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A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

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Abstract

The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems.

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References

  1. Adleman, L.: A subexponential algorithm for discrete logarithm problem with applications to cryptography. In: Proc. 20th IEEE Found. Comp. Sci. Symp., pp. 55–60 (1979)

  2. Adleman, L., DeMarrais, J.: A subexponential algorithm for discrete logarithms over all finite fields. Math. Comp. 61, 1–15 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adleman, L., DeMarrais, J., Huang, M.: A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields. In: Proc. of Algorithmic Number Theory, LNCS, vol. 877, pp. 28–40 (1994)

  4. Adleman, L., Huang, M.: Primality testing and abelian varieties over finite fields. In: LNM 1512. Springer, Berlin Heidelberg New York (1992)

    Google Scholar 

  5. American National Standards Institute: Public key cryptography for the financial services industry: The elliptic curve digital signature algoritm (ECDSA), ANSI X9.62-1998.

  6. Blake, I., Seroussi, G., Smart, N.: Elliptic curves in cryptography, vol. 265. London Math. Soc. (1999)

  7. Boneh, D.: Twenty years of attacks on the RSA cryptosystem. Notices Amer. Math. Soc. 46(2), 203–213 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Brickell, E., Odlyzko, A.: Cryptanalysis: A survey of recent results. Proc. IEEE 76, 578–593 (1988)

    Article  MATH  Google Scholar 

  9. Le Brigand, D.: Decoding of codes on hyperelliptic curves. In: Proc. of Eurocode'90, LNCS, vol. 514, pp. 126–134 (1991)

  10. Brown, M., Cheung, D., Hankerson, D., Hernandez, J., Kirkup, M., Menezes, A.: PGP in constrained wireless devices. In: Proceedings of the Ninth USENIX Security Symposium, pp. 247–261 (2000)

  11. Buchmann, J., Williams, H.: A key-exchange system based on imaginary quadratic fields. J. Cryptology. 1, 107–118 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buhler, J.P., Lenstra Jr., H.W., Pomerance, C.: Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The Development of the Number Field Sieve, LNM 1554, pp. 50–94. Springer, Berlin Heidelberg New York (1993)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Cardona, G.: On the number of curves of genus 2 over a finite field. Finite Fields Appl. 9, 505–526 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cardona, G., Nart, E., Pujolás, J.: Curves of genus two over fields of even characteristic. Available at http://arxiv.org/abs/math.NT/0210105.

  16. Cavalar, S., et al.: Factorization of a \(512\)-bit RSA modulus. In: Proc. of Eurocrypt'2000, LNCS, vol. 1807, pp. 1–18 (2000)

  17. Choie, Y., Jeong, E.: Isomorphism classes of hyperelliptic curves of genus 2 over \(\mathbb{F}_{2^{n}}\). Cryptology ePrint Archive 2003/213.

  18. Choie, Y., Yun, D.: Isomorphism classes of hyperelliptic curves of genus 2 over \(\mathbb{F}_{q}\). In: Proc. of ACISP'02, LNCS, vol. 2384, pp. 190–202 (2002)

  19. Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton, FL (2006)

    MATH  Google Scholar 

  20. Coppersmith, D.: Fast evaluation of logarithms in finite fields of characteristic two. IEEE Trans. Inform. Theory 30(4), 587–594 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Coppersmith, D., Odlyzko, A., Schroeppel, R.: Discrete logarithms in \(GF(p)\). Algorithmica 1, 1–15 (1986)

  22. Deng, Y., Liu, M.: Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2. Sci. China Ser. A 49(2), 173–184 (2005)

    Article  MathSciNet  Google Scholar 

  23. Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inform. Theory 22, 644–654 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  24. ElGamal, T.: A public-key cryptosystem and a signature scheme based on discrete logarithm. IEEE Trans. Inform Theory 31, 469–472 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Enge, A.: Computing discrete logarithms in high-genus hyperelliptic Jacobians in probably subexponential time. Math. Comp. 71, 729–742 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Enge, A., Gaudry, P.: A general framework for subexponential discrete logarithm algorithms. Acta Arith. 1, 83–103 (2002)

    MATH  MathSciNet  Google Scholar 

  27. 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 

  28. Galbratih, S., Menezes, A.: Algebraic curves and cryptography. Finite Fields Appl. 11, 544–577 (2005)

    Article  MathSciNet  Google Scholar 

  29. Galbrait, S., Paulus, S., Smart, N.: Arithmetic of superelliptic curves. Math. Comp. 71, 393–405 (2002)

    Article  MathSciNet  Google Scholar 

  30. Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Proc. of Eurocrypt 2000, LNCS, vol. 1807, pp. 19–34 (2000)

  31. Gaudry, P., Thériault, N., Thomé, E.: A double large prime variation for small genus hyperelliptic index calculus. Cryptology ePrint Archive 2004/153

  32. Gilbert, H., Gupta, D., Odlyzko, A., Quisquater, J.J.: Attacks on Shamir's ‘RSA for paranoids’. Inform Process. Lett. 68(4), 197–199 (1998)

    Article  Google Scholar 

  33. Hartshorne, R.: Algebraic Geometry. Springer, Berlin Heidelberg New York (1977)

    MATH  Google Scholar 

  34. Hernández Encinas, L., Menezes, A.J., Muñoz Masqué, J.: Isomorphism classes of genus-2 hyperelliptic curves over finite fields. Appl. Algebra Engrg. Comm. Comput. 13, 57–65 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hernández Encinas, L., Muñoz Masqué, J.: Isomorphism classes of hyperelliptic curves of genus 2 in characteristic 5. Technical Report CORR2002-07, Centre For Applied Cryptographic Research (CACR), University of Waterloo (2002)

  36. Hernández Encinas, L., Muñoz Masqué, J.: Isomorphism classes of genus-2 hyperelliptic curves over finite fields \(\mathbb{F}_{5^{m}}\). Information 8(6), 8 pp. (2005)

  37. Jacobson Jr., M., Menezes, A., Stein, A.: Hyperelliptic curves and cryptography. Technical Report CORR 2003-13, Centre for Applied Cryptography Research (CACR), University of Waterloo (2003)

  38. Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  39. Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptology 1, 139–150 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  40. Koblitz, N.: Algebraic Aspects of Cryptography. Springer, Berlin Heidelberg New York (1998)

    MATH  Google Scholar 

  41. Kurosawa, K., Ito, T., Takeuchi, M.: Public key cryptosystem using a reciprocal number with the same intractability as factoring a large number. Cryptologia 12, 225–233 (1988)

    MATH  MathSciNet  Google Scholar 

  42. Lange, T.: Efficient arithmetic on genus 2 hyperelliptic curves over finite fields via explicit formulae. Cryptology ePrint Archive, Report 2002/121.

  43. Lenstra, A., Verheul, E.: Selecting cryptographic key sizes. In: Proc. of PKC 2000, LNCS, vol. 1751, pp. 446–465 (2000)

  44. Lenstra, H.W., Pila, J., Pomerance, C.: A hyperelliptic smoothness test. I. R. Soc. Philos. Trans., Ser. A Math. Phys. Eng. Sci. 345, 397–408 (1993)

    MATH  MathSciNet  Google Scholar 

  45. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  46. Lockhart, P.: On the discriminant of a hyperelliptic curve. Trans. Amer. Math. Soc. 342(2), 729–752 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  47. McCurley, K.: A key distribution system equivalent to factoring. J. Cryptology 1, 95–105 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  48. Menezes, A.: Elliptic Curve Public Key Cryptosystems. Kluwer Academic, Dordrecht (1993)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  50. Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC, Boca Raton, FL (1997)

    MATH  Google Scholar 

  51. Menezes, A., Wu, Y., Zuccherato, R.: An elementary introduction to hyperelliptic curves. In: Koblitz, N. (ed.) Algebraic Aspects of Cryptography, pp. 155–178. Springer, Berlin Heidelberg New York (1998)

    Google Scholar 

  52. Miller, V.: Uses of elliptic curves in cryptography. In: Proc. of Crypto'85, LNCS, vol. 218, pp. 417–426 (1986)

  53. Müller, V., Stein, A., Thiel, C.: Computing discrete logarithms in real quadratic congruence function fields of large genus. Math. Comp. 68, 807–822 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  54. National Institute for Standards and Technology, Digital Signature Standard, FIPS PUB 186 (1993)

  55. National Institute for Standards and Technology, Digital Signature Standard (DSS), FIPS PUB 186-2 (2000)

  56. National Institute for Standards and Technology, Elliptic curve Diffie-Hellman or elliptic curve MQV, Draft NIST Special Publication 800-56 (2003)

  57. Pelzl, J., Wollinger, T., Guajardo, J., Paar, C.: Hyperelliptic curve cryptosystems: closing the performance gap to elliptic curves (update). Cryptology ePrint Archive, Report 2003/026

  58. Pelzl, J., Wollinger, T., Guajardo, J., Paar, C.: Low cost security: Explicit formulae for genus 4 hyperelliptic curves. Cryptology ePrint Archive, Report 2003/097.

  59. Pelzl, J., Wollinger, T., Paar, C.: Low cost security: explicit formulae for genus-4 hyperelliptic curves. In: Proc. Tenth Annual Workshop on Selected Areas in Cryptography, LNCS, vol. 3006, pp. 1–16 (2004)

  60. Petersen, M.: Hyperelliptic Cryptosystems. Technical Report, University of Aarhus, Denmark (1994)

  61. Pollard, J.: Monte Carlo methods for index computation \(\operatorname{mod}p\). Math. Comp. 32, 918–924 (1978)

    Google Scholar 

  62. Rabin, M.O.: Digitalized signatures and public key functions as intractable as factorization. Technical Report TM-212, Lab. for Computer Science, M.I.T. (1979)

  63. Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  64. Schirokauer, O., Weber, D., Denny, T.: Discrete logarithms: the effectiveness of the index calculus method. In: Proc. of Algorithmic Number Theory, LNCS, vol. 1122, pp. 337–361 (1996)

  65. Schnorr, C.: Efficient signature generation by smart cards. J. Cryptology 4, 161–174 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  66. Schoof, R.: Elliptic curves over finite fields and the computation of square roots \(\operatorname{mod}p\). Math. Comp. 44, 483–494 (1985)

    Google Scholar 

  67. Shamir, A.: RSA for paranoids. CryptoBytes 1(3), 1, 3 and 4 (1995)

  68. Shamir, A.: Factoring large numbers with the TWINKLE device, (Extended abstract) Eurocrypt'99, Rump session

  69. Silverman, J.: The Arithmetic of Elliptic Curves. Springer, Berlin Heidelberg New York (1986)

    MATH  Google Scholar 

  70. Silverman, J.: Advanced Topics in the Arithmetic of Elliptic Curves. Springer, Berlin Heidelberg New York (1994)

    MATH  Google Scholar 

  71. Smart, N.: On the performance of hyperelliptic cryptosystems. In: Proc. of Eurocrypt 1999, LNCS, vol. 1592, pp. 165–175 (1999)

  72. Takagi, T.: Fast RSA-type cryptosystems using \(N\)-adic expansion. Lecture Notes in Comput. Sci. 1294, 372–384 (1997)

    MATH  MathSciNet  Google Scholar 

  73. Takagi, T.: Fast RSA-type cryptosystem modulo \(p^{k}q\). Lecture Notes in Comput. Sci. 1462, 318–326 (1998)

    Google Scholar 

  74. Thériault, N.: Index calculus attack for hyperelliptic curves of small genus. In: Proc. of Asiacrypt 2003, LNCS, vol. 2894, pp. 75–92 (2003)

  75. Williams, H.C.: A modification of the RSA public-key encryption procedure. IEEE Trans. Inform. Theory 26, 726–729 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  76. Williams, H.C.: An \(M^{3}\) public-key encryption scheme. Lecture Notes in Comput. Sci. 218, 358–368 (1986)

    Article  MATH  Google Scholar 

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  1. Department of Information Processing and Coding, Applied Physics Institute, CSIC, C/ Serrano 144, E-28006, Madrid, Spain

    J. Espinosa García, L. Hernández Encinas & J. Muñoz Masqué

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  1. J. Espinosa García
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  2. L. Hernández Encinas
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  3. J. Muñoz Masqué
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Espinosa García, J., Hernández Encinas, L. & Muñoz Masqué, J. A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point. Acta Appl Math 93, 299–318 (2006). https://doi.org/10.1007/s10440-006-9045-2

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