• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2002 Fiscal Year Final Research Report Summary

Number theory for positive characteristics and its application to elliptic curve cryptography

Research Project

Project/Area Number 12640009
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionSaitama University

Principal Investigator

SATOH Takakazu  Saitama Univ., Dept. of Mathematics, Assoc. Prof., 理学部, 助教授 (70215797)

Co-Investigator(Kenkyū-buntansha) GON Yasuro  Saitama Univ., Dept. of Mathematics, Assistant, 理学部, 助手 (30302508)
YANAI Hisae  Saitama Univ., Dept. of Mathematics, Lecturer, 理学部, 講師 (10008865)
TAKEICHI Kisao  Saitama Univ., Dept. of Mathematics, Professor, 理学部, 教授 (00011560)
Project Period (FY) 2000 – 2002
Keywordsthe Frobenius substitutions / finite fields / elliptic curves / order counting
Research Abstract

We establish and develop a p-adic point counting algorithm for elliptic curves over finite fields of small characteristics. Let p be a fixed small prime and put q to be the N-th power of p. For a given ordinal elliptic curve E defined over the finite field k of q elements, we construct a fast algorithm to compute the number of k-rational points of E. When a small prime p is fixed and N tends to infinity, our algorithm is faster than the so-called SEA algorithm.
Our algorithm is based on the canonical lifts of elliptic curves. First we lift a given ordinal elliptic curve to its canonical lift. We use the fact that two j-invariants of lifted curves are related by the p-th modular polynomial. So, construction of the canonical lifts is reduced to find a solution to a certain system of non-linear equations. Second, we compute the leading coefficient of the dual of the lift of the p-th Frobenius morphism. This should not be confused with the inverse Frobenius substitution, since we are working over the field of characteristic zero once the curve is lifted. Third, by looking at the action of the dual of the lifted Frobenius morphism, we can compute the trace of the q-th Frobenius endomorphism. Using well-known Hasse's equality, we obtain the number of the rational points and we are done.
We further construct a faster algorithm, with some precomputations which depends on only on q. The precomputation is quite feasible for the case that N is less than, say, 500. Hence the cost of precomputation is no problem for practical applications. On the other hand, thanks to the precomputation, we can evaluate the Frobenius substitution quickly. This ameliorates the growth rate of time complexity with respect to a number of bit operations by a factor of at least the square root of N.

  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] T.Satoh, B.Skjernaa, Y.Taguchi: "Fast computation of canonical lifts of elliptic curves and its application to point counting"Finite fields and their appl.. 9. 89-101 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Gon, M.Tsuzuki: "The resolvent trace formula for rank one Lie group"Asian J. Math.. 6. 227-252 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Satoh: "On p-adic point counting algorithms for elliptic curves over finite fields"Lect. Notes in Comput. Sci.. 2369. 43-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Satoh: "The canonical lift of elliptic curve over a finite field and its point counting"J. Ramanujan Math. Soc.. 15. 247-270 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Gon: "Generalized Whittaker functions on SU(2,2) with respect to the Siegel parabolic subgroup"AMS. viii+116 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y. Gon: "Generalized Whittaker functions on SU(2, 2) with respect to the Siegel parabolic subgroup"Memor. Amer. Math. Soc.. 155. viii+116 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Gon, M. Tsuzuki: "The resolvent trace formula for rank one Lie groups"Asian J. Math.. 6. 227-252 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Satoh: "The canonical lift of elliptic curve over a finite field and its point counting"J. Rmanujan Math. Soc.. 15. 247-270 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Satoh: "On p-adic point counting algorithms for elliptic curves over finite fields"Proc. ANTS-V, Lecture Notes in Comput. Sci.. 2369. 43-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Satoh, B. Skjernaa, Y. Taguchi: "Fast computation of canonical lifts of elliptic curves and its application to point counting"Finite Fields and Their Appl.. 9. 89-101 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Koike: "A new method of construction of ε-optimal feedback controls from Hamilton-Jacobi equations"Proceedings of the Ninth Tokyo Conference on Nonlinear PDE 1999. 14-22 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Koike: "Tiny results on L^p- viscosity solutions of fully nonlinear uniformly elliptic equations"Proceedings of the Tenth Tokyo Conference on Nonlinear PDE 2000. 10-19 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ishibashi ・ S.Koike: "On fully nonlinear PDEs derived from variational problems of L^p norms"SIAM Journal on Mathematical Analysis. 33・3. 545-569 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] O.Alvarez ・ S.Koike ・ I.Nakayama: "Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem"SIAM Jouranal on Control and Optimization. 38・2. 470-481 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Bardi ・ S.Koike ・ P.Soravia: "Pursuit-evasion games with state constraints : dynamic programming and discrete-time approximations"Discrete and Continuous Dynamical Systems. 6・2. 361-380 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Ishii ・ S.Koike: "On ε-optimal controls for state constraint problem"Annales de l'Institut Henri Poincare, Analyses Non Lineaire. 17・4. 473-502 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Koike: "ε-optimal controls for state constraint problem"数理解析研究所講究録. 1135. 110-119 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2004-04-14  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi