1. Feige-Fiat-Shamir Zero Knowledge Proof Based on difficulty of computing square roots mod a composite n Given two large primes p, q and n=p * q, computing x mod n is very hard without knowing p, q But there exist efficient algorithms for computing square roots modulo a prime number, and therefore x mod n can be computed efficiently if p and q are known

2. Feige-Fiat-Shamir Zero Knowledge Proof ProverVerifier

3. p, q, S n=p*q V=S*S mod n Feige-Fiat-Shamir Zero Knowledge Proof ProverVerifier

4. p, q, S n=p*q V=S*S mod n Feige-Fiat-Shamir Zero Knowledge Proof Protocol: Prover: Generate random r, send x = r * r mod n x = r * r mod n ProverVerifier

5. p, q, S n=p*q V=S*S mod n Feige-Fiat-Shamir Zero Knowledge Proof Protocol: Prover: Generate random r, send x = r * r mod n Verifier: Select e ∈ {0,1}, ask to prove it knows x mod n e ProverVerifier

6. p, q, S n=p*q V=S*S mod n Feige-Fiat-Shamir Zero Knowledge Proof Protocol: Prover: Generate random r, send x = r * r mod n Verifier: Select e ∈ {0,1}, ask to prove it knows x mod n Prover: Send y = r * S e mod n y ProverVerifier

7. p, q, S n=p*q V=S*S mod n Feige-Fiat-Shamir Zero Knowledge Proof Protocol: Prover: Generate random r, send x = r * r mod n Verifier: Select e ∈ {0,1}, ask to prove it knows x mod n Prover: Send y = r * S e mod n Verifier: Checks y * y = x * V e mod n ProverVerifier

8. Parallel Zero Knowledge Protocols Send m commitments in one message {c(1), c(2),...,c(m)} ProverVerifier

9. Monitor is the Trust Center Trust Center with Key Dictionary public key ProverVerifier