Digital Signatures with Minimal Overhead from Indifferentiable Random Invertible Functions

In a digital signature scheme with message recovery, rather than transmitting the message m and its signature σ, a single enhanced signature τ is transmitted. The verifier is able to recover m from τ and at the same time verify its authenticity. The two most important parameters of such a scheme are its security and overhead |τ | − |m|. A simple argument shows that for any scheme with “n bits security” |τ | − |m| ≥ n, i.e., the overhead is lower bounded by the security parameter n. Currently, the best known constructions in the random oracle model are far from this lower bound requiring an overhead of n + log qh, where qh is the number of queries to the random oracle. In this paper we give a construction which basically matches the n bit lower bound. We propose a simple digital signature scheme with n + o(log qh) bits overhead, where qh denotes the number of random oracle queries. Our construction works in two steps. First, we propose a signature scheme with message recovery having optimal overhead in a new ideal model, the random invertible function model. Second, we show that a four-round Feistel network with random oracles as round functions is tightly “public-indifferentiable” from a random invertible function. At the core of our indifferentiability proof is an almost tight upper bound for the expected number of edges of the densest “small” subgraph of a random Cayley graph, which may be of independent interest.