Non-Malleable Codes for Small-Depth Circuits

Marshall Ball, Dana Dachman-Soled, Siyao Guo, Tal Malkin, 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. AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> tampering functions), our codes have codeword length n = k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+0(1)</sup> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(√k)</sup> . Our construction remains efficient for circuit depths as large as Θ(log(n)/loglog(n)) (indeed, our codeword length remains n ≤ k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+ε</sup> ), and extending our result beyond this would require separating P from NC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . 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.