Paper 2018/207

Non-Malleable Codes for Small-Depth Circuits

Marshall Ball, Dana Dachman-Soled, Siyao Guo, Tal Malkin, and Li-Yang Tan

Abstract

We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e.~${\mathsf{AC}}^{\mathsf{0}}$ tampering functions), our codes have codeword length $n=k^{1+o(1)}$ for a $k$-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length $2^{O(\sqrt{k})}$. Our construction remains efficient for circuit depths as large as $\mathrm{\Theta }(\log (n)/\log \log (n))$ (indeed, our codeword length remains $n\le k^{1+ϵ})$, and extending our result beyond this would require separating $\mathsf{P}$ from ${\mathsf{NC}}^{\mathsf{1}}$. We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction.