Abstract
The jacobian of hyperelliptic curves, including elliptic curves as a special case, offers a good primitive for cryptosystems, since cryptosystems (discrete logarithms) based on the jacobians seem to be more intractable than those based on conventional multiplicative groups. In this paper, we show that the problem to determine the group structure of the jacobian can be characterized to be in NP ∩ co-NP, when the jacobian is a non-degenerate type (“non-half-degenerate”). We also show that the hyperelliptic discrete logarithm can be characterized to be in NP ∩ co-NP, when the group structure is non-half-degenerate. Moreover, we imply the reducibility of the hyperelliptic discrete logarithm to a multiplicative discrete logarithm. The extended Weil pairing over the jacobian is the key tool for these algorithms.
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References
D. Cantor, “Computing in the Jacobian of a Hyperelliptic Curve”, Math. Comp., 48, pp.95–101 (1987).
D. Coppersmith, “Fast evaluation of logarithms in fields of characteristic two”, IEEE Transaction on Information Theory, IT-30, 587–594 (1984).
D. Coppersmith, A. Odlyzko and R. Schroeppel, “Discrete logarithms in GF(p)”, Algorithmica, 1 (1986), 1–15.
W. Diffie and M. E. Hellman, “New Directions in Cryptography”, IEEE Transactions on Information Theory, Vol.IT-22, No. 6, pp.644–654 (1976).
T. ElGamal, “A subexponential-time algorithm for computing discrete logarithms over GF(p 2)”, IEEE Transactions on Information Theory, IT-31, pp.473–481 (1985).
S. Even, A.L. Selman and Y. Yacobi, “The Complexity of Promise Problems with Applications to Public-Key Cryptography”, Information and Control, 61, pp.159–173 (1984).
W. Fulton, “Algebraic Curves,” Benjamin, New York, 1969.
B. Kaliski, “A pseudorandom bit generator based on elliptic logarithms”, Advances in Cryptology: Proceedings of Crypto’ 86, Lecture Notes in Computer Science, 293, Springer-Verlag, pp.84–103 (1987).
N. Koblitz, “Elliptic Curve Cryptosystems”, Math. Comp., 48, pp.203–209 (1987).
N. Koblitz, “Hyperelliptic Cryptosystems”, Journal of Cryptology, Vol.1, pp.139–150 (1989).
D. Knuth, The Art of Computer Programming, Vol. 2. Reading, MA: Addison-Wesley, 1981.
S. Lang, Abelian Varieties, Interscience, New York, 1959.
A.J. Menezes, T. Okamoto, S.A. Vanstone, “Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field”, Proc. of STOC’91, pp.80–89 (1991).
V. Miller, “Uses of elliptic curves in cryptography”, Proc. of Crypto’ 85, pp.417–426 (1986).
V. Miller, “Short programs for functions on curves”, unpublished manuscript, 1986.
A. Odlyzko, “Discrete logarithms and their cryptographic significance”, Proc. of Eurocrypt’ 84, pp.224–314, (1985).
J. Pila, “Frobenius Maps of Abelian Varieties and Finding Roots of Unity in Finite Fields”, PhD Thesis of Stanford Univ., (1988)
R. Schoof, “Elliptic curves over finite fields and the computation of square roots mod p”, Mathematics of Computation, 44, pp.483–494 (1985).
R. Schoof, “Nonsingular plane cubic curves over finite fields”, Journal of Combinatorial Theory, A 46, pp.183–211 (1987).
J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
H. Shizuya, T. Itoh, K. Sakurai, “On the Complexity of Hyperelliptic Discrete Logarithm Problem”, to appear in Proc. of Eurocrypt’ 91.
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© 1992 Springer-Verlag Berlin Heidelberg
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Okamoto, T., Sakurai, K. (1992). Efficient Algorithms for the Construction of Hyperelliptic Cryptosystems. In: Feigenbaum, J. (eds) Advances in Cryptology — CRYPTO ’91. CRYPTO 1991. Lecture Notes in Computer Science, vol 576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46766-1_21
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DOI: https://doi.org/10.1007/3-540-46766-1_21
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