Subquadratic-time factoring of polynomials over finite fields

Mathematics of Computation

Volumn 67, Issue 223, 1998, Pages 1179-1197

  Kaltofen, Erich ^a,c   Shoup, Victor ^b,d,e  

^a Rensselaer Polytechnic Institute   (United States)

^b SAARLAND UNIVERSITY   (Germany)

^c NORTH CAROLINA STATE UNIVERSITY   (United States)

^d TELCORDIA TECHNOLOGIES   (United States)

^e IBM RESEARCH ZURICH   (Switzerland)

Author keywords

Factoring; Finite fields; Normal bases; Polynomials; Randomized algorithms

Indexed keywords

EID: 0032365366     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-98-00944-2     Document Type: Article

References (39)

1. Aho, A., Hopcroft, J. and Ullman, J., Design and Analysis of Computer Algorithms, Addison and Wesley, Reading, MA, 1974. MR 54:1706
2. Baur, W. and Strassen, V., The complexity of partial derivatives, Theoretical Comp. Sci., vol. 22. 317-330. 1983. MR 84c:68027
3. Ben-Or, M., Probabilistic algorithms in finite fields, Proc. 22nd IEEE Symp. Foundations Comp. Sci., 394-398, 1981.
4. Berlekamp, E. R., Factoring polynomials over large finite fields, Math. Comp., 24, 1970, 713-735. MR 43:1948
5. Brent, R. P., Gustavson, F. G., and Yun, D. Y. Y., Fast solution of Toeplitz systems of equations and computation of Padé approximants, J. Algorithms, vol. 1, 259-295, 1980. MR 82d:65033
6. Brent, R. P. and Kung, H. T., Fast algorithms for manipulating formal power series, J. ACM. vol. 25, no. 4, 581-595, 1978. MR 58:25090
7. Canny, J., Kaltofen, E. and Lakshman Yagati, Solving systems of non-linear polynomial equations faster, Proc. ACM-SIGSAM 1989 Internat. Symp. Symbolic Algebraic Comput., 121-128, ACM Press, 1989.
8. Cantor, D. G. and Kaltofen, E., On fast multiplication of polynomials over arbitrary algebras, Acta Inform., vol. 28, no. 7, 693-701, 1991. MR 92i:68068
9. Cantor, D. G. and Zassenhaus, H., A new algorithm for factoring polynomials over finite fields, Math. Comp., vol. 36, 587-592, 1981. MR 82e:12020
10. Coppersmith, D., Rapid multiplication of rectangular matrices, SIAM J. Comput., vol. 11, no. 3, 467-471, 1982. MR 83j:68047a
11. Coppersmith, D. and Winograd, S., Matrix multiplication via arithmetic progressions, J. Symbolic Comput., vol. 9, no. 3, 251-280, 1990. MR 91i:68058
12. Dornstetter, J. L., On the equivalence between Berlekamp's and Euclid's algorithms, IEEE Trans. Inf. Theory, vol. 33, no. 3, 428-431, 1987. MR 88j:94018
13. Fiduccia, C. M., On obtaining upper bounds on the complexity of matrix multiplication, Complexity of Computer Computations, (R. E. Miller and J. W. Thatcher). Plenum Press, New York, 1972, 31-40. MR 52:12398
14. Fiduccia, C. M., On the Algebraic Complexity of Matrix Multiplication. Ph.D. Thesis, Center Comput. Inform. Sci., Div. Engin., Brown Univ., Providence, Rhode Island, June 1973.
15. q, Linear Algebra and Applications, vol. 192, 101-108, 1993. MR 94f:11129
16. von zur Gathen, J., and Giesbrecht, M., Constructing normal bases in finite fields, J. Symbolic Comput., vol. 10, no. 6, 547-570, 1990. MR 92e:11142
17. von zur Gathen, J., and Shoup, V., Computing Frobenius maps and factoring polynomials, Comput. Complexity, vol. 2, 187-224, 1992. MR 94d:12011
18. Giesbrecht, M., Nearly optimal algorithms for canonical matrix forms, Ph.D. Thesis, Dept. Comput. Science, University of Toronto, Toronto, Canada, 1993.
19. Griewank, A., Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization Methods and Software, Gordon and Breach Science Publishers, vol. 1, 35-54, 1992.
20. Kaltofen, E., and Lobo, A., Factoring high-degree polynomials by the black box Berlekamp algorithm, Proc. Internat. Symp. Symbolic Algebraic Comput. ISSAC '94. (J. von zur Gathen and M. Giesbrecht), ACM Press, New York, N. Y., 90-98, 1994.
21. Kaltofen, E., and Pan, V., Processor efficient parallel solution of linear systems over an abstract field, Proc. 3rd Ann. ACM Symp. Parallel Algor. Architecture, ACM Press, 1991, 180-191.
22. Kaminski, M., Kirkpatrick, D. G. and Bshouty, N. H., Addition requirements for matrix and transposed matrix products, J. Algorithms, vol. 9, 354-364, 1988 MR 89m:68061
23. Knuth, D. E., The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Ed. 2, Addison Wesley, Reading, MA, 1981. MR 83i:68003
24. Lidl, R. and Niederreiter, H., Finite Fields, Addison-Wesley, Reading, MA, 1983. MR 86c:11106
25. Linnainmaa, S., Taylor expansion of the accumulated rounding error, BIT. vol. 16, 146-160, 1976. MR 54:9070
26. Lotti, G., and Romani, F., On the asymptotic complexity of rectangular matrix multiplication, Theoretical Comput. Sci.,, vol. 23, 171-185, 1933. MR 84g:68029
27. Massey, J. L., Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, vol. 15, 122-127, 1969. MR 39:3887
28. Niederreiter, H., A new efficient factorization algorithm for polynomials over small finite fields, Applic. Algebra Engin., Commun. Comput., vol. 4, 81-87, 1993. MR 94h:11112
29. Niederreiter, H. and Göttfert, R., Factorization of polynomials over finite fields and characteristic sequences. J. Symbolic Comput., vol. 16, no. 5, 401-412, 1993. MR 95d:68072
30. Ostrowski, G. M., Wolin, Ju. M. and Borisow, W. W., Über die Berechnung von Ableitungen, Wissenschaftliche Zeitschrift Techn. Hochsch. Chem. Leuna-Merseburg, vol. 13, no. 4, 382-384, 1971.
31. Rabin, M. O., Probabilistic algorithms in finite fields, SIAM J. Comp., vol. 9, 273-280, 1980. MR 81g:12002
32. Schwarz, Št., On the reducibility of polynomials over a finite field, Quart. J. Math. Oxford Ser. (2), vol. 7, 110-124, 1956. MR 20:3162
33. Shoup, V., On the deterministic complexity of factoring polynomials over finite fields, Inform. Process. Letters, vol. 33, no. 5, 261-267, 1990. MR 91f:11088
34. Shoup, V., Fast construction of irreducible polynomials over finite fields, J. Symbolic Comput., vol. 17, no. 5, 371-391, 1994. MR 95k:11156
35. Shoup, V., A new polynomial factorization algorithm and its implementation, J. Symbolic Comput., vol. 20, 363-397, 1995. MR 97d:12011
36. Wiedemann, D., Solving sparse linear equations over finite fields, IEEE Trans. Inf. Theory, vol. 32, 54-62, 1986. MR 87g:11166
37. Bürgisser, P., Clausen, M. and Shokrollahi, M. A., Algebraic Complexity Theory, Springer-Verlag, Heidelberg, Germany, 1997. CMP 97:10
38. Huang, X. and Pan, V., Fast rectangular matrix multiplications and improving parallel matrix computations, In Proc. Second Internat. Symp. Parallel Symbolic Comput. PASCO '97, M. Hitz and E. Kaltofen, editors, pages 11-23, New York, N.Y., 1997. ACM Press.
39. Kaltofen, E. and Shoup, V., Fast polynomial factorization over high algebraic extensions of finite fields. In ISSAC 97 Proc. 1997 Internat. Symp. Symbolic Algebraic Comput., W. Küchlin, editor, pages 184-188, New York, N.Y., 1997. ACM Press.