Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

In this paper we present a new multi-authority secret-ballot election scheme that guarantees privacy, universal verifiability, and robustness. It is the first scheme for which the performance is optimal in the sense that time and communication complexity is minimal both for the individual voters and the authorities. An interesting property of the scheme is that the time and communication complexity for the voter is independent of the number of authorities. A voter simply posts a single encrypted message accompanied by a compact proof that it contains a valid vote. Our result is complementary to the result by Cramer, Franklin, Schoenmakers, and Yung in the sense that in their scheme the work for voters is linear in the number of authorities but can be instantiated to yield information-theoretic privacy, while in our scheme the voter's effort is independent of the number of authorities but always provides computational privacy-protection. We will also point out that the majority of proposed voting schemes provide computational privacy only (often without even considering the lack of information-theoretic privacy), and that our new scheme is by far superior to those schemes.

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A secure and optimally efficient multi-authority election scheme

A publicly verifiable secret sharing (PVSS) scheme is a verifiable secret sharing scheme with the property that the validity of the shares distributed by the dealer can be verified by any party; hence verification is not limited to the respective participants receiving the shares. We present a new construction for PVSS schemes, which compared to previous solutions by Stadler and later by Fujisaki and Okamoto, achieves improvements both in efficiency and in the type of intractability assumptions. The running time is O(nk), where k is a security parameter, and n is the number of participants, hence essentially optimal. The intractability assumptions are the standard Diffie-Hellman assumption and its decisional variant. We present several applications of our PVSS scheme, among which is a new type of universally verifiable election scheme based on PVSS. The election scheme becomes quite practical and combines several advantages of related electronic voting schemes, which makes it of interest in its own right.

A simple publicly verifiable secret sharing scheme and its application to electronic voting

A publicly verifiable secret sharing (PVSS) scheme is a veri fiable secret sharing scheme with the property that the validity of the shares distributed by the dealer can be verified by any party; hence verification is not limited to the respective participants receiving the shares. We present a new construction for PVSS schemes, which compared to previous solutions by Stadler and later by Fujisaki and Okamoto, achieves improvements both in efficiency and in the type of intractability assumptions. The running time is O(nk), where k is a security parameter, and n is the number of participants, hence essentially optimal. The intractability assumptions are the standard Diffie-Hellman assumption and its decisional variant. We present several applications of our PVSS scheme, among which is a new type of universally verifiable election scheme based on PVSS. The election scheme becomes quite practical and combines several advantages of related electronic voting schemes, which makes it of interest in its own right.

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A Simple Publicly Verifiable Secret Sharing Scheme and Its Application to Electronic

Looking at current cryptographic-based e-voting protocols, one can distinguish three basic design paradigms (or approaches): (a) Mix-Networks based, (b) Homomorphic Encryption based, and (c) Blind Signatures based. Each of the three possesses different advantages and disadvantages w.r.t. the basic properties of (i) efficient tallying, (ii) universal verifiability, and (iii) allowing write-in ballot capability (in addition to predetermined candidates). In fact, none of the approaches results in a scheme that simultaneously achieves all three. This is unfortunate, since the three basic properties are crucial for efficiency, integrity and versatility (flexibility), respectively. Further, one can argue that a serious business offering of voting technology should offer a flexible technology that achieves various election goals with a single user interface. This motivates our goal, which is to suggest a new ”vector-ballot” based approach for secret-ballot e-voting that is based on three new notions: Provably Consistent Vector Ballot Encodings, Shrink-and-Mix Networks and Punch-Hole-Vector-Ballots. At the heart of our approach is the combination of mix networks and homomorphic encryption under a single user interface; given this, it is rather surprising that it achieves much more than any of the previous approaches for e-voting achieved in terms of the basic properties. Our approach is presented in two generic designs called ”homomorphic vector-ballots with write-in votes” and ”multi-candidate punch-hole vector-ballots”; both of our designs can be instantiated over any homomorphic encryption function.

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The Vector-Ballot E-voting approach

Secret sharing allows a secret key to be distributed among n persons, such that k(1 <= k <= n) of these must be present in order to recover it at a later time. This report first shows how this can be done such that every person can verify (by himself) that his part of the secret is correct even though fewer than k persons get no Shannon information about the secret. However, this high level of security is not needed in public key schemes, where the secret key is uniquely determined by a corresponding public key. It is therefore shown how such a secret key (which can be used to sign messages or decipher cipher texts) can be distributed. This scheme has the property, that even though everybody can verify his own part, sets of fewer than k persons cannot sign/decipher unless they could have done so given just the public key. This scheme has the additional property that more than k persons can use the key without compromising their parts of it. Hence, the key can be reused. This technique is further developed to be applied to undeniable signatures. These signatures differ from traditional signatures as they can only be verified with the signer's assistance. The report shows how the signer can authorize agents who can help verifying signatures, but they cannot sign (unless the signer permits it).

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Torben Pryds Pedersen

Papers

Distributed Provers and Verifiable Secret Sharing Based on the Discrete Logarithm Problem