The knowledge complexity of quadratic residuosity languages

AbstractNoninteractive perfect zero-knowledge (ZK) proofs are very elusive objects. In fact, since the introduction of the noninteractive model of Blum . (1988), the only perfect zero-knowledge proof known was the one for quadratic nonresiduosity of Blum . (1991). The situation is no better in the interactive case where perfect zero-knowledge proofs are known only for a handful of particular languages.In this work, we show that a large class of languages related to quadratic residuosity admits noninteractive perfect zero-knowledge proofs. More precisely, we give a protocol for the language of thresholds of quadratic residuosity.Moreover, we develop a new technique for converting noninteractive zero-knowledge proofs into round-optimal zero-knowledge proofs for an even wider class of languages. The transformation preserves perfect zero knowledge in the sense that, if the noninteractive proof we started with is a perfect zero-knowledge proof, then we obtain a round-optimal perfect zero-knowledge proof. The noninteractive perfect zero-knowledge proofs presented in this work can be transformed into 4-round (which is optimal) interactive perfect zero-knowledge proofs. Until now, the only known 4-round perfect ZK proof systems were the ones for quadratic nonresiduosity (Goldwasser et al., 1989) and for graph nonisomorphism (Goldreich et al., 1986) and no 4-round perfect zero-knowledge proof system was known for the simple case of the language of quadratic residues

On formal verification of arithmetic-based cryptographic primitives

Cryptographic primitives are fundamental for information security: they are used as basic components for cryptographic protocols or public-key cryptosystems. In many cases, their security proofs consist in showing that they are reducible to computationally hard problems. Those reductions can be subtle and tedious, and thus not easily checkable. On top of the proof assistant Coq, we had implemented in previous work a toolbox for writing and checking game-based security proofs of cryptographic primitives. In this paper we describe its extension with number-theoretic capabilities so that it is now possible to write and check arithmetic-based cryptographic primitives in our toolbox. We illustrate our work by machine checking the game-based proofs of unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and semantic security of the public-key cryptographic scheme of Goldwasser and Micali.Comment: 13 page

Adiabatic Quantum State Generation and Statistical Zero Knowledge

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure

On server trust in private proxy auctions

We investigate proxy auctions, an auction model which is proving very successful for on-line businesses (e.g.http://www.ebay.com), where a trusted server manages bids from clients by continuously updating the current price of the item and the currently winning bid as well as keeping private the winning client’s maximum bid. We propose techniques for reducing the trust in the server by defining and achieving a security property, called server integrity. Informally, this property protects clients from a novel and large class of attacks from a corrupted server by allowing them to verify the correctness of updates to the current price and the currently winning bid. Our new auction scheme achieves server integrity and satisfies two important properties that are not enjoyed by previous work in the literature: it has minimal interaction, and only requires a single trusted server. The main ingredients of our scheme are two minimal-round implementations of zero-knowledge proofs for proving lower bounds on encrypted values: one based on discrete logarithms that is more efficient but uses the random oracle assumption, and another based on quadratic residuosity that only uses standard intractability assumptions but is less efficient.Postprint (published version

On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed

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

Relating non-local quantum computation to information theoretic cryptography

Non-local quantum computation (NLQC) is a cheating strategy for position-verification schemes, and has appeared in the context of the AdS/CFT correspondence. Here, we connect NLQC to the wider context of information theoretic cryptography by relating it to a number of other cryptographic primitives. We show one special case of NLQC, known as $f$-routing, is equivalent to the quantum analogue of the conditional disclosure of secrets (CDS) primitive, where by equivalent we mean that a protocol for one task gives a protocol for the other with only small overhead in resource costs. We further consider another special case of position verification, which we call coherent function evaluation (CFE), and show CFE protocols induce similarly efficient protocols for the private simultaneous message passing (PSM) scenario. By relating position-verification to these cryptographic primitives, a number of results in the cryptography literature give new implications for NLQC, and vice versa. These include the first sub-exponential upper bounds on the worst case cost of $f$-routing of $2^{O(\sqrt{n\log n})}$ entanglement, the first example of an efficient $f$-routing strategy for a problem believed to be outside $P/poly$, linear lower bounds on entanglement for CDS in the quantum setting, linear lower bounds on communication cost of CFE, and efficient protocols for CDS in the quantum setting for functions that can be computed with quantum circuits of low $T$ depth