With increasing demands for internet security, the demand for distribution of public key certificates and certificate authorities spreads. This brings up the new problem of public key validation. This problem can arise in different contexts and can be adressed in some of them using novel zero knowledge protocols, while in other contexts it can be only poorly solved. The purpose of this paper is to give an overview of the topic of key validation and to discuss the possible solutions to the different instances in which it appears. We also present a new zero knowledge protocol for correctness of an RSA public exponent and a simple protocol (not zero-knowledge) for primality of a discrete logarithm. We use the latter to give a proof protocol (not zero knowledge) that an RSA key is the product of strong primes These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Let n be a large composite number. Without factoring n, the computation of a 2 t (mod n)given a, t with gcd(a# n) = 1 and t!n can be done in t squarings modulo n.For t n (e.g., n?2 1024 and t!2 100 ), no lower complexity than t squarings is known to fulfill this task. Rivest et al suggested to use such constructions as good candidates for realising timed-release crypto problems. We argue the necessity for a zero-knowledge proof of the correctness of such constructions and propose the first practically efficient protocol for a realisation. Our protocol proves, in log 2 t standard crypto operations, the correctness of (a e ) 2 t (mod n) with respect to a e where e is an RSA encryption exponent. With such a proof, a Timed-release Encryption of a message M can be given as a 2 t M (mod n) with the assertion that the correct decryption of the RSA ciphertext M e (mod n) can be obtained by performing t squarings modulo n starting from a. Timed-release RSA signatures can be constructed analogously. Keywords Timed-release cryptography, Time-lock puzzles, Non-parallelisability, Efficient zero-knowledge protocols. 1
Following Dwork, Naor, and Sahai (30th STOC, 1998), we consider concurrent execution of protocols in a semi-synchronized network. Specifically, we assume that each party holds a local clock such that a constant bound on the relative rates of these clocks is a-priori known, and consider protocols that employ time-driven operations (i.e., time-out in-coming messages and delay out-going messages). We show that the constant-round zero-knowledge proof for NP of Goldreich and Kahan (Jour. of Crypto., 1996) preserves its security when polynomially-many independent copies are executed concurrently under the above timing model. We stress that our main result establishes zero-knowledge of interactive proofs, whereas the results of Dwork et. al. are either for zero-knowledge arguments or for a weak notion of zero-knowledge (called ffl-knowledge) proofs. Our analysis identifies two extreme schedulings of concurrent executions under the above timing model: the first is the case of parallel execution of polynomially-many copies, and the second is of concurrent execution of polynomially-many copies such the number of copies that are simultaneously active at any time is bounded by a constant (i.e., bounded simultaneity). Dealing with each of these extreme cases is of independent interest, and the general result (regarding concurrent executions under the timing model) is obtained by combining the two treatments.
A family $(S_t)$ of sets is $p$-bounded Diophantine if $S_t$ has a representing $p$-bounded polynomial $R_{S,t}$, s.t.~$x\\in S_t \\iff (\\exists y)[R_{S}(x;y)=0]$. We say that $(S_t)$ is unbounded Diophantine if additionally, $R_{S,t}$ is a fixed $t$-independent polynomial. We show that $p$-bounded (resp., unbounded) Diophantine set has a polynomial-size (resp., constant-size) statistical zero-knowledge proof system that a committed tuple $x$ belongs to $S$. We describe efficient SZK proof systems for several cryptographically interesting sets. Finally, we show how to prove in SZK that an encrypted number belongs to $S$.
We propose a zero-knowledge interactive proof based identification and signature scheme. The protocol is based on Euler's totient function and discrete logarithms over the ring Z/nZ , and can be applied to smart cards. A prover keeps a signed subgroup generator provided by a trusted center as its secret information. Our scheme has symmetricity in the sense that the same computational complexity and the same hardware both for Prover and for Verifier are required. Also, it requires minimal amount of computation and communications for secret information. The protocol is versatile enough to be applicable to digital signature scheme, multiple digital signature scheme and key exchange protocol. We outline those protocols to show the versatility of our protocol.
This paper presents a technique to speed up practical zero-knowledge cryptographic protocols. We introduce a new technique of batching proofs of knowledge. Its security is analysed and its applicability to some cryptographic protocols is described. The use of the batching technique can significantly reduce the number of online computations required in the proofs and consequently helps to lower the costs associated with the corresponding transactions. This technique is particularly useful for e-commerce related cryptographic protocols.
In this paper we show how to construct noninteractive zero knowledge proofs for any NP statement under general (rather than number theoretic) assumptions, and how to enable polynomially many provers to give polynomially many such proofs based on a single random string. Our constructions can be used in cryptographic applications in which the prover is restricted to polynomial time.
The wide applicability of zero-knowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and/or parallel) composition of zero-knowledge protocols is zero-knowledge too. We demonstrate the limitations of the composition of zero-knowledge protocols by proving that the original definition of zero-knowledge is not closed under sequential composition; and that even the strong formulations of zero-knowledge (e.g., black-box simulation) are not closed under parallel execution. We present lower bounds on the round complexity of zero-knowledge proofs, with significant implications for the parallelization of zero-knowledge protocols. We prove that three-round interactive proofs and constant-round Arthur-Merlin proofs that are black-box simulation zero-knowledge exist only for languages in BPP. In particular, it follows that the “parallel versions” of the first interactive proofs systems presented for quadratic residuosity, graph isomorphism, and any language in NP, are not black-box simulation zero-knowledge, unless the corresponding languages are in BPP Whether these parallel versions constitute zero-knowledge proofs was an intriguing open questions arising from the early works on zero-knowledge. Other consequences are a proof of optimality for the round complexity of various known zero-knowledge protocols and the necessity of using secret coins in the design of “parallelizable” constant-round zero-knowledge proofs.
The present paper investigates the possibility of designing zero-knowledge identification schemes based on hard problems from coding theory. Zero-knowledge proofs were introduced by Goldwasser, Micali, and Rackoff (1985). Their practical significance was soon demonstrated in the work of Fiat and Shamir [1986], who turned zero-knowledge proofs of quadratic residuosity into efficient means of establishing user identities. In the present paper, we propose a new identification scheme, based on error-correcting codes, which is zero-knowledge and seems of practical value. Furthermore, we describe several variants, including one which has an identity-based character. The security of our schemes depends on the hardness of finding a word of given syndrome and prescribed (small) weight with respect to some randomly generated binary linear error-correcting code. This is, of course, not the first attempt to design a cryptographic scheme using tools from coding theory. The difference is that identification protocols do not follow the public key paradigm based on trap-door functions and described in the seminal Diffie-Hellman paper [1976]. Rather, they only require one-way functions, which opens the way to using, in a rather direct manner, simple combinatorial problems of the kind provided by coding theory. The resulting schemes compare favorably to their number-theoretic analogs.
The communication complexity of zero-knowledge proof systems is improved. Let C be a Boolean circuit of size n. Previous zero-knowledge proof systems for the satisfiability of C require the use of Omega (kn) bit commitments in order to achieve a probability of undetected cheating not greater than 2/sup -k/. In the case k=n, the communication complexity of these protocols is therefore Omega (n/sup 2/) bit commitments. A zero-knowledge proof is given for achieving the same goal with only O(n/sup m/+k square root n/sup m/) bit commitments, where m=1+ epsilon /sub n/ and epsilon /sub n/ goes to zero as n goes to infinity. In the case k=n, this is O(n square root n/sup m/). Moreover, only O(k) commitments need ever be opened, which is interesting if committing to a bit is significantly less expensive than opening a commitment.>