Abstract
We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e. in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. Also a deterministic version of the algorithm is discussed whose running time is polynomial in the degree of the input polynomial and the size of the field.
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References
A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms. Addison-Wesley, Reading MA, 1974.
M. Ben-Or, Probabilistic algorithms in finite fields. Proc. 22nd Symp. Foundations Comp. Sci. IEEE, 1981, 394–398.
E.R. Berlekamp, Factoring polynomials over finite fields. Bell System Tech. J. 46 (1967), 1853–1859.
E.R. Berlekamp, Factoring polynomials over large finite fields. Math. Comp. 24 (1970), 713–735.
W.S. Brown, On Euclid's algorithm and the computation of polynomial Greatest Common Divisors. J. ACM 18 (1971), 478–504.
D.G. Cantor and H. Zassenhaus, On algorithms for factoring polynomials over finite fields. Math. Comp. 36 (1981), 587–592.
A.L. Chistov and D.Yu. Grigoryev, Polynomial-time factoring of the multivariable polynomials over a global field. LOMI preprint E-5-82, Leningrad, 1982.
J.H. Davenport and B.M. Trager, Factorization over finitely generated fields. Proc. 1981 ACM Symp. Symbolic and Algebraic Computation, ed. by P. Wang, 1981, 200–205.
J. von zur Gathen, Hensel and Newton methods in valuation rings. Tech. Report 155(1981), Dept. of Computer Science, University of Toronto. To appear in Math. Comp.
J. von zur Gathen, Parallel algorithms for algebraic problems. Proc. 15th ACM Symp. Theory of Computing, Boston, 1983.
J. von zur Gathen [83a], Factoring sparse multivariate polynomials. Manuscript, 1983.
G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Clarendon Press, Oxford, 1962.
E. Kaltofen, A Polynomial Time Reduction from Bivariate to Univariate Integral Polynomial Factorization. Proc. 23rd Symp. Foundations of Comp. Sci., IEEE, 1982, 57–64.
E. Kaltofen, Polynomial-time Reduction from Multivariate to Bivariate and Univariate Integer Polynomial Factorization. Manuscript, 1983, submitted to SIAM J. Comput.
D.E. Knuth, The Art of Computer Programming, Vol.2, 2nd Ed. Addison-Wesley, Reading MA, 1981.
A. Lempel, G. Seroussi and S. Winograd, On the complexity of multiplication in finite fields. Theor. Comp. Science 22 (1983), 285–296.
A.K. Lenstra, Factoring multivariate polynomials over finite fields. Proc. 15th ACM Symp. Theory of Computing, Boston, 1983.
A.K. Lenstra, H.W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.
D.R. Musser, Algorithms for Polynomial Factorization. Ph.D. thesis and TR 134, Univ. of Wisconsin, 1971.
M.O. Rabin, Probabilistic algorithms in finite fields. SIAM J. Comp. 9 (1980), 273–280.
T. Schönemann, Grundzüge einer allgemeinen Theorie der höheren Congruenzen, deren Modul eine reelle Primzahl ist. J. f. d. reine u. angew. Math. 31 (1846), 269–325.
B.L. van der Waerden, Modern Algebra, vol. 1. Ungar, New York, 1953.
H. Weyl, Algebraic theory of numbers. Princeton University Press, 1940.
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von zur Gathen, J., Kaltofen, E. (1983). Polynomial-time factorization of multivariate polynomials over finite fields. In: Diaz, J. (eds) Automata, Languages and Programming. ICALP 1983. Lecture Notes in Computer Science, vol 154. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036913
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DOI: https://doi.org/10.1007/BFb0036913
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