Random Oracles in a Quantum World

The interest in post-quantum cryptography - classical systems that remain secure in the presence of a quantum adversary - has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have classical access to the random oracle. We argue that to prove post-quantum security one needs to prove security in the quantum-accessible random oracle model where the adversary can query the random oracle with quantum states. We begin by separating the classical and quantum-accessible random oracle models by presenting a scheme that is secure when the adversary is given classical access to the random oracle, but is insecure when the adversary can make quantum oracle queries. We then set out to develop generic conditions under which a classical random oracle proof implies security in the quantum-accessible random oracle model. We introduce the concept of a history-free reduction which is a category of classical random oracle reductions that basically determine oracle answers independently of the history of previous queries, and we prove that such reductions imply security in the quantum model. We then show that certain post-quantum proposals, including ones based on lattices, can be proven secure using history-free reductions and are therefore post-quantum secure. We conclude with a rich set of open problems in this area.Comment: 38 pages, v2: many substantial changes and extensions, merged with a related paper by Boneh and Zhandr

Security of signed ELGamal encryption

Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel one-more-decyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts

Fully Adaptive Schnorr Threshold Signatures

We prove adaptive security of a simple three-round threshold Schnorr signature scheme, which we call Sparkle. The standard notion of security for threshold signatures considers a static adversary – one who must declare which parties are corrupt at the beginning of the protocol. The stronger adaptive adversary can at any time corrupt parties and learn their state. This notion is natural and practical, yet not proven to be met by most schemes in the literature. In this paper, we demonstrate that Sparkle achieves several levels of security based on different corruption models and assumptions. To begin with, Sparkle is statically secure under minimal assumptions: the discrete logarithm assumption (DL) and the random oracle model (ROM). If an adaptive adversary corrupts fewer than t/2 out of a threshold of t + 1 signers, then Sparkle is adaptively secure under a weaker variant of the one-more discrete logarithm assumption (AOMDL) in the ROM. Finally, we prove that Sparkle achieves full adaptive security, with a corruption threshold of t, under AOMDL in the algebraic group model (AGM) with random oracles. Importantly, we show adaptive security without requiring secure erasures. Ours is the first proof achieving full adaptive security without exponential tightness loss for any threshold Schnorr signature scheme; moreover, the reduction is tight

Pairing-based identification schemes

We propose four different identification schemes that make use of bilinear pairings, and prove their security under certain computational assumptions. Each of the schemes is more efficient and/or more secure than any known pairing-based identification scheme

Revisiting the security model for aggregate signature schemes

Aggregate signature schemes combine the digital signatures of multiple users on different messages into one single signature. The Boneh-Gentry-Lynn-Shacham (BGLS) aggregate signature scheme is one such scheme, based on pairings, where anyone can aggregate the signatures in any order. We suggest improvements to its current chosen-key security model. In particular, we argue that the scheme should be resistant to attackers that can adaptively choose their target users, and either replace other users' public keys or expose other users' private keys. We compare these new types of forgers to the original targeted-user forger, building up to the stronger replacement-and-exposure forger. Finally, we present a security reduction for a variant of the BGLS aggregate signature scheme with respect to this new notion of forgery. Recent attacks by Joux and others on the discrete logarithm problem in small-characteristic finite fields dramatically reduced the security of many type I pairings. Therefore, we explore security reductions for BGLS with type III rather than type I pairings. Although our reductions are specific to BGLS, we believe that other aggregate signature schemes could benefit from similar changes to their security models

A tight security reduction in the quantum random oracle model for code-based signature schemes

Quantum secure signature schemes have a lot of attention recently, in particular because of the NIST call to standardize quantum safe cryptography. However, only few signature schemes can have concrete quantum security because of technical difficulties associated with the Quantum Random Oracle Model (QROM). In this paper, we show that code-based signature schemes based on the full domain hash paradigm can behave very well in the QROM i.e. that we can have tight security reductions. We also study quantum algorithms related to the underlying code-based assumption. Finally, we apply our reduction to a concrete example: the SURF signature scheme. We provide parameters for 128 bits of quantum security in the QROM and show that the obtained parameters are competitive compared to other similar quantum secure signature schemes

Making Existential-Unforgeable Signatures Strongly Unforgeable in the Quantum Random-Oracle Model

Strongly unforgeable signature schemes provide a more stringent security guarantee than the standard existential unforgeability. It requires that not only forging a signature on a new message is hard, it is infeasible as well to produce a new signature on a message for which the adversary has seen valid signatures before. Strongly unforgeable signatures are useful both in practice and as a building block in many cryptographic constructions. This work investigates a generic transformation that compiles any existential-unforgeable scheme into a strongly unforgeable one, which was proposed by Teranishi et al. and was proven in the classical random-oracle model. Our main contribution is showing that the transformation also works against quantum adversaries in the quantum random-oracle model. We develop proof techniques such as adaptively programming a quantum random-oracle in a new setting, which could be of independent interest. Applying the transformation to an existential-unforgeable signature scheme due to Cash et al., which can be shown to be quantum-secure assuming certain lattice problems are hard for quantum computers, we get an efficient quantum-secure strongly unforgeable signature scheme in the quantum random-oracle model.Comment: 15 pages, to appear in Proceedings TQC 201

Fiat-Shamir for highly sound protocols is instantiable

The Fiat–Shamir (FS) transformation (Fiat and Shamir, Crypto '86) is a popular paradigm for constructing very efficient non-interactive zero-knowledge (NIZK) arguments and signature schemes from a hash function and any three-move interactive protocol satisfying certain properties. Despite its wide-spread applicability both in theory and in practice, the known positive results for proving security of the FS paradigm are in the random oracle model only, i.e., they assume that the hash function is modeled as an external random function accessible to all parties. On the other hand, a sequence of negative results shows that for certain classes of interactive protocols, the FS transform cannot be instantiated in the standard model. We initiate the study of complementary positive results, namely, studying classes of interactive protocols where the FS transform does have standard-model instantiations. In particular, we show that for a class of “highly sound” protocols that we define, instantiating the FS transform via a q-wise independent hash function yields NIZK arguments and secure signature schemes. In the case of NIZK, we obtain a weaker “q-bounded” zero-knowledge flavor where the simulator works for all adversaries asking an a-priori bounded number of queries q; in the case of signatures, we obtain the weaker notion of random-message unforgeability against q-bounded random message attacks. Our main idea is that when the protocol is highly sound, then instead of using random-oracle programming, one can use complexity leveraging. The question is whether such highly sound protocols exist and if so, which protocols lie in this class. We answer this question in the affirmative in the common reference string (CRS) model and under strong assumptions. Namely, assuming indistinguishability obfuscation and puncturable pseudorandom functions we construct a compiler that transforms any 3-move interactive protocol with instance-independent commitments and simulators (a property satisfied by the Lapidot–Shamir protocol, Crypto '90) into a compiled protocol in the CRS model that is highly sound. We also present a second compiler, in order to be able to start from a larger class of protocols, which only requires instance-independent commitments (a property for example satisfied by the classical protocol for quadratic residuosity due to Blum, Crypto '81). For the second compiler we require dual-mode commitments. We hope that our work inspires more research on classes of (efficient) 3-move protocols where Fiat–Shamir is (efficiently) instantiable