computational complexity

, Volume 22, Issue 3, pp 463–516 | Cite as

Modular Composition Modulo Triangular Sets and Applications

  • Adrien Poteaux
  • Éric Schost


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.


Triangular set modular composition power projection finite fields complexity 

Subject classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Accettella C. J., DelCorso G. M., Manzini G. (2003) Inversion of two level circulant matrices over Zp. Linear Algebra and its Applications 366: 5–23MathSciNetzbMATHCrossRefGoogle Scholar
  2. A. V. Aho, J. E. Hopcroft & J. D. Ullman (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley.Google Scholar
  3. 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.Google Scholar
  4. A. O. L. Atkin (1992). The number of points on an elliptic curve modulo a prime (II). Available at
  5. 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
  6. I. Blake, G. Seroussi & N. Smart (1999). Elliptic curves in cryptography, volume 265 of London Mathematical Society Lecture Notes Series. Cambridge University Press.Google Scholar
  7. A. Bostan, P. Flajolet, B. Salvy & É. Schost (2006). Fast computation of special resultants. Journal of Symbolic Computation 41(1), 1–29.Google Scholar
  8. A. Bostan, G. Lecerf &. Schost (2003). Tellegen’s Principle into Practice. In ISSAC’03, 37–44. ACM.Google Scholar
  9. 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
  10. F. Boulier, F. Lemaire & M. Moreno Maza (2001). PARDI! In ISSAC’01, 38–47. ACM.Google Scholar
  11. R. P. Brent & H. T. Kung (1978). Fast algorithms for manipulating formal power series. Journal of the ACM 25(4), 581–595.Google Scholar
  12. P. B"urgisser, M. Clausen & M. A. Shokrollahi (1997). Algebraic Complexity Theory. Springer.Google Scholar
  13. L. S. Charlap, R. Coley & D. P. Robbins (1991). Enumeration of rational points on elliptic curves over finite fields. Draft.Google Scholar
  14. D. Coppersmith & S. Winograd (1990). Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9(3), 251–280.Google Scholar
  15. X. Dahan, M. Moreno Maza, Schost & Y. Xie (2006). On the complexity of the D5 principle. In Transgressive Computing, 149–168.Google Scholar
  16. N. Elkies (1992). Explicit isogenies. Draft.Google Scholar
  17. J. von zur Gathen (1990). Functional decomposition of polynomials: the tame case. Journal of Symbolic Computation 9, 281–299.Google Scholar
  18. J. von zur Gathen & J. Gerhard (1999). Modern Computer Algebra. Cambridge University Press.Google Scholar
  19. J. von zur Gathen & V. Shoup (1992). Computing Frobenius maps and factoring polynomials. Computational Complexity 2(3), 187–224.Google Scholar
  20. 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.Google Scholar
  21. 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
  22. 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
  23. X. Huang & V. Y. Pan (1998). Fast rectangular matrix multiplication and applications. Journal of Complexity 14(2), 257–299.Google Scholar
  24. 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.Google Scholar
  25. M. Kalkbrener (1993). A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation 15, 143–167.Google Scholar
  26. E. Kaltofen (1988). Greatest common divisors of polynomials given by straight-line programs. Journal of the ACM 35(1), 231–264.Google Scholar
  27. E. Kaltofen (2000). Challenges of symbolic computation: my favorite open problems. Journal of Symbolic Computation 29(6), 891–919.Google Scholar
  28. 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.Google Scholar
  29. K. S. Kedlaya & C. Umans (2011). Fast Polynomial Factorization and Modular Composition. SIAM J. Computing 40(6), 1767–1802.Google Scholar
  30. R. Lercier & T. Sirvent (2008). Elkies subgroups of elliptic curve -torsion points. Journal de Th orie des Nombres de Bordeaux 20(3), 783–797.Google Scholar
  31. 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.Google Scholar
  32. 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.Google Scholar
  33. M. Moreno Maza (1999). On Triangular Decompositions of Algebraic Varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK.
  34. V. Y. Pan (1994). Simple multivariate polynomial multiplication. Journal of Symbolic Computation 18(3), 183–186.Google Scholar
  35. C. Pascal & É. Schost (2006). Change of order for bivariate triangular sets. In ISSAC’06, 277–284. ACM.Google Scholar
  36. C. Peters (2006). Bestimmung des Elkies-Faktors im Schoof-Elkies-Atkin-Algorithmus. Diploma Thesis, Universität Paderborn.Google Scholar
  37. D. Reischert (1997). Asymptotically Fast Computation of Subresultants. In ISSAC’97, 233–240. ACM.Google Scholar
  38. F. Rouillier (1999). Solving zero-dimensional systems through the Rational Univariate Representation. Applicable Algebra in Engineering, Communication and Computing 9(5), 433–461.Google Scholar
  39. R. Schoof (1985). Elliptic curves over finite fields and the computation of square roots mod p. Mathematics of Computation 44, 483–494.Google Scholar
  40. É. Schost (2003). Complexity results for triangular sets. Journal of Symbolic Computation 36(3–4), 555–594.Google Scholar
  41. V. Shoup (1990). New algorithms for finding irreducible polynomials over finite fields. Mathematics of Computation 54(189), 435–447.Google Scholar
  42. V. Shoup (1991). A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In ISSAC’91, 14–21. ACM.Google Scholar
  43. V. Shoup (1994). Fast construction of irreducible polynomials over finite fields. Journal of Symbolic Computation 17(5), 371–391.Google Scholar
  44. A. Stothers (2010). On the Complexity of Matrix Multiplication. Ph.D. thesis, University of Edinburgh.Google Scholar
  45. C. Umans (2008). Fast polynomial factorization and modular composition in small characteristic. In STOC, 481–490.Google Scholar
  46. V. Vassilevska Williams (2012). Multiplying matrices faster than coppersmith-winograd. In STOC, 887–898.Google Scholar
  47. 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): 33–49MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.LIFL, UMR-CNRS 8022LilleFrance
  2. 2.Computer Science DepartmentThe University of Western, OntarioLondonCanada

Personalised recommendations