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Compactly Hiding Linear Spans: Tightly Secure Constant-Size Simulation-Sound QA-NIZK Proofs and Applications

Benoît Libert 1 Thomas Peters 2, 3, 4, 5 Marc Joye 6 Moti Yung 7, 8
1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
2 CASCADE - Construction and Analysis of Systems for Confidentiality and Authenticity of Data and Entities
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR 8548
Abstract : Quasi-adaptive non-interactive zero-knowledge (QA-NIZK) proofs is a powerful paradigm, suggested recently by Jutla and Roy (Asiacrypt '13), which is motivated by the Groth-Sahai seminal techniques for efficient non-interactive zero-knowledge (NIZK) proofs. In this paradigm, the common reference string may depend on specific language parameters, a fact that allows much shorter proofs in important cases. It even makes certain standard model applications competitive with the Fiat-Shamir heuristic in the Random Oracle idealization (such QA-NIZK proofs were recently optimized to constant size by Jutla and Roy (Crypto '14) and Libert et al. (Eurocrypt '14) for the important case of proving that a vector of group elements belongs to a linear subspace). While, e.g., the QA-NIZK arguments of Libert et al. provide unbounded simulation-soundness and constant proof length, their simulation-soundness is only loosely related to the underlying assumption (with a gap proportional to the number of adversarial queries) and it is unknown how to alleviate this limitation without sacrificing efficiency. Here, we deal with the basic question of whether and to what extent we can simultaneously optimize the proof size and the tightness of security reductions, allowing for important applications with tight security (which are typically to date quite lengthy) to be of shorter size. In this paper, we resolve this question by describing a novel simulation-sound QA-NIZK argument showing that a vector v ∈ G n belongs to a subspace of rank t < n using a constant number of group elements. Unlike previous constant-size QA-NIZK proofs of such statements, the unbounded simulation-soundness of our system is nearly tightly related (i.e., the reduction only loses a factor proportional to the security parameter) to the standard Decision Linear assumption. To show simulation-soundness in the constrained context of tight reductions, we employ a number of techniques, and explicitly point at a technique – which may be of independent interest – of hiding the linear span of a structure-preserving homomorphic signature (which is part of an OR proof). As an application, we design a public-key cryptosystem with almost tight CCA2-security in the multi-challenge, multiuser setting with improved length (asymptotically optimal for long messages). We also adapt our scheme to provide CCA security in the key-dependent message scenario (KDM-CCA2) with ciphertext length reduced by 75% when compared to the best known tightly secure KDM-CCA2 system so far.
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https://hal.inria.fr/hal-01225363
Contributor : Benoit Libert <>
Submitted on : Friday, November 6, 2015 - 10:43:33 AM
Last modification on : Friday, December 18, 2020 - 6:46:05 PM
Long-term archiving on: : Monday, February 8, 2016 - 12:47:26 PM

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Benoît Libert, Thomas Peters, Marc Joye, Moti Yung. Compactly Hiding Linear Spans: Tightly Secure Constant-Size Simulation-Sound QA-NIZK Proofs and Applications. Advances in Cryptology - Asiacrypt 2015, IACR, Nov 2015, Auckland, New Zealand. ⟨hal-01225363⟩

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