**Digital Signatures with Minimal Overhead from Indifferentiable Random Invertible Functions**

*Eike Kiltz and Krzysztof Pietrzak and Mario Szegedy*

**Abstract: **In a digital signature scheme with message recovery, rather than transmitting the message $m$ and its signature $\sigma$, a single enhanced signature $\tau$ is transmitted. The verifier is able to recover $m$ from $\tau$ and at the same time
verify its authenticity. The two most important parameters of such a scheme are its security and overhead $|\tau|-|m|$. A simple argument shows that for any scheme with ``$n$ bits security" $|\tau|-|m|\ge n$, 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 $n+\log q_h$, where $q_h$ 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 $n+o(\log q_h)$ bits overhead, where $q_h$ 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.

**Category / Keywords: **Digital signatures, indifferentiability, Feistel, Additive combinatorics, Cayley graph.

**Publication Info: **A preliminary version appears in CRYPTO 2013. This is the full version.

**Date: **received 19 Nov 2012, last revised 12 Jun 2013

**Contact author: **krzpie at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20130612:195821 (All versions of this report)

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