Basing PRFs on Constant-Query Weak PRFs: Minimizing Assumptions for Efficient Symmetric Cryptography

Although it is well known that all basic private-key cryptographic primitives can be built from one-way functions, finding weak assumptions from which practical implementations of such primitives exist remains a challenging task. Towards this goal, this paper introduces the notion of a constant-query weak PRF , a function with a secret key which is computationally indistinguishable from a truly random function when evaluated at a constant number s of known random inputs, where s can be as small as two. We provide iterated constructions of (arbitrary-input-length) PRFs from constant-query weak PRFs that even improve the efficiency of previous constructions based on the stronger assumption of a weak PRF (where polynomially many evaluations are allowed). One of our constructions directly provides a new mode of operation using a constant-query weak PRF for IND-CPA symmetric encryption which is essentially as efficient as conventional PRF-based counter-mode encryption. Furthermore, our constructions yield efficient modes of operation for keying hash functions (such as MD5 and SHA-1) to obtain iterated PRFs (and hence MACs) which rely solely on the assumption that the underlying compression function is a constant-query weak PRF, which is the weakest assumption ever considered in this context.