A deterministic algorithm for factorizing polynomials ofF q[X]. Ann. Discrete Math.17, 149–157 (1983)Google Scholar
- 3.Knuth, D. E.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley 1981Google Scholar
- 4.Lidl, R., Niederreiter, H.: Finite Fields. Reading, MA: Addison-Wesley 1983Google Scholar
- 5.Mignotte, M.: Mathématiques pour le calcul formel. Paris: Presses Universitaires de France 1989Google Scholar
- 6.Willett, M.: Factoring polynomials over a finite field. SIAM J. Appl. Math.35, 333–337 (1978)Google Scholar
- 7.Göttfert, R.: The Niederreiter factorization algorithm is polynomial time in characteristic 2. Preprint, 1992Google Scholar
- 8.Niederreiter, H.: Factorization of polynomials and some linear algebra problems over finite fields. Preprint, 1992Google Scholar
- 9.Niederreiter, H.: Factoring polynomials over finite fields using differential equations and normal bases. Preprint, 1992Google Scholar
- 10.Niederreiter, H., Göttfert, R.: Factorization of polynomials over finite fields and characteristic sequences. Preprint, 1992Google Scholar