Domain extension of public random functions: beyond the birthday Barrier

• Authors: Ueli Maurer;Stefano Tessaro
• Venue: CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
• Year: 2007

A public random function is a random function that is accessible by all parties, including the adversary. For example, a (public) random oracle is a public random function {0,1}* → {0,1}n. The natural problem of constructing a public random oracle from a public random function {0, 1}m → {0, 1}n (for some m n) was first considered at Crypto 2005 by Coron et al. who proved the security of variants of the Merkle-Damgård construction against adversaries issuing up to O(2n/2) queries to the construction and to the underlying compression function. This bound is less than the square root of n2m, the number of random bits contained in the underlying random function. In this paper, we investigate domain extenders for public random functions approaching optimal security. In particular, for all Ɛ ∈ (0, 1) and all functions m and l (polynomial in n), we provide a construction CƐ,m,l(ċ) which extends a public random function R : {0, 1}n → {0, 1}n to a function CƐ,m,l(R) : {0, 1}m(n) → {0, 1}l(n) with time-complexity polynomial in n and 1/Ɛ and which is secure against adversaries which make up to Θ(2n(1-Ɛ)) queries. A central tool for achieving high security are special classes of unbalanced bipartite expander graphs with small degree. The achievability of practical (as opposed to complexity-theoretic) efficiency is proved by a non-constructive existence proof. Combined with the iterated constructions of Coron et al., our result leads to the first iterated construction of a hash function {0, 1}* → {0, 1}n from a component function {0, 1}n → {0, 1}n that withstands all recently proposed generic attacks against iterated hash functions, like Joux's multi-collision attack, Kelsey and Schneier's second-preimage attack, and Kelsey and Kohno's herding attacks.