Abstract
Computational security proofs in cryptography, without unproven intractability assumptions, exist today only if one restricts the computational model. For example, one can prove a lower bound on the complexity of computing discrete logarithms in a cyclic group if one considers only generic algorithms which can not exploit the properties of the representation of the group elements.
We propose an abstract model of computation which allows to capture such reasonable restrictions on the power of algorithms. The algorithm interacts with a black-box with hidden internal state variables which allows to perform a certain set of operations on the internal state variables, and which provides output only by allowing to check whether some state variables satisfy certain relations. For example, generic algorithms correspond to the special case where only the equality relation, and possibly also an abstract total order relation, can be tested.
We consider several instantiation of the model and different types of computational problems and prove a few known and new lower bounds for computational problems of interest in cryptography, for example that computing discrete logarithms is generically hard even if an oracle for the decisional Diffie-Hellman problem and/or other low degree relations were available.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Boneh, D., Lipton, R.J.: Algorithms for black-box fields and their application to cryptography. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 283–297. Springer, Heidelberg (1996)
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Transactions on Information Theory 22(6), 644–654 (1976)
Maurer, U.: Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 271–281. Springer, Heidelberg (1994)
Maurer, U., Wolf, S.: Lower bounds on generic algorithms in groups. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 72–84. Springer, Heidelberg (1998)
Maurer, U., Wolf, S.: On the complexity of breaking the Diffie-Hellman protocol. SIAM Journal on Computing 28, 1689–1721 (1999)
Nechaev, V.I.: Complexity of a deterministic algorithm for the discrete logarithm. Mathematical Notes 55(2), 91–101 (1994)
Pohlig, S.C., Hellman, M.E.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Transactions on Information Theory 24(1), 106–110 (1978)
Pollard, J.M.: Monte Carlo methods for index computation mod p. Mathematics of Computation 32, 918–924 (1978)
Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM 27(3), 701–717 (1980)
Shoup, V.: Lower bounds for discrete logarithms and related problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Maurer, U. (2005). Abstract Models of Computation in Cryptography. In: Smart, N.P. (eds) Cryptography and Coding. Cryptography and Coding 2005. Lecture Notes in Computer Science, vol 3796. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11586821_1
Download citation
DOI: https://doi.org/10.1007/11586821_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30276-6
Online ISBN: 978-3-540-32418-8
eBook Packages: Computer ScienceComputer Science (R0)