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International Conference on Cryptology in India

INDOCRYPT 2005: Progress in Cryptology - INDOCRYPT 2005 pp 1–12Cite as

Abelian Varieties and Cryptography

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 3797))

Abstract

Let A be an Abelian variety over a finite field \(\mathbb{F}\). The possibility of using the group \(A(\mathbb{F})\) of points on A in \(\mathbb{F}\) as the basis of a public-key cryptography scheme is still at an early stage of exploration. In this article, we will discuss some of the issues and their current staus. In particular, we will discuss arithmetic on Abelian varieties, methods for point counting, and attacks on the Discrete Logarithm Problem, especially those that are peculiar to higher-dimensional varieties.

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References

  1. Miri, S.A., Murty, V.K.: An application of sieve methods to elliptic curves. In: Pandu Rangan, C., Ding, C. (eds.) INDOCRYPT 2001. LNCS, vol. 2247, pp. 91–98. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  2. Arita, S., Miura, S., Sekiguchi, T.: An addition algorithm on the Jacobian varieties of curves. J. Ramanujan Math. Soc. 19, 235–251 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Blake, I., Kumar Murty, V., Xu, G.: Refinements of Miller’s algorithm for computing the Weil/Tate pairing. J. Algorithms (to appear)

    Google Scholar 

  4. Cantor, D.: Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48, 95–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chao, J., Matsuo, K., Tsujii, S.: Baby step giant step algorithms in point counting of hyperelliptic curves. IEICE Trans. Fundamentals E86-A(4) (2003)

    Google Scholar 

  6. Cojocaru, A.: Bounded number of prime factors for the orders of the reductions of a CM elliptic curve, Preprint (2004)

    Google Scholar 

  7. Denef, J., Vercauteren, F.: An extension of Kedlaya’s algorithm to Artin-Schreier curves in characteristic 2. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 308–323. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  8. Fouquet, M., Gaudry, P., Harley, R.: An extension of Satoh’s algorithm and its implementation. J. Ramanujan Math. Soc. 15, 281–318 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Frey, G., Rück, H.: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp. 62, 865–874 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gaudry, P., Harley, R.: Counting points on hyperelliptic curves over finite fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 297–312. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  11. Guyot, C., Kaveh, K., Patankar, V.: Explicit algorithm for the arithmetic on the hyperelliptic Jacobians of genus 3. J. Ramanujan Math. Soc. 19, 75–115 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, New York (2004)

    MATH  Google Scholar 

  13. Izadi, F., Kumar Murty, V.: Counting points on an Abelian variety over a finite field. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 323–333. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  14. Kedlaya, K.: Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001) See also: Errata 18, 417–418 (2003)

    Google Scholar 

  15. Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory 39(5), 1639–1646 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Miller, V.: The Weil pairing and its efficient calculation. J. Cryptology 17, 235–261 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Murty, V.K.: Splitting of Abelian varieties: a new local-global problem. In: Tandon, R. (ed.) Algebra and Number Theory. Hindustan Book Agency, Delhi (2005)

    Google Scholar 

  18. Mumford, D.: Abelian Varieties, Oxford

    Google Scholar 

  19. Patankar, V.: Splitting of Abelian varieties, Ph.D Thesis, University of Toronto (2005)

    Google Scholar 

  20. Satoh, T.: The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc. 15, 247–270 (2000)

    MATH  MathSciNet  Google Scholar 

  21. Steuding, J., Weng, A.: On the number of prime divisors of the order of elliptic curves modulo p. Acta Arith. 117, 341–352 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Thériault, N.: Index calculus attack for hyperelliptic curves of small genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  23. Zarhin, Y.: Abelian varieties of K3-type and ℓ-adic representations. In: Algebraic Geometry and Analytic Geometry, pp. 231–255. Springer, Tokyo (1991)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 3G3, Canada

    V. Kumar Murty

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  1. V. Kumar Murty
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Editors and Affiliations

  1. Indian Statistical Institute, 203 B T Road, 700 108, Kolkata, India

    Subhamoy Maitra

  2. Indian Institute of Science, Bangalore

    C. E. Veni Madhavan

  3. Microsoft Research India Private Limited, “Scientia”, No:196/36, 2nd Main Road, Sadashivnagar, 560080, Bangalore, India

    Ramarathnam Venkatesan

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Murty, V.K. (2005). Abelian Varieties and Cryptography. In: Maitra, S., Veni Madhavan, C.E., Venkatesan, R. (eds) Progress in Cryptology - INDOCRYPT 2005. INDOCRYPT 2005. Lecture Notes in Computer Science, vol 3797. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11596219_1

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  • DOI: https://doi.org/10.1007/11596219_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30805-8

  • Online ISBN: 978-3-540-32278-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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