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. AC0 tampering
functions), our codes have codeword length n=k1+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
2O(k). Our construction remains efficient for circuit depths as
large as Θ(log(n)/loglog(n)) (indeed, our codeword length remains
n≤k1+ϵ), and extending our result beyond this would require
separating P from NC1.
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.Comment: 26 pages, 4 figure