Non-Malleable Codes for Small-Depth Circuits

TL;DR

This paper presents a new construction of non-malleable codes secure against small-depth circuit tampering, achieving exponential improvements in code length over previous methods and utilizing advanced derandomization techniques.

Contribution

The authors introduce an efficient non-malleable reduction from small-depth tampering to split-state tampering, incorporating derandomization techniques to improve code length and efficiency.

Findings

Achieved exponential improvement in code length for small-depth circuit tampering

Extended the construction to circuits up to logarithmic depth with efficient code length

Utilized a pseudorandom switching lemma from circuit complexity in the analysis.

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. $\mathsf{AC^0}$ 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 $\Theta(\log(n)/\log\log(n))$ (indeed, our codeword length remains $n\leq k^{1+\epsilon})$, and extending our result beyond this would require separating $\mathsf{P}$ from $\mathsf{NC^1}$. 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.