1. Commitment schemes Commit stage Reveal stage m m S mm

2. Parallel repetition might not reduce soundness error b à {0,1} P b’ = b V accepts if b’ = b, and the safes are different Safes are realized as commitment schemes Soundness error ½ b’ b b’ bb b’b’

3. Two repetitions do not improve soundness b 1 Ã {0,1} b1b1b1b1 P*P* b 2 Ã {0,1} b2b2b2b2 12 Both verifiers accept if b 1 = b 2 ) soundness error ½ Can be extended to any (# of repetitions) k There exists a single protocol whose soundness error remains ½ for any (poly) k b1_b1_b1_b1_ b1_b1_b1_b1_ b2_b2_b2_b2_ b2_b2_b2_b2_