A density lemma

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In Chapter 6 we show how highly efficient zero-knowledge protocols can be constructed. As we have discussed, security in the presence of malicious adversaries is typically achieved by forcing the parties to behave honestly. The immediate way to do this is to force the parties to prove in zero-knowledge that they are following the protocol specification. Needless to say, a straightforward implementation of this idea is very inefficient. For this reason, many try to stay clear of explicit zero-knowledge proofs for enforcing honest behavior.

Outputs: The honest party always outputs the output value it obtained from the trusted party. The corrupted party outputs nothing. The adversary A outputs any arbitrary (probabilistic polynomial-time computable) function of the initial inputs of the corrupted party, the auxiliary input z, and the value fi (x′ , y ′ ) obtained from the trusted party. The output of the honest party and the adversary in an execution of the above ideal model is denoted by idealscϵf,S(z),i (x, y, n). 4 Security in the Presence of Covert Adversaries 35 Notice that there are two types of “cheating” here.

Mit ), where w ∈ {x, y} (its value depending on the value of i), r i equals the contents of the ith party’s internal random tape, and mij represents the jth message that it received. • The output of the ith party during an execution of π on (x, y) and security parameter n is denoted by outputπi (x, y, n) and can be computed from its own view of the execution. We denote the joint output of both parties by outputπ (x, y, n) = (outputπ1 (x, y, n), outputπ2 (x, y, n)). t. semi-honest behavior): Let f = (f1 , f2 ) be a functionality.

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