Efficient String-Commitment from Weak Bit-Commitment

We study security amplification for commitment schemes and improve the efficiency of black-box security amplification in the computational setting, where the security holds against PPT active adversaries. We show that ω(log s) black-box calls to a weak bit-commitment scheme with constant security is sufficient to construct a commitment scheme with standard negligible security, where s denotes the security parameter and ω(log s) denotes any super-logarithmic function of s. Furthermore, the resulting scheme is a string commitment scheme that can commit to O(log s)-bit strings. This improves on previous work of Damg̊ard et al. [DKS99] and Halevi and Rabin [HR08], whose transformations require ω(log s) black-box calls to commit a single bit. As a byproduct of our analysis, we also improve the efficiency of security amplification for message authentication codes, digital signatures, and pseudorandom functions studied in [DIJK09]. This is from an improvement of the “Chernoff-type Theorems” of dynamic weakly-verifiable puzzles of [DIJK09].