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.

/pdf/the-knowledge-complexity-of-interactive-proof-systems-31apre0ecf.pdf

The knowledge complexity of interactive proof-systems

We present a zero-knowledge proof system [19] for any NP language L, which allows showing that x in L with error probability less than 2^−k using communication corresponding to O(|x|^c) + k bit commitments, where c is a constant depending only on L. The proof can be based on any bit commitment scheme with a particular set of properties. We suggest an efficient implementation based on factoring. We also present a 4-move perfect zero-knowledge interactive argument for any NP-language L. On input x in L, the communication complexity is O(|x|^c) max(k; l) bits, where l is the security parameter for the prover. Again, the protocol can be based on any bit commitment scheme with a particular set of properties. We suggest efficient implementations based on discrete logarithms or factoring. We present an application of our techniques to multiparty computations, allowing for example t committed oblivious transfers with error probability 2^−k to be done simultaneously using O(t+k) commitments. Results for general computations follow from this. As a function of the security parameters, our protocols have the smallest known asymptotic communication complexity among general proofs or arguments for NP. Moreover, the constants involved are small enough for the protocols to be practical in a realistic situation: both protocols are based on a Boolean formula Phi containing and- , or- and not-operators which verifies an NP-witness of membership in L. Let n be the number of times this formula reads an input variable. Then the communication complexity of the protocols when using our concrete commitment schemes can be more precisely stated as at most 4n + k + 1 commitments for the interactive proof and at most 5nl +5l bits for the argument (assuming k the number of commitments required for the proof is linear in n. Both protocols are also proofs of knowledge of an NP-witness of membership in the language involved.

/pdf/linear-zero-knowledgde-a-note-on-efficient-zero-knowledge-3071gbx1nv.pdf

Linear Zero-Knowledgde. A Note on Efficient Zero-Knowledge Proofs and Arguments

We consider the problem of basing Oblivious Transfer (OT) and Bit Commitment (BC), with information theoretic security, on seemingly weaker primitives.We introduce a general model for describing such primitives, called Weak Generic Transfer (WGT). This model includes as important special cases Weak Oblivious Transfer (WOT), where both the sender and receiver may learn too much about the other party's input, and a new, more realistic model of noisy channels, called unfair noisy channels. An unfair noisy channel has a known range of possible noise levels; protocols must work for any level within this range against adversaries who know the actual noise level.

We give a precise characterization for when one can base OT on WOT. When the deviation of the WOT from the ideal is above a certain threshold, we show that no information-theoretic reductions from OT (even against passive adversaries) and BC exist; when the deviation is below this threshold, we give a reduction from OT (and hence BC) that is information-theoretically secure against active adversaries.

For unfair noisy channels we show a similar threshold phenomenon for bit commitment. If the upper bound on the noise is above a threshold (given as a function of the lower bound) then no information-theoretic reduction from OT (even against passive adversaries) or BC exist; when it is below this threshold we give a reduction from BC. As a partial result, we give a reduction from OT to UNC for smaller noise intervals.

/pdf/on-the-im-possibility-of-basing-oblivious-transfer-and-bit-1lszetsiln.pdf

On the (im)possibility of basing oblivious transfer and bit commitment on weakened security assumptions

We present a new constant round protocol for non-malleable zero-knowledge. Using this protocol as a subroutine, we obtain a new constant-round protocol for non-malleable commitments. Our constructions rely on the existence of (standard) collision resistant hash functions. Previous constructions either relied on the existence of trapdoor permutations and hash functions that are collision resistant against sub-exponential sized circuits, or required a super-constant number of rounds.Additional results are the first construction of a non-malleable commitment scheme that is statistically hiding (with respect to opening), and the first non-malleable protocols that satisfy a strict polynomial-time simulation requirement. The latter are constructed by additionally assuming the existence of trapdoor permutations.Our approach differs from the approaches taken in previous works in that we view non-malleable zero-knowledge as a building-block rather than an end goal. This gives rise to a modular construction of non-malleable commitments and results in a somewhat simpler analysis.The techniques that we use to construct our zero-knowl-edge protocol are non black-box, but are different than the non black-box techniques previously used in the context of non-malleable coin-tossing.

/pdf/new-and-improved-constructions-of-non-malleable-220dev3rdg.pdf

New and improved constructions of non-malleable cryptographic protocols

In this paper we show that three apparently unrelated problems are in fact very closely related. We sketch these problems at a high level. The selective decommitment problem first arose in a slightly different form, selective decryption, in the context of Byzantine agreement, no later than 1985. Instead of seeing encryptions of plaintexts the adversary is given commitments to the plaintexts. This problem is poorly understood even in strong-receiver commitments, which leak no information about the plaintext values information-theoretically. The second problem is in complexity theory: what can be proved in (a possibly weakened form of) zero-knowledge in a 3-round argument (interactive proof in which the prover is polynomial-time bounded)? The Fiat-Shamir Methodology is cryptographic, and addresses a methodology suggested by Fiat and Shamir (1987) to construct a (non-interactive) signature scheme from any 3-round (not necessarily zero-knowledge) public-coin identification scheme.

http://ieeexplore.ieee.org/iel5/6604/17631/00814626.pdf

Magic functions

``Perfect zero-knowledge arguments'' is a cryptographic primitive which allows one polynomial-time player to convince another polynomial-time player of the validity of an NP statement, without revealing any additional information (in the information-theoretic sense). Here the security achieved is on-line: in order to cheat and validate a false theorem, the prover must break a cryptographic assumption on-line during the conversation, while the verifier cannot find (ever) any information unconditionally. Despite their practical and theoretical importance, it was only known how to implement zero-knowledge arguments based on specific algebraic assumptions.

In this paper we show a general construction which can be based on any one-way permutation. The result is obtained by a construction of an information-theoretic secure bit-commitment protocol. The protocol is efficient (both parties are polynomial time) and can be based on any one-way permutation.

http://www.wisdom.weizmann.ac.il/%7Enaor/PAPERS/novy.pdf

Rafail Ostrovsky

Papers

Perfect Zero-Knowledge Arguments for NP Using Any One-Way Permutation