Abstract
We show how the discrete logarithm problem in some finite cyclic groups can easily be reduced to the discrete logarithm problem in a finite field. The cyclic groups that we consider are the set of points on a singular elliptic curve over a finite field, the set of points on a genus 0 curve over a finite field given by the Pell equation, and certain subgroups of the general linear group.
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References
Ben-Or, M.: Probabilistic algorithms in finite fields, 22nd Annual Symposium on Foundations of Computer Science, 394–398 (1981)
Buchmann, J., Williams, H.: A key-exchange system based on imaginary quadratic fields. J. Cryptol.1, 107–118 (1988)
Coppersmith, D.: Fast evaluation of logarithms in fields of characteristic two. IEEE Trans. Inf. Theory30, 587–594 (1984)
Coppersmith, D., Odlyzko, A., Schroeppel, R.: Discrete logarithms inGF(p). Algorithmica1, 1–15 (1986)
Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory22, 644–654 (1976)
ElGamal, T.: A subexponential-time algorithm for computing discrete logarithms over GF(p2). IEEE Trans. Inf. Theory31, 473–481 (1985)
Hoffman, K., Kunze, R.: Linear Algebra. New York: Prentice-Hall, NJ 1971
Husemöller, D.: Elliptic Curves. Berlin, Heidelberg, New York: Springer 1987
Koblitz, N.: Elliptic curve cryptosystems. Math. Comput.48, 203–209 (1987)
Koblitz, N.: Hyperelliptic cryptosystems. J. Cryptol.1, 139–150 (1989)
McCurley, K.: The discrete logarithm problem. Cryptol. Comput. Number Theory. Proc. Symp. Appl. Math.42, 49–74 (1990)
Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. Proceedings of the 22nd Annual ACM Symposium on the Theory of Computing, 80–89 (1991)
Miller, V.: Uses of elliptic curves in cryptography. Advances in Cryptology — Proceedings of Crypto '85. Lecture Notes in Computer Science vol. 218, pp. 417–426. Berlin, Heidelberg, New York: Springer 1986
Odlyzko, A.: Discrete logarithms in finite fields and their cryptographic significance. Advances in Cryptology — Proceedings of Eurocrypt '84. Lecture Notes in Computer Science vol. 209, pp. 224–314. Berlin, Heidelberg, New York: Springer 1985
Odoni, R., Varadharajan, V., Sanders, R.: Public key distribution in matrix rings, Electronic Letters,20, 386–387 (1984)
Rosser, J., Schoenfield, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math.6, 64–94 (1962)
Shallit, J.: personal communication (1991)
Silverman, J.: The Arithmetic of Elliptic Curves. Berlin, Heidelberg, New York: Springer 1986
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Menezes, A.J., Vanstone, S.A. A note on cyclic groups, finite fields, and the discrete logarithm problem. AAECC 3, 67–74 (1992). https://doi.org/10.1007/BF01189025
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DOI: https://doi.org/10.1007/BF01189025