The knowledge complexity of interactive proof systems. However later on when defining a more general form of. Signature schemes and anonymous credentials from bilinear. Quantum attacks on classical proof systems the hardness of. A protocol for propertybased attestation proceedings of.
Discrete logarithms for finite groups, computing 10. The discrete logarithm problem is the computational task of. Pdf efficient noninteractive zeroknowledge proofs in cross. In cryptography, the fiatshamir heuristic is a technique for taking an interactive proof of knowledge and creating a digital signature based on it. Computation of discrete logarithms in prime fields lamacchia, odlyzdo.
In this paper, employing a bloom filter, we propose a multiparty private set intersection cardinality mpsica, where the number of participants in. Sigma protocols are proof systems that focus on proving algebraic statements, i. In george blakley and david chaum, editors, advancesincryptology, volume 196 of. Nevertheless, the most important examples of publickey cryptography using discrete logarithms, in terms of wide use and. E cient proofs of knowledge of discrete logarithms and. Proof systems for general statements about discrete logarithms 1997. We identify the basic underlying techniques, generalize these techniques to prove linear relations among discrete logarithms, and propose a notation for describing complex and general statements about knowledge of discrete logarithms. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Technical report tr 1997, institute for theoretical computer science, eth. However, centralized frameworks suffer from the issues of unscalability and singlepointoffailure. In any of the cryptographic systems that are based on discrete logarithms, p must be chosen such that p 1 has at least one large prime factor. Thanks for contributing an answer to mathematics stack exchange. In particular, if g hgi, then hgui generates the subgroup of uth powers in g, which has order v, and similarly hgvi generates the subgroup of vth powers, which has order u. Discrete logarithms carl pomerance, dartmouth college.
In x3, we describe our computation of discrete logarithms in the 4841bit eld f 36 509. Proof systems for general statements about discrete logarithms 1997 camenisch, stadler. Nizk proof of knowledge n of m discrete logarithms threshold. Algebraic macs and keyedverification anonymous credentials. Almost exclusively, these proof systems focus on proving algebraic statements, i. Batch proofs of partial knowledge centre for applied. The following table gives a summary of the logarithm properties. A preliminary version of this paper appeared in the proceedings of the 35th annual symposium on foundations of computer science, santa fe, nm, nov. For the cryptographic foundations of this framework see proof systems for. This is the same as a schnorr signature with public key h. Schnorrs digital signature scheme 18 and systems for proving the. In this paper we present, for the first time, efficient zeroknowledge proofs of. Typical examples are efficient proofs of knowledge of a discrete logarithm which are based on schnorrs digital signature scheme 18. Statements in a proof can include the axioms something assumed to be true, the premises, and previously proved theorems rules of inference, and definitions of terms, are used to draw intermediate conclusions from the other statements, tying the steps of a proof final step is usually the conclusion of theorem 3.
Zeroknowledge proof systems are an essential building block used in many privacypreserving systems, e. Floor and ceiling 164 definition and basic properties. E cient noninteractive zeroknowledge proofs in cross. Blind attributebased encryption and oblivious transfer. Specification of the identity mixer cryptographic library. Practical verifiable encryption and decryption of discrete logarithms. Proving in zeroknowledge that a number is the product of. This example implements linkable ring signatures lrs generically using the camenischstadler proof framework and hashprover. Proof in new multisignatures in the plain publickey. Polynomialtime algorithms for prime factorization and.
We use a generalized version of an or proof of knowledge of one out of two discrete logarithms and other standard proofs of knowledge about discrete logarithms 1. Discrete logarithms for finite groups discrete logarithms for finite groups klingler, lee. Public key cryptography using discrete logarithms in. One is a proof of the equality of two discrete logarithms as used in 9 for a signature scheme. Of particular interest is a method for proving the knowledge of a signature. In most of the cases, psi allows two parties to securely determine the intersection of their private input sets, and no other information. Secure and efficient multiparty private set intersection. There is an explicit formula 1 for discrete logarithms established by mullen and white. Proof systems for general statements about discrete logarithms.
Constantround perfect zeroknowledge computationally. An integer is a primitive root modulo p if for every relatively prime to p there is an integer x such that x mod p. A general, flexible and efficient proof of inclusion and exclusion. How to generate cryptographically strong sequences of.
Discrete logarithm commitment scheme common input knowledge extractor group. To avoid confusion with ordinary logs, we sometimes call this the. The general method on designing zeroknowledge proofs can be seen in 29. Introduction much excitement was caused when it was discovered in 1986 by goldreich, micali and wigderson that all statements in np have computational zeroknowledge interac tive proof systems under the assumption that secure encryption functions exist. A shorter proof for an explicit formula for discrete. Boneh, editor, crypto, volume 2729 of lecture notes in computer science, pages 126144. Proofofknowledge of representation of committed value and its application 2010 au, susilo, mu. Quantum attacks on classical proof systems the hardness.
E cient proofs of knowledge of discrete logarithms and representations in groups with hidden order endre bangerter1, jan camenisch1. It also has important applications in computer science. Signature schemes and anonymous credentials from bilinear maps. Contradiction and contraposition 171 proof by contradiction. Crypto 2016 proposed an interactive zk proof system for this. To show algorithms always produce the correct results. Proofs that yield nothing but their validity or all languages in np have zero. Most of the existing iot infrastructures are centralized, in which the presence of a cloud server is mandatory. Although we restrict ourselves to statements about discrete logarithms. The main reason is that zk proofs for general statements are usually ine cient. It is the basis of the correct mathematical arguments, that is, the proofs.
One could, of course, express any np relation as a combination of algebraic statements, for example by expressing the. Proving in zeroknowledge that a number is the product of two. Proofs of logarithm properties solutions, examples, games. The internet of things iot is experiencing explosive growth and has gained extensive attention from academia and industry in recent years. Ergoscript, a cryptocurrency scripting language supporting. In contrast to real numbers that have the property of varying smoothly, the objects studied in discrete mathematics such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values. Nov 11, 2018 proof systems for general statements about discrete logarithms 1997 camenisch, stadler. This is the second computation of discrete logarithms in a cryptographicallyinteresting nite eld that was purported to provide 128 bits of security against coppersmiths attack. Girault 26 suggests an e cient proof of knowledge for discrete. The presumed difficulty of computing discrete logarithms in finite fields is the basis of several popular public key cryptosystems. For the cryptographic foundations of this framework see proof systems for general statements about discrete logarithms at. Based on this proof of knowledge of a discrete logarithm, several other systems have been proposed. On factoring integers and evaluating discrete logarithms a thesis presented by john aaron gregg to the departments of mathematics and computer science in partial ful. An axiom is a statement that is assumed to be true, or in the case of a.
C protocols to prove knowledge of and relations among discrete logarithms 38. Pdf cryptanalysis of an efficient proof of knowledge of. Since the sigmaprotocol is a classical proof of knowledge, it is also a classical proof. This way, some fact for example, knowledge of a certain secret number can be publicly proven without revealing underlying information.
An improved protocol for demonstrating possession of discrete logarithms and some generalizations. Specification of the identity mixer cryptographic library version 2. But a classical proof is also a quantum proof, because an unlimited classical adversary can simulate a quantum adversary. The technique is due to amos fiat and adi shamir 1986. Some other forms of argument fallacies can lead from true statements to an incorrect conclusion. Proof systems for general statements about discrete. These compilations provide unique perspectives and applications you wont find anywhere else. Polynomialtime algorithms for prime factorization and discrete logarithms on a quantum computer thanks. In the field of privacy preserving protocols, private set intersection psi plays an important role. Relation between proof by contradiction and proof by contraposition. A ring signature proves that the signer owns one of a list of public keys, without revealing anything about which public key.
Take a guided, problemsolving based approach to learning algebra. Simon singh a proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true. In fact in these schemes, the authors often employ the protocol with nonbinary challenges, sometimes wrongly relying on them to be proofs of knowledge in this setting as well. Public key cryptography using discrete logarithms in finite. How can we prove that two discrete logarithms are equal. We identify the basic underlying techniques, generalize these techniques to prove linear relations among discrete logarithms, and propose a notation for describing complex and generalstatements about knowledge of discrete logarithms. A linkable ring signature lrs is the same but includes a linkage tag, which the signer proves to correspond 1to1. More generally, one can prove that two discrete logarithms satisfy a linear equation. Efficient proofs of knowledge of discrete logarithms and. Proof systems for knowledge of discrete logarithms are an important primitive in cryptography. A ring signature proves that the signer owns one of a list of public keys, without revealing anything about which public key the signer actually owns. An introduction to discrete math with a focus on proof, logic, proof by induction, cryptography and networks of processors.
In advances in cryptology eurocrypt87, volume 304 of lncs, pages 127. In drynx, we choose to verify computation integrity by using the proofs for general statements about discrete logarithms, introduced by camenisch and stadler 44. But avoid asking for help, clarification, or responding to other answers. The discrete logarithm in finite fields has interesting applications in crytography. Unfortunately, only a few systems have been used in practice. Efficient zeroknowledge proof of algebraic and nonalgebraic. A public key cryptosystem and a signature scheme based on discrete logarithms.
Scroll down the page for more explanations and examples on how to proof the logarithm properties. The author deeply regrets that, due to space and time constraints, it is not exhaustive. Then, in x4, we show that a recent idea of guillevic 25 can be used to compute. In particular, given proof systems for single statements, they show how to construct a proof system for any monotone boolean formula over these statements. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Proof systems for general statements about discrete logarithms 1997 cached. On factoring integers and evaluating discrete logarithms. Typical examples are efficient proofs of knowledge of a discrete logarithm which are based on. However, this argument does not apply when we consider computationally limited provers, see section 6. The main reason is that zk proofs for general statements are usually ine. If p 1 has only small prime fac tors, then computing discrete logarithms is easy see a. Here is an interactive proof of knowledge of a discrete. But then computing logg t is really solving the congruence ng.
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