Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TATE PAIRING COMPUTATION ON THE DIVISORS OF HYPERELLIPTIC CURVES OF GENUS 2

  • Published : 2008.07.31
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Abstract

We present an explicit Eta pairing approach for computing the Tate pairing on general divisors of hyperelliptic curves $H_d$ of genus 2, where $H_d\;:\;y^2+y=x^5+x^3+d$ is defined over ${\mathbb{F}}_{2^n}$ with d=0 or 1. We use the resultant for computing the Eta pairing on general divisors. Our method is very general in the sense that it can be used for general divisors, not only for degenerate divisors. In the pairing-based cryptography, the efficient pairing implementation on general divisors is significantly important because the decryption process definitely requires computing a pairing of general divisors.

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References

  1. P. S. L. M. Barreto, S. D. Galbraith, C. O'hEigeartaigh, and M. Scott, Efficient pairing computation on supersingular abelian varieties, Des. Codes Cryptogr. 42 (2007), no. 3, 239-271 https://doi.org/10.1007/s10623-006-9033-6
  2. P. S. L. M. Barreto, H. Y. Kim, B. Lynn, and M. Scott, Efficient algorithms for pairing-based cryptosystems, Advances in cryptology-RYPTO 2002, 354-368, Lecture Notes in Comput. Sci., 2442, Springer, Berlin, 2002
  3. P. S. L. M. Barreto, B. Lynn, and M. Scott, On the selection of pairing-friendly groups, Selected areas in cryptography, 17-25, Lecture Notes in Comput. Sci., 3006, Springer, Berlin, 2004
  4. D. Boneh and M. Franklin, Identity-based encryption from the Weil pairing, SIAM J. Comput. 32 (2003), no. 3, 586-615 https://doi.org/10.1137/S0097539701398521
  5. D. Boneh, B. Lynn, and H. Shacham, Short signatures from the Weil pairing, Advances in cryptology-SIACRYPT 2001 (Gold Coast), 514-532, Lecture Notes in Comput. Sci., 2248, Springer, Berlin, 2001 https://doi.org/10.1007/3-540-45682-1_30
  6. L. Chen and C. Kudla, Identity Based Authenticated Key Agreement Protocols from Pairings, Cryptology eprint Archives, Number 2002/184
  7. Y. Choie and E. Lee, Implementation of Tate pairing on hyperelliptic curves of genus 2, Information security and cryptology-CISC 2003, 97-111, Lecture Notes in Comput. Sci., 2971, Springer, Berlin, 2004
  8. I. Duursma and H. Lee, Tate pairing implementation for hyperelliptic curves $y^{2}$ = $x^{p}$ - x + d, Advances in cryptology-SIACRYPT 2003, 111-123, Lecture Notes in Comput. Sci., 2894, Springer, Berlin, 2003 https://doi.org/10.1007/978-3-540-40061-5_7
  9. G. Frey and H.-G. Ruck, A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp. 62 (1994), no. 206, 865-874 https://doi.org/10.2307/2153546
  10. S. Galbraith, Supersingular curves in cryptography, Advances in cryptology-SIACRYPT 2001 (Gold Coast), 495-513, Lecture Notes in Comput. Sci., 2248, Springer, Berlin, 2001
  11. S. Galbraith, K. Harrison, and D. Soldera, Implementing the Tate pairing, Algorithmic number theory (Sydney, 2002), 324-337, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002
  12. R. Granger, F. Hess, R. Oyono, N. Theriault, and F. Vercauteren, Ate pairing on hyperelliptic curves, Proceedings of Euro 2007, 430-447, Lecture Notes in Comput. Sci., 4515, Springer, Berlin, 2007
  13. M. Katagi, I. Kitamura, T. Akishita, and T. Takagi, Novel efficient implementations of hyperelliptic curve cryptosystems using degenerate divisors, In Information Security Applications-WISA'2004, 345-359, Lecture Notes in Comput. Sci., 3325, Springer, Berlin, 2005
  14. N. Koblitz, Algebraic Aspects of Cryptography, With an appendix by Alfred J. Menezes, Yi-Hong Wu and Robert J. Zuccherato. Algorithms and Computation in Mathematics, 3. Springer-Verlag, Berlin, 1998
  15. N. Koblitz and A. Menezes, Pairing-based cryptography at high security levels, Cryptography and coding, 13-36, Lecture Notes in Comput. Sci., 3796, Springer, Berlin, 2005
  16. A. J. Menezes, T. Okamoto, and S. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Trans. Inform. Theory 39 (1993), no. 5, 1639-1646 https://doi.org/10.1109/18.259647
  17. D. Mumford, Tata Lectures on Theta. II, Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. Progress in Mathematics, 43. Birkhauser Boston, Inc., Boston, MA, 1984
  18. K. Rubin and A. Silverberg, Using Abelian Varieties to Improve Pairing-Based Cryptography, to appear in Journal of Cryptology
  19. M. Scott and P. S. Barreto, Compressed pairings, Advances in cryptology-RYPTO 2004, 140-156, Lecture Notes in Comput. Sci., 3152, Springer, Berlin, 2004
  20. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986
  21. E. R. Verheul, Evidence that XTR is more secure than supersingular elliptic curve cryptosystems, Advances in cryptology-UROCRYPT 2001 (Innsbruck), 195-210, Lecture Notes in Comput. Sci., 2045, Springer, Berlin, 2001 https://doi.org/10.1007/3-540-44987-6_13
  22. C. K. Yap, Fundamental Problems of Algorithmic Algebra, Oxford University Press, New York, 2000
  23. A.Weimerskirch, D. Stebila, and S. Shantz, Generic GF($2^{m}$) arithmetic in software and its application to ECC, Proceedings of ACISP 2003, 79-92, Lecture Notes in Comput. Sci., 2727, Springer, Berlin, 2003