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Modular Composition Modulo Triangular Sets and Applications

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Abstract

We generalize Kedlaya and Umans’ modular composition algorithm to the multivariate case. As a main application, we give fast algorithms for many operations involving triangular sets (over a finite field), such as modular multiplication, inversion, or change of order. For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables. As a further application, we show that, from the complexity viewpoint, Charlap, Coley, and Robbins’ approach to elliptic curve point counting can be competitive with the better known approach due to Elkies.

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References

  • Accettella C. J., DelCorso G. M., Manzini G. (2003) Inversion of two level circulant matrices over Zp. Linear Algebra and its Applications 366: 5–23

    Article  MathSciNet  MATH  Google Scholar 

  • A. V. Aho, J. E. Hopcroft & J. D. Ullman (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley.

  • M. E. Alonso, E. Becker, M.-F. Roy & T. Wörmann (1996). Zeros, multiplicities and idempotents for zerodimensional systems. In MEGA 94, volume 142 of Progress in Mathematics, 1–15. Birkhäuser.

  • A. O. L. Atkin (1992). The number of points on an elliptic curve modulo a prime (II). Available at http://listserv.nodak.edu/archives/nmbrthry.html.

  • P. Aubry, D. Lazard & M. Moreno Maza (1999). On the theories of triangular sets. Journal of Symbolic Computation 28(1, 2), 45–124.

    Google Scholar 

  • I. Blake, G. Seroussi & N. Smart (1999). Elliptic curves in cryptography, volume 265 of London Mathematical Society Lecture Notes Series. Cambridge University Press.

  • A. Bostan, P. Flajolet, B. Salvy & É. Schost (2006). Fast computation of special resultants. Journal of Symbolic Computation 41(1), 1–29.

  • A. Bostan, G. Lecerf &. Schost (2003). Tellegen’s Principle into Practice. In ISSAC’03, 37–44. ACM.

  • A. Bostan, M. F. I. Chowdhury, J. van der Hoeven &. Schost (2011). Homotopy techniques for multiplication modulo triangular sets. Journal of Symbolic Computation 46(12), 1378Gȴ1402.

    Google Scholar 

  • F. Boulier, F. Lemaire & M. Moreno Maza (2001). PARDI! In ISSAC’01, 38–47. ACM.

  • R. P. Brent & H. T. Kung (1978). Fast algorithms for manipulating formal power series. Journal of the ACM 25(4), 581–595.

    Google Scholar 

  • P. B"urgisser, M. Clausen & M. A. Shokrollahi (1997). Algebraic Complexity Theory. Springer.

  • L. S. Charlap, R. Coley & D. P. Robbins (1991). Enumeration of rational points on elliptic curves over finite fields. Draft.

  • D. Coppersmith & S. Winograd (1990). Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280.

    Google Scholar 

  • X. Dahan, M. Moreno Maza, Schost & Y. Xie (2006). On the complexity of the D5 principle. In Transgressive Computing, 149–168.

  • N. Elkies (1992). Explicit isogenies. Draft.

  • J. von zur Gathen (1990). Functional decomposition of polynomials: the tame case. Journal of Symbolic Computation 9, 281–299.

    Google Scholar 

  • J. von zur Gathen & J. Gerhard (1999). Modern Computer Algebra. Cambridge University Press.

  • J. von zur Gathen & V. Shoup (1992). Computing Frobenius maps and factoring polynomials. Computational Complexity 2(3), 187–224.

    Google Scholar 

  • P. Gianni & T. Mora (1989). Algebraic Solution of systems of polynomial equations using Gröbner bases. In AAECC’5, volume 356 of Lecture Notes in Computer Science, 247–257. Springer Verlag.

  • M. Giusti, J. Heintz, J. E. Morais, J. Morgenstern & L. M. Pardo (1998). Straight-Line Programs in Geometric Elimination Theory. Journal of Pure and Applied Algebra 124, 101–146.

    Google Scholar 

  • M. Giusti, G. Lecerf & B. Salvy (2001). A Gröbner free alternative for polynomial system solving. Journal of Complexity 17(1), 154–211. ISSN 0885-064X.

    Google Scholar 

  • X. Huang & V. Y. Pan (1998). Fast rectangular matrix multiplication and applications. Journal of Complexity 14(2), 257–299.

    Google Scholar 

  • É. Hubert (2003). Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems. In Symbolic and numerical scientific computation, volume 2630 of Lecture Notes in Computer Science, 1–39. Springer.

  • M. Kalkbrener (1993). A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation 15, 143–167.

    Google Scholar 

  • E. Kaltofen (1988). Greatest common divisors of polynomials given by straight-line programs. Journal of the ACM 35(1), 231–264.

    Google Scholar 

  • E. Kaltofen (2000). Challenges of symbolic computation: my favorite open problems. Journal of Symbolic Computation 29(6), 891–919.

    Google Scholar 

  • E. Kaltofen & Y. Laskhman (1989). Improved sparse multivariate polynomial interpolation algorithms. In ISSAC’88, volume 358 of Lecture Notes in Computer Science, 467–474. Springer Verlag.

  • K. S. Kedlaya & C. Umans (2011). Fast Polynomial Factorization and Modular Composition. SIAM J. Computing 40(6), 1767–1802.

    Google Scholar 

  • R. Lercier & T. Sirvent (2008). Elkies subgroups of elliptic curve ℓ-torsion points. Journal de Th orie des Nombres de Bordeaux 20(3), 783–797.

    Google Scholar 

  • X. Li, M. Moreno Maza & É. Schost (2009). Fast Arithmetic for triangular sets: from theory to practice. Journal of Symbolic Computation 44(7), 891–907.

  • F. Morain, P. Mihailescu & É. Schost (2007). Computing the eigenvalue in the Schoof-Elkies-Atkin algorithm using Abelian lifts. In ISSAC’07, 285–292. ACM.

  • M. Moreno Maza (1999). On Triangular Decompositions of Algebraic Varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK. http://www.csd.uwo.ca/~moreno/.

  • V. Y. Pan (1994). Simple multivariate polynomial multiplication. Journal of Symbolic Computation 18(3), 183–186.

    Google Scholar 

  • C. Pascal & É. Schost (2006). Change of order for bivariate triangular sets. In ISSAC’06, 277–284. ACM.

  • C. Peters (2006). Bestimmung des Elkies-Faktors im Schoof-Elkies-Atkin-Algorithmus. Diploma Thesis, Universität Paderborn.

  • D. Reischert (1997). Asymptotically Fast Computation of Subresultants. In ISSAC’97, 233–240. ACM.

  • F. Rouillier (1999). Solving zero-dimensional systems through the Rational Univariate Representation. Applicable Algebra in Engineering, Communication and Computing 9(5), 433–461.

  • R. Schoof (1985). Elliptic curves over finite fields and the computation of square roots mod p. Mathematics of Computation 44, 483–494.

    Google Scholar 

  • É. Schost (2003). Complexity results for triangular sets. Journal of Symbolic Computation 36(3–4), 555–594.

  • V. Shoup (1990). New algorithms for finding irreducible polynomials over finite fields. Mathematics of Computation 54(189), 435–447.

    Google Scholar 

  • V. Shoup (1991). A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In ISSAC’91, 14–21. ACM.

  • V. Shoup (1994). Fast construction of irreducible polynomials over finite fields. Journal of Symbolic Computation 17(5), 371–391.

    Google Scholar 

  • A. Stothers (2010). On the Complexity of Matrix Multiplication. Ph.D. thesis, University of Edinburgh.

  • C. Umans (2008). Fast polynomial factorization and modular composition in small characteristic. In STOC, 481–490.

  • V. Vassilevska Williams (2012). Multiplying matrices faster than coppersmith-winograd. In STOC, 887–898.

  • Yang L., Hou X., Xia B. (2001) A complete algorithm for automated discovering of a class of inequality-type theorems. Science in China. Series F. Information Sciences 44(1): 33–49

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LIFL, UMR-CNRS 8022, Université Lille 1, Lille, France

    Adrien Poteaux

  2. Computer Science Department, The University of Western, Ontario, London, ON, Canada

    Éric Schost

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  1. Adrien Poteaux
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  2. Éric Schost
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Correspondence to Éric Schost.

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Poteaux, A., Schost, É. Modular Composition Modulo Triangular Sets and Applications. comput. complex. 22, 463–516 (2013). https://doi.org/10.1007/s00037-013-0063-y

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  • DOI: https://doi.org/10.1007/s00037-013-0063-y

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