ETRI Journal

  • 제27권5호
  • /
  • Pages.617-627
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  • 2005
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  • 1225-6463(pISSN)
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  • 2233-7326(eISSN)
한국전자통신연구원 (Electronics and Telecommunications Research Institute)
권호 표지

Speeding up Scalar Multiplication in Genus 2 Hyperelliptic Curves with Efficient Endomorphisms

  • 투고 : 2004.11.22
  • 발행 : 2005.10.31
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초록

This paper proposes an efficient scalar multiplication algorithm for hyperelliptic curves, which is based on the idea that efficient endomorphisms can be used to speed up scalar multiplication. We first present a new Frobenius expansion method for special hyperelliptic curves that have Gallant-Lambert-Vanstone (GLV) endomorphisms. To compute kD for an integer k and a divisor D, we expand the integer k by the Frobenius endomorphism and the GLV endomorphism. We also present improved scalar multiplication algorithms that use the new expansion method. By our new expansion method, the number of divisor doublings in a scalar multiplication is reduced to a quarter, while the number of divisor additions is almost the same. Our experiments show that the overall throughputs of scalar multiplications are increased by 15.6 to 28.3 % over the previous algorithms when the algorithms are implemented over finite fields of odd characteristics.

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참고문헌

  1. Information Security and Cryptology-ICISC 2003 v.2971 of LNCS Efficient Scalar Multiplication in Hyperelliptic Curves Using a New Frobenius Expansion Park, T.;Lee, M.;Park, K.
  2. IEEE Trans. on Information Theory v.IT-22 no.6 New Directions in Cryptography Diffie, W.;Hellman, M.E.
  3. J. of Cryptology v.1 no.3 Hyperelliptic Cryptosystems Koblitz, N.
  4. Cryptographic Hardware and Embedded Systems-CHES 2004 v.3156 of LNCS Aspects of Hyperelliptic Curves over Large Prime Fields in Software Implementations Avanzi, R.M.
  5. Advances in Cryptology-CRYPTO 91 v.576 of LNCS CM-Curves with Good Cryptographic Properties Koblitz, N.
  6. Advances in Cryptology-CRYPTO 92 v.740 of LNCS Eficient Multiplication on Certain Non-Supersingular Elliptic Curves Meier, W.;Staffelbach, O.
  7. J. of Cryptology v.1 Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two Muller, V.
  8. ETRI J. v.21 no.1 Scalar Multiplication on Elliptic Curves by Frobenius Expansions Cheon, J.H.;Park, S.;Park, C.;Hahn, S.G.
  9. J. of Cryptology v.12 Elliptic Curve Cryptosystems over Small Fields of Odd Characteristic Smart, N.P.
  10. ETRI J. v.26 no.3 Improved Scalar Multiplication on Elliptic Curves Defined over $F_2^{mn}$ Lee, D.H.;Chee, S.;Hwang, S.C.;Ryou, J.C.
  11. Selected Areas in Cryptography-SAC 2001 v.2012 of LNCS Speeding up the Arithmetic on Koblitz Curves of Genus 2 Gunter, C.;Lange, T.;Stein, A.
  12. Efficient Arithmetic on Hyperelliptic Koblitz Curves Lange, T.
  13. Finite Fields and their Applications v.11 Koblitz Curve Cryptosystems Lange, T.
  14. Progress in Cryptology-INDOCRYPT 2002 v.2551 of LNCS Speeding up the Scalar Multiplication in the Jacobians of cHyperelliptic Curves Using Frobenius Map Choie, Y.;Lee, J.
  15. Advances in Cryptology-CRYPTO 2001 v.2139 of LNCS Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms Gallant, R.;Lambert, R.;Vanstone, S.
  16. Selected Areas in Cryptography-SAC 2002 v.2595 of LNCS Analysis of the Gallant-Lambert-Vanstone Method Based on Efficient Endomorphisms: Elliptic and Hyperelliptic Curves Sica, F.;Ciet, M.;Quisquater, J.J.
  17. Advances in Cryptology-EUROCRYPT 2002 v.2332 of LNCS Speeding up Point Multiplication on Hyperelliptic Curves with Efficient-Computable Endomorphisms Park, Y.;Jeong, S.;Lim, J.
  18. Information Security and Cryptology-ICISC 2002 v.2587 of LNCS New Frobenius Expansions for Elliptic Curves with Efficient Endomorphisms Park, T.;Lee, M.;Park, K.
  19. An Elementary Introduction to Hyperelliptic Curves, Technical Report CORR 96-19 Menezes, A.J.;Wu, Y.H.;Zuccherato, R.J.
  20. Algebraic Geometry Hartshone, R.
  21. Tata Lectures on Theta I Mumford, D.
  22. Mathematics of Computation v.48 Computing in the Jacobian of a Hyperelliptic Curve Cantor, D.
  23. AAECC Formulae for Arithmetic on Genus 2 Hyperelliptic Curves Lange, T.
  24. Bull. Austral. Math. Soc. v.58 Lattice Basis Reduction, Jacobi Sums and Hyperelliptic Cryptosystems Buhler, J.;Koblitz, N.
  25. Advances in Cryptology -ASIACRYPT 99 v.1716 of LNCS Speeding up the Discrete Log Computation on Curves with Automorphisms Duursma, L.;Gaudry, P.;Morain, F.
  26. The Arithmetic of Elliptic Curves Silverman, J.
  27. Information Security and Cryptology-ICISC 2003 v.2971 of LNCS A General Expansion Method Using Efficient Endomorphisms Park, T.;Lee, M;Kim, E.;Park, K.
  28. Invent. Math. v.2 Endomorphisms of Abelian Varieties over Finite Fields Tate, J.
  29. An Introduction to the Theory of Numbers Hardy, G.;Wright, E.
  30. Advances in Cryptology-EUROCRYPT 99 v.1592 of LNCS Fast Elliptic Curve Algorithm Combining Frobenius Map and Table Reference to Adapt to Higher Characteristic Kobayashi, T.;Morita, H.;Kobayashi, K.;Hoshino, F.
  31. IEICE Trans. Fundamentals v.E83-A Base-${\varphi}$ Method for Elliptic Curves over OEF Kobayashi, T.
  32. Advances in Cryptology-CRYPTO 97 v.1294 of LNCS An Improved Algorithm for Arithmetic on a Family of Elliptic Curves Solinas, J.
  33. Designs, Codes and Cryptography v.19 Efficient Arithmetic on Koblitz Curves Solinas, J.
  34. FrobSelf Lange, T.
  35. MAGMA V2.10 -The Magma Computational Algebra System MAGMA Group
  36. Low-Weight Binary Representations for Pairs of Integers, Technical Report CORR 2001-41 Solinas, J.
  37. Information Security and Cryptology-ICISC 99 v.1787 of LNCS Speeding up Elliptic Scalar Multiplication with Precomputation Lim, C.;Hwang, H.
  38. J. of Cryptology v.15 Constructive and Destructive Facets of Weil Descent on Elliptic Curves Gaudry, P.;Hes, F.;Smart, N.
  39. Progress in Cryptology-INDOCRYPT 2001 v.2247 of LNCS Elliptic Curves of Prime Order over Optimal Extension Fields for Use in Cryptography Baier, H.