We show that only languages in BPP have public-coin, black-box zero-knowledge protocols that are secure under an unbounded (polynomial) number of parallel repetitions. This result holds both in the plain model (without any set-up) and in the Bare Public-Key Model (where the prover and the verifier have registered public keys). We complement this result by showing the existence of a public-coin black-box zero-knowledge proof that remains secure under any a-priori bounded number of concurrent executions.
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