The randomness complexity of parallel repetition

Consider a m-round interactive protocol with soundness error 1/2. How much extra randomness is required to decrease the soundness error to δ through parallel repetition? Previous work, initiated by Bell are, Goldreich and Gold wasser, shows that for public-coin interactive protocols with statistical soundness, m·O(log (1/δ)) bits of extra randomness suffices. In this work, we initiate a more general study of the above question. • We establish the first derandomized parallel repetition theorem for public-coin interactive protocols with computational soundness (a.k.a. arguments). The parameters of our result essentially matches the earlier works in the information-theoretic setting. • We show that obtaining even a sub-linear dependency on the number of rounds m (i.e., o(m)·log(1/δ)) is impossible in the information-theoretic, and requires the existence of one-way functions in the computational setting. • We show that non-trivial derandomized parallel repetition for private-coin protocols is impossible in the information-theoretic setting and requires the existence of one-way functions in the computational setting. \end{itemize} These results are tight in the sense that parallel repetition theorems in the computational setting can trivially be derandomized using pseudorandom generators, which are implied by the existence of one-way functions.