Compact Structure-preserving Signatures with Almost Tight Security
Masayuki Abe¹
Dennis Hofheinz*²
Ryo Nishimaki¹
Miyako Ohkubo³
Jiaxin Pan†²
¹ Secure Platform Laboratories, NTT Corporation, Japan {abe.masayuki, nishimaki.ryo}@lab.ntt.co.jp
² Karlsruhe Institute of Technology, Germany {dennis.hofheinz, jiaxin.pan}@kit.edu
³ Security Fundamentals Laboratory, CSR, NICT, Japan m.ohkubo@nict.go.jp
Abstract
In structure-preserving cryptography, every building block shares the same bilinear groups. These groups must be generated for a specific, a priori fixed security level, and thus it is vital that the security reduction of all involved building blocks is as tight as possible. In this work, we present the first generic construction of structure-preserving signature schemes whose reduction cost is independent of the number of signing queries. Its chosen-message security is almost tightly reduced to the chosen- plaintext security of a structure-preserving public-key encryption scheme and the security of Groth-Sahai proof system. Technically, we adapt the adaptive partitioning technique by Hofheinz (Eurocrypt 2017) to the setting of structure-preserving signature schemes. To achieve a structure-preserving scheme, our new variant of the adaptive partitioning technique relies only on generic group operations in the scheme itself. Interestingly, however, we will use non-generic operations during our security analysis. Instantiated over asymmetric bilinear groups, the security of our concrete scheme is reduced to the external Diffie-Hellman assumption with linear reduction cost in the security parameter, independently of the number of signing queries. The signatures in our schemes consist of a larger number of group elements than those in other non-tight schemes, but can be verified faster, assuming their security reduction loss is compensated by increasing the security parameter to the next standard level.
Keywords: Structure-preserving signatures, Tight reduction, Adaptive partitioning
1 Introduction
BACKGROUND. A structure-preserving signature (SPS) scheme [4] is designed over bilinear groups, and features public keys, messages, and signatures that only consist of source group elements. Furthermore, signature verification only uses group membership testing and relations that can be expressed as pairing product equations. Coupled with the Groth-Sahai non-interactive proof system [34] (GS proofs for short), SPS schemes are a powerful tool in constructing a wide range of cryptographic applications. Various SPS schemes based on compact standard assumptions exist in the literature [33, 20, 4, 21, 18, 5, 3, 45, 41, 39]. When looking at schemes from standard assumptions the state-of-the-art scheme in [39] yields signatures as compact as consisting of six source group elements.
In this paper, we address the tightness of security proofs for SPS schemes with compact parameters, i.e., constant-size signatures and standard (non $q$-type) assumptions. Formally, a security reduction constructs an adversary $\mathcal{A}$ on a computational assumption out of an adversary $\mathcal{A}'$ on the security of a cryptographic scheme. If we let $\epsilon$ and $t$ denote the success probability and runtime of $\mathcal{A}$, and $\epsilon'$ and $t'$ the success probability and runtime of $\mathcal{A}'$, then we define the security loss of the reduction, or simply the reduction cost, as $(\epsilon't)/(\epsilon t')$ [22]. The reduction is tight if the security loss is a small constant or almost tight if it grows only (as a preferably small function) in the security parameter $\lambda$. In particular, we are concerned with the independence of the security loss from the number $q_s$ of $\mathcal{A}'$'s signing queries in a chosen-message attack. We note that in practice, $q_s$ can be as large as $2^{30}$ while $\lambda$ is typically 128.
*Supported by DFG grants HO 4534/4-1 and HO 4534/2-2. †Supported by the DFG grant HO 4534/4-1.