Linear recurrences with polynomial coefficients and application to integer factorization and cartier-manin operator

SIAM Journal on Computing

Volumn 36, Issue 6, 2007, Pages 1777-1806

  Bostan, Alin ^a   Gaudry, Pierrick ^b   Schost, Éric ^b  

^a INRIA   (France)

^b ÉCOLE POLYTECHNIQUE   (France)

Author keywords

Cartier Manin operator; Factorization; Linear recurrences

Indexed keywords

INTEGER PROGRAMMING; MATHEMATICAL OPERATORS; POLYNOMIALS;

CARTIER-MANIN OPERATOR; LINEAR RECURRENCES; POLYNOMIAL COEFFICIENTS;

LINEAR PROGRAMMING;

EID: 36348972668     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539704443793     Document Type: Article

References (50)

1. J. BERKOWITZ, On computing the determinant in small parallel time using a small number of processors, Inform. Process. Lett., 18 (1984), pp. 147-150.
2. J. L. BORDEWIJK, Inter-reciprocity applied to electrical networks, Appl. Sci. Res. B., 6 (1956), pp. 1-74.
3. A. BORODIN, Time space tradeoffs (getting closer to the barrier?), in Proceedings of the 4th International Symposium on Algorithms and Computation, Lecture Notes in Comput. Sci. 762, Springer-Verlag, London, 1993, pp. 209-220.
4. A. BORODIN AND R. T. MOENCK, Fast modular transforms, Comput. Systems Sci., 8 (1974), pp. 366-386.
5. W. BOSMA, J. CANNON, AND C. PLAYOUST, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), pp. 235-265. See also http://www.maths.usyd.edu.au.
6. A. BOSTAN, P. GAUDRY, AND É. SCHOST, Linear recurrences with polynomial coefficients and computation of the Cartier-Manin operator on hyperelliptic curves, in Proceedings of the International Conference on Finite Fields and Applications (Toulouse, 2003), Lecture Notes in Comput. Sci. 2948, Springer, Berlin, 2004, pp. 40-58.
7. A. BOSTAN, G. LECERF, AND É. SCHOST, Tellegen's principle into practice, in Proceedings of the International Conference on Symbolic and Algebraic Computation, ACM Press, New York, 2003, pp. 37-44.
8. P. BÜRGISSER, M. CLAUSEN, AND M. A. SHOKROLLAHI, Algebraic Complexity Theory, Grundlehren der Math. Wiss. 315, Springer-Verlag, Berlin, 1997.
9. D. G. CANTOR AND E. KALTOFEN, On fast multiplication of polynomials over arbitrary algebras, Acta Inform., 28 (1991), pp. 693-701.
10. P. CARTIER, Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris, 244 (1957), pp. 426-428.
11. D. V. CHUDNOVSKY AND G. V. CHUDNOVSKY, Approximations and complex multiplication according to Ramanujan, in Ramanujan Revisited (Urbana-Champaign, IL, 1987), Academic Press, Boston, MA, 1988, pp. 375-472.
12. S. COOK, On the Minimum Computation Time of Functions, Ph.D. thesis, Harvard University, Cambridge, MA, 1966.
13. D. COPPERSMITH AND S. WINOGRAD, Matrix multiplication via arithmetic progressions, J. Symbolic Comput., 9 (1990), pp. 251-280.
14. P. FLAJOLET AND B. SALVY, The SIGSAM challenges: Symbolic asymptotics in practice, SIGSAM Bull., 31 (1997), pp. 36-47.
15. J. VON ZUR GATHEN AND J. GERHARD, Modern Computer Algebra, Cambridge University Press, Cambridge, UK, 1999.
16. J. VON ZUR GATHEN AND V. SHOUP, Computing Frobenius maps and factoring polynomials, Comput. Complexity, 2 (1992), pp. 187-224.
17. P. GAUDRY AND N. GÜREL, Counting points in medium characteristic using Kedlaya's algorithm, Experiment. Math., 12 (2003), pp. 395-402.
18. P. GAUDRY AND R. HARLEY, Counting points on hyperelliptic curves over finite fields, in Algorithmic Number Theory (ANTS-IV), Lecture Notes in Comput. Sci. 1838, Springer, Berlin, 2000, op. 313-332.
19. P. GAUDRY AND É. SCHOST, Construction of secure random curves of genus 2 over prime fields, in Advances in Cryptology (EUROCRYPT 2004), C. Cachin and J. Camenisch, eds., Lecture Notes in Comput. Sci. 3027, Springer, Berlin, 2004, pp. 239-256.
20. G. HANROT, M. QUERCIA, AND P. ZIMMERMANN, The middle product algorithm, I. Speeding up the division and square root of power series, Appl. Algebra Engrg. Comm. Comput., 14 (2004), pp. 415-438.
21. H. HASSE AND E. WITT, Zyklische unverzweigte Erweiterungskörper vom Primzahlgrade p über einem algebraischen Funktionenkörper der Charakteristik p, Monatsch. Math. Phys., 43 (1936), pp. 477-492.
22. E. HOROWITZ, A fast method for interpolation using preconditioning, Inform. Process. Lett., 1 (1972), pp. 157-163.
23. E. KALTOFEN, R. M. CORLESS, AND D. J. JEFFREY, Challenges of symbolic computation: My favorite open problems, J. Symbolic Comput., 29 (2000), pp. 891-919.
24. K. S. KEDLAYA, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology, J. Ramanujan Math. Soc., 16 (2001), pp. 323-338.
25. D. E. KNUTH, The analysis of algorithms, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, 1971, pp. 269-274.
26. A. K. LENSTRA, H. W. LENSTRA, JR., M. S. MANASSE, AND J. M. POLLARD, The number field sieve, in Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, ACM, New York, 1990, pp. 564-572.
27. H. W. LENSTRA, JR., AND C. POMERANÈE, A rigorous time bound for factoring integers, J. Amer. Math. Soc., 5 (1992), pp. 483-516.
28. J. I. MANIN, The Hasse-Witt matrix of an algebraic curve, Trans. Amer. Math. Soc., 45 (1965), pp. 245-264.
29. K. MATSUO, J. CHAO, AND S. TSUJII, An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields, in Algorithmic Number Theory (ANTSV), Lecture Notes in Comput. Sci. 2369, Springer, Berlin, 2002, pp. 461-474.
30. J. MCKEE AND R. PINCH, Old and new deterministic factoring algorithms, in Algorithmic Number Theory (Talence, 1996), Lecture Notes in Comput. Sci. 1122, Springer, Berlin, 1996, pp. 217-224.
31. R. T. MOENCK AND A. BORODIN, Fast modular transforms via division, in Proceedings of the Thirteenth Annual IEEE Symposium on Switching and Automata Theory (University of Maryland, College Park, MD), 1972, pp. 90-96.
32. P. L. MONTGOMERY, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp., 48 (1987), pp. 243-264.
33. P. L. MONTGOMERY, An FFT Extension of the Elliptic Curve Method of Factorization, Ph.D. thesis, University of California, Los Angeles CA, 1992.
34. J. PILA, Frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp., 55 (1990), pp. 745-763.
35. J. M. POLLARD, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc., 76 (1974), pp. 521-528.
36. C. POMERANÈE, Analysis and comparison of some integer factoring algorithms, in Computational Methods in Number Theory, Part I, Math. Centre Tracts 154, Math. Centrum, Amsterdam, 1982, pp. 89-139.
37. T. SATOH, The canonical lift of an ordinary elliptic curve over a finite field and its point counting, J. Ramanujan Math. Soc., 15 (2000), pp. 247-270.
38. A. SCHÖNHAGE, Schnelle Berechnung von Kettenbruchentwicklungen, Acta Inform., 1 (1971), pp. 139-144.
39. A. SCHÖNHAGE, Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2, Acta Inform., 7 (1977), pp. 395-398.
40. A. SCHÖNHAGE, A. F. W. GROTEFELD, AND E. VETTER, Fast Algorithms, Bibliographisches Institut, Mannheim, 1994.
41. A. SCHÖNHAGE AND V. STRASSEN, Schnelle Multiplikation groβer Zahlen, Computing, 7 (1971), pp. 281-292.
42. V. SHOUP, NTL: A library for doing number theory, http://www.shoup.net/ntl (2005).
43. V. SHOUP, A fast deterministic algorithm for factoring polynomials over finite fields of small characteristi, in Proceedings of the International Conference on Symbolic and Algebraic Computation, ACM Press, New York, 1991, pp. 14-21.
44. J. H. SILVERMAN, The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, SpringerVerlag, New York, 1996.
45. A. STEIN AND H. WILLIAMS, Some methods for evaluating the regulator of a real quadratic function field, Experiment. Math., 8 (1999), pp. 119-133.
46. V. STRASSEN, Gaussian elimination is not optimal, Numer. Math., 13 (1969), pp. 354-356.
47. V. STRASSEN, Die Berechnungskomplexitât von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten, Numer. Math., 20 (1972/73), pp. 238-251.
48. V. STRASSEN, Einige Resultate über Berechnungskomplexität, Jber. Deutsch. Math.-Verein., 78 (1976/77), pp. 1-8.
49. B. TELLEGEN, A general network theorem, with applications, Philips Res. Rep., 7 (1952), pp. 259-269.
50. N. YUI, On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2, J. Algebra, 52 (1978), pp. 378-410.