Abstract Models of Computation in Cryptography

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.