TY - GEN
T1 - A tight parallel repetition theorem for partially simulatable interactive arguments via smooth kl-divergence
AU - Berman, Itay
AU - Haitner, Iftach
AU - Tsfadia, Eliad
N1 - Publisher Copyright: © International Association for Cryptologic Research 2020.
PY - 2020
Y1 - 2020
N2 - Hardness amplification is a central problem in the study of interactive protocols. While “natural” parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS ’97]) and public-coin protocols (Håstad, Pass, Wikström, and Pietrzak [TCC ’10], Chung and Liu [TCC ’10] and Chung and Pass [TCC ’15]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikström [TCC ’07]). The only known round-preserving approach that applies to all interactive arguments is Haitner’s random-terminating transformation [SICOMP ’13], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original m-round protocol has soundness error 1-ε, then the n-parallel repetition of its random-terminating variant has soundness error (1-ε)ε n/m4 (omitting constant factors). Håstad et al. have generalized this result to partially simulatable interactive arguments, showing that the n-fold repetition of an m-round δ-simulatable argument of soundness error 1-ε has soundness error (1-ε)ε δ2 n/m2 . When applied to random-terminating arguments, the Håstad et al. bound matches that of Haitner. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the n parallel repetition is (1-ε)n/m, only an m factor from the optimal rate of (1-ε)n achievable in public-coin and three-message arguments. The result generalizes to δ-simulatable arguments, for which we prove a bound of (1-ε)δ n/m. This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol.
AB - Hardness amplification is a central problem in the study of interactive protocols. While “natural” parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS ’97]) and public-coin protocols (Håstad, Pass, Wikström, and Pietrzak [TCC ’10], Chung and Liu [TCC ’10] and Chung and Pass [TCC ’15]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikström [TCC ’07]). The only known round-preserving approach that applies to all interactive arguments is Haitner’s random-terminating transformation [SICOMP ’13], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original m-round protocol has soundness error 1-ε, then the n-parallel repetition of its random-terminating variant has soundness error (1-ε)ε n/m4 (omitting constant factors). Håstad et al. have generalized this result to partially simulatable interactive arguments, showing that the n-fold repetition of an m-round δ-simulatable argument of soundness error 1-ε has soundness error (1-ε)ε δ2 n/m2 . When applied to random-terminating arguments, the Håstad et al. bound matches that of Haitner. In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the n parallel repetition is (1-ε)n/m, only an m factor from the optimal rate of (1-ε)n achievable in public-coin and three-message arguments. The result generalizes to δ-simulatable arguments, for which we prove a bound of (1-ε)δ n/m. This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol.
KW - Interactive argument
KW - Parallel repetition
KW - Partially simulatable
KW - Smooth KL-divergence
UR - http://www.scopus.com/inward/record.url?scp=85089721142&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/978-3-030-56877-1_19
DO - https://doi.org/10.1007/978-3-030-56877-1_19
M3 - منشور من مؤتمر
SN - 9783030568764
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 544
EP - 573
BT - Advances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, Proceedings
A2 - Micciancio, Daniele
A2 - Ristenpart, Thomas
T2 - 40th Annual International Cryptology Conference, CRYPTO 2020
Y2 - 17 August 2020 through 21 August 2020
ER -