Near-optimal secret sharing and error correcting codes in AC⁰

We study the question of minimizing the computational complexity of (robust) secret sharing schemes and error correcting codes. In standard instances of these objects, both encoding and decoding involve linear algebra, and thus cannot be implemented in the class AC⁰. The feasibility of non-trivial secret sharing schemes in AC⁰ was recently shown by Bogdanov et al. (Crypto 2016) and that of (locally) decoding errors in AC⁰ by Goldwasser et al. (STOC 2007). In this paper, we show that by allowing some slight relaxation such as a small error probability, we can construct much better secret sharing schemes and error correcting codes in the class AC⁰. In some cases, our parameters are close to optimal and would be impossible to achieve without the relaxation. Our results significantly improve previous constructions in various parameters. Our constructions combine several ingredients in pseudorandomness and combinatorics in an innovative way. Specifically, we develop a general technique to simultaneously amplify security threshold and reduce alphabet size, using a two-level concatenation of protocols together with a random permutation. We demonstrate the broader usefulness of this technique by applying it in the context of a variant of secure broadcast.