Abstract
In this paper we consider generic algorithms for computational problems in cyclic groups. The model of a generic algorithm was proposed by Shoup at Eurocrypt '97. A generic algorithm is a generalpurpose algorithm that does not make use of any particular property of the representation of the group elements. Shoup proved the hardness of the discrete logarithm problem and the Diffie-Hellman problem with respect to such algorithms for groups whose order contains a large prime factor. By extending Shoup's technique we prove lower bounds on the complexity of generic algorithms solving different problems in cyclic groups, and in particular of a generic reduction of the discrete logarithm problem to the Diffie-Hellman problem. It is shown that the two problems are not computationally equivalent in a generic sense for groups whose orders contain a multiple large prime factor. This complements earlier results which stated this equivalence for all other groups. Furthermore, it is shown that no generic algorithm exists that computes p-th roots efficiently in a group whose order is divisible by p2 if p is a large prime.
Chapter PDF
Keywords
References
D. Boneh and R. J. Lipton, Algorithms for black-box fields and their application to cryptography, Advances in Cryptology — CRYPTO '96, Lecture Notes in Computer Science, Vol. 1109, pp. 283–297, Springer-Verlag, 1996.
B. den Boer, Diffie-Hellman is as strong as discrete log for certain primes, Advances in Cryptology — CRYPTO '88, Lecture Notes in Computer Science, Vol. 403, pp. 530–539, Springer-Verlag, 1989.
W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, Vol. 22, No. 6, pp. 644–654, 1976.
J. L. Massey, Advanced Technology Seminars Short Course Notes, pp. 6.66–6.68, Zürich, 1993.
U. M. Maurer, Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms, Advances in Cryptology — CRYPTO '94, Lecture Notes in Computer Science, Vol. 839, pp. 271–281, Springer-Verlag, 1994.
U. M. Maurer and S. Wolf, The relationship between breaking the Diffie-Hellman protocol and computing discrete logarithms, to appear in SIAM Journal of Computing, 1998.
U. M. Maurer and S. Wolf, Diffie-Hellman oracles, Advances in Cryptology — CRYPTO '96, Lecture Notes in Computer Science, Vol. 1109, pp. 268–282, Springer-Verlag, 1996.
K. S. McCurley, The discrete logarithm problem, in Cryptology and computational number theory, C. Pomerance (Ed.), Proc. of Symp. in Applied Math., Vol. 42, pp. 49–74, American Mathematical Society, 1990.
A. J. Menezes, Elliptic curve public key cryptosystems, Kluwer Academic Publishers, 1993.
S. C. Pohlig and M. E. Hellman, An improved algorithm for computing logarithms over GF(p) and its cryptographic significance, IEEE Transactions on Information Theory, Vol. 24, No. 1, pp. 106–110, 1978.
J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, Journal of the ACM, Vol. 27, No. 4, pp. 701–717, 1980.
V. Shoup, Lower bounds for discrete logarithms and related problems, Advances in Cryptology — EUROCRYPT '97, Lecture Notes in Computer Science, Vol. 1233, pp. 256–266, Springer-Verlag, 1997.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Maurer, U., Wolf, S. (1998). Lower bounds on generic algorithms in groups. In: Nyberg, K. (eds) Advances in Cryptology — EUROCRYPT'98. EUROCRYPT 1998. Lecture Notes in Computer Science, vol 1403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054118
Download citation
DOI: https://doi.org/10.1007/BFb0054118
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64518-4
Online ISBN: 978-3-540-69795-4
eBook Packages: Springer Book Archive