Abstract
As we have seen in Section 6.1, the elements of a finite cyclic group G may be used to implement several cryptographic schemes, provided that finding logarithms of elements in G is infeasible. We may take G to be a cyclic subgroup of E(F q ), the group of F q -rational points of an elliptic curve defined over F q ; this was first suggested by N. Koblitz [10] and V. Miller [17]. Since the addition in this group is relatively simple, and moreover the discrete logarithm problem in G is believed to be intractable, elliptic curve cryptosystems have the potential to provide security equivalent to that of existing public key schemes, but with shorter key lengths. Having short key lengths is a factor that can be crucial in some applications, for example the design of smart card systems.
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References
G. Agnew, T. Beth, R. Mullin and S. Vanstone, Arithmetic operations in GF(2 m ), J. of Cryptology, to appear.
G. Agnew, R. Mullin, I. Onyszchuk and S. Vanstone, “An. implementation for a fast public key cryptosystem”, J. of Cryptology, 3 (1991), 63–79.
M. Ben-Or, “Probabilistic algorithms in finite fields”, 22nd Annual Symposium on Foundations of Computer Science (1981), 394–398.
D. Coppersmith, “Fast evaluation of logarithms in fields of characteristic two”, IEEE Trans. Info. Th., 30 (1984), 587–594.
D. Coppersmith, A. Odlyzko and R. Schroeppel, “Discrete logarithms in Gf(p)”, Algorithmica, 1 (1986), 1–15.
T. Elgamal, “A public key cryptosystem and a signature scheme based on discrete logarithms”, IEEE Trans. Info. Th., 31 (1985), 469–472.
T. Elgamal, “A subexponential-time algorithm for computing discrete logarithms over Gf(p 2)”, IEEE Trans. Info. Th., 31 (1985), 473–481.
D. Husemoller, Elliptic Curves, Springer-Verlag, New York, 1987.
B. Kaliski, Elliptic Curves and Cryptography: A PseudorAndom Bit Generator and other Tools, Ph.D. thesis, M.I.T., January 1988.
N. Koblitz, “Elliptic curve cryptosystems”, Math. Comp., 48 (1987), 203–209.
N. Koblitz, “Constructing elliptic curve cryptosystems in characteristic 2”, Advances in Cryptology: Proceedings of Crypto ’90, Lecture Notes in Computer Science, 537 (1991), Springer-Verlag, 156–167.
N. Koblitz, “Elliptic curve implementation of zero-knowledge blobs”, J. of Cryptology, 4 (1991), 207–213.
N. Koblitz, “Cm-Curves with good cryptographic properties”, Advances in Cryptology: Proceedings of Crypto ’91, Lecture Notes in Computer Science, 576 (1992), Springer-Verlag, 279–287.
A. Lenstra, H.W. Lenstra, M. Manasse and J. Pollard, “The number field sieve” , Proceedings of the 22nd Annual Acm Symposium on Theory of Computing (1990), 564–572.
A. Menezes, T. Okamoto and S. Vanstone, “Reducing elliptic curve logarithms to logarithms in a finite field”, Proceedings of the 23rd Annual Acm Symposium on Theory of Computing (1991), 80–89.
A. Menezes, S. Vanstone and R. Zuccherato, “Counting points on elliptic curves over F2m”, Math. Comp., to appear.
V. Miller, “Uses of elliptic curves in cryptography”, Advances in Cryptology: Proceedings of Crypto ’85, Lecture Notes in Computer Science, 218 (1986), Springer-Verlag, 417–426.
V. Miller, “Short programs for functions on curves”, unpublished manuscript, 1986.
A. Odlyzko, “Discrete logarithms and their cryptographic significance”, in Advances in Cryptology: Proceedings of Eurocrypt ’84, Lecture Notes in Computer Science, 209 (1985), Springer-Verlag, 224–314.
C. Pomerance, “Fast, rigorous factorization and discrete logarithms al-gorithms”, in Discrete Algorithms and Complexity, Academic Press, 1987, 119–143.
J. Rosser and L. Schoenfield, “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6 (1962), 64–94.
R.J. Schoof, “Elliptic curves over finite fields and the computation of square roots mod p”, Math. Comp., 44 (1985), 483–494.
J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
R. Silverman, “The multiple polynomial quadratic sieve”, Math. Comp., 48 (1987), 329–339.
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Blake, I.F., Gao, X., Mullin, R.C., Vanstone, S.A., Yaghoobian, T. (1993). Elliptic Curve Cryptosystems. In: Menezes, A.J. (eds) Applications of Finite Fields. The Springer International Series in Engineering and Computer Science, vol 199. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2226-0_8
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