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The motivation, to me, is that in reality you can consider any router on the internet to be successfully executing an "intruder-in-the-middle" attack just by forwarding messages unchanged. After a successful execution of the identification scheme, Bob knows that someone on the channel is Alice, which is all the protocol was hoping to achieve. It was ...


Without a sign the verifier learns that the number he received is a QR modulo n. Whether a number is a QR is a hard problem as he does not know the factors of n.


In context of interactive proof systems (including zero-knowledge proofs) completeness means the same as the term correctness as used for many other (interactive) cryptographic schemes or protocols. I guess that's mainly due to historical reasons (there are even some people that use correctness instead of completeness in context of zero-knowledge proofs). ...


A straightforward way to prove this when you can prove AND as well as OR statements about discrete logarithms is to take all the $K=\binom{M}{N}$ subsets $A_i=\{A_{i_1},\ldots,A_{i_N}\}$ with $N$ elements of points from the set of your $M$ points and prove the statement $$PK\{(\alpha_1,\ldots,\alpha_N): \bigvee_{j\in K} \big( \bigwedge_{A_{j_i}\in A_j} ...

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