In this thesis we follow two directions: Zero Knowledge Protocols and the Discrete Logarithm Problem. In each direction we present the necessary background and we give a new approach for some parts of the existing protocols. / The new parts are dedicated to the soundness property of the Schnorr Identification Scheme and to the security of the sum+-Protocol. Since both directions are very well-known and studied in the field of cryptography, they are presented with many details so that the new results are easy to follow. / In writing this thesis we have tried to present the material in a specific order and in a manner easy to read even by beginners in cryptography.

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With current state-of-the-art privacy-preserving blockchain solutions, users can submit transactions to a blockchain while maintaining full anonymity and not leaking the contents of the transaction through cryptographic techniques like zero-knowledge proofs and homomorphic encryption. However, the architecture of a blockchain consists of a decentralized network where every network participant maintains their own local copy of the blockchain and updates it upon every added transaction. As a result, the volume of blockchain transactions and the timestamp of each blockchain transaction for an application is publicly available. This is problematic for applications with time-sensitive or volume-sensitive outcomes because users may want this information to be privatized, such as not leaking the lateness of student examinations. However, this is not possible with existing blockchain research. In this thesis, we propose a blockchain system for multi-party applications that does not leak any useful information from the volume and timing metadata of the application's transactions, including maintaining the privacy of a time-sensitive or volume-sensitive outcome. We achieve this by adding sufficient noise using indistinguishable decoy transactions such that an adversary cannot deduce which transactions actually impacted the outcome of the application. This is facilitated in a manner where anyone can publicly verify the application's execution to be correct, fair, and honest. We demonstrate and evaluate our approach by implementing a Dutch auction that supports decoy bid transactions on a private Ethereum blockchain network. / Master of Science / Blockchains are distributed, append-only, digital ledgers whose current state is continuously agreed upon through the consensus of network participants and not by any centralized party. These characteristics make them unique for many applications because they enable the application to be facilitated and executed in a public, verifiable, decentralized, and tamper-proof manner. For example, Bitcoin, the most popular cryptocurrency, uses blockchains to continuously maintain a permanent, verifiable ledger of payment transactions. However, one downside of this public architecture is that the volume of transactions and the timestamp of each transaction can always be publicly observed (e.g. the timestamp of every Bitcoin payment is public). This is problematic for applications with time-sensitive or volume-sensitive outcomes because users may want this volume and timing information to be privatized, such as not leaking the lateness of student examinations which could have severe consequences like violating student privacy laws. But currently with state-of-the-art blockchain research, privatizing this information is not possible. In this thesis, we demonstrate our approach that enables these time-sensitive and volume-sensitive applications to be implemented on blockchains in a manner that can maintain the privacy of these time-sensitive or volume-sensitive outcomes without sacrificing the application's integrity or verifiability. We then demonstrate and evaluate our approach through implementing a Dutch auction that supports decoy bid transactions on a private blockchain network.

We show that the long-standing assumption of "no-communication" between the provers of the two-prover model is not sufficiently precise to guarantee the security of a bit commitment scheme against malicious adversaries. Indeed, we show how a simple correlated random variable, which does not allow to communicate, can be used to cheat a simplified version (sBGKW) of the bit commitment scheme of Ben-Or, Goldwasser, Kilian, and Wigderson [BGKW88]. Instead we propose a stronger notion of separation between the two provers which takes into account correlated computations. To emphasize the risk that entanglement still represents for the security of a commitment scheme despite the stronger notion of separation, we present two variations of the sBGKW scheme that can be cheated by quantum provers with probability (almost) one. A complete proof of security against quantum adversaries is then given for the sBGKW scheme. By reduction we also obtain the security of the original BGKW scheme against quantum provers. For the unfamiliar reader, basic notions of quantum processing are provided to facilitate the understanding of the proofs presented.

Thesis (M.S.)--University of North Texas, Dec., 2009. / Title from title page display. Includes bibliographical references.

Zero-knowledge proofs protocols are effective interactive methods to prove a node's identity without disclosing any additional information other than the veracity of the proof. They are implementable in several ways. In this thesis, I investigate the graph isomorphism based zero-knowledge proofs protocol. My experiments and analyses suggest that graph isomorphism can easily be solved for many types of graphs and hence is not an ideal solution for implementing ZKP.

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Zero-knowledge protocols enable one party, called a prover, to "convince" another party, called a verifier, the validity of a mathematical statement such that the verifier "learns nothing" other than the fact that the proven statement is true. The different ways of formulating the terms "convince" and "learns nothing" gives rise to four classes of languages having zero-knowledge protocols, which are: statistical zero-knowledge proof systems, computational zero-knowledge proof systems, statistical zero-knowledge argument systems, and computational zero-knowledge argument systems. We establish complexity-theoretic characterization of the classes of languages in NP having zero-knowledge argument systems. Using these characterizations, we show that for languages in NP: -- Instance-dependent commitment schemes are necessary and sufficient for zero-knowledge protocols. Instance-dependent commitment schemes for a given language are commitment schemes that can depend on the instance of the language, and where the hiding and binding properties are required to hold only on the YES and NO instances of the language, respectively. -- Computational zero knowledge and computational soundness (a property held by argument systems) are symmetric properties. Namely, we show that the class of languages in NP intersect co-NP having zero-knowledge arguments is closed under complement, and that a language in NP has a statistical zero-knowledge **argument** system if and only if its complement has a **computational** zero-knowledge proof system. -- A method of transforming any zero-knowledge protocol that is secure only against an honest verifier that follows the prescribed protocol into one that is secure against malicious verifiers. In addition, our transformation gives us protocols with desirable properties like having public coins, being black-box simulatable, and having an efficient prover. The novelty of our results above is that they are **unconditional**, meaning that they do not rely on any unproven complexity assumptions such as the existence of one-way functions. Moreover, in establishing our complexity-theoretic characterizations, we give the first construction of statistical zero-knowledge argument systems for NP based on any one-way function.

Thesis (M.S.C.S.) PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This paper proposes a novel distributed authentication system that uses robust alternatives in cryptographic algorithms to grant a third-party access to personal data without compromising a user's credentials. The paper examines briefly the concept of distributed authentication systems, and discusses how elliptic curve cryptography and Lamport's hash chain can operate in a zero-knowledge proof to establish and manage trust. The paper also discusses how this design avoids some of the most common flaws in distributed authentication systems. Finally, based on results from tests conducted with included source codes, the paper argues that increasing number of rounds of zero-knowledge proof yields substantially faster performance than increasing the modulus for elliptic curve calculations while maintaining comparable levels of security. / 2031-01-01

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