Non-Malleable Codes for Small-Depth Circuits

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 tampering functions), our codes have codeword length n = k 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 √ . Our construction remains efficient for circuit depths as large as Θ(log(n)/ log log(n)) (indeed, our codeword length remains n ≤ k), and extending our result beyond this would require separating P from NC. 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. ∗ [email protected], Columbia University. Supported in part by the Defense Advanced Research Project Agency (DARPA) and Army Research Office (ARO) under Contract W911NF-15-C-0236, NSF grants CNS1445424 and CCF-1423306, ISF grant no. 1790/13, the Leona M. & Harry B. Helmsley Charitable Trust, and the Check Point Institute for Information Security. Part of this research was done while visiting the FACT Center at IDC Herzliya. Any opinions, findings and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the Defense Advanced Research Projects Agency, Army Research Office, the National Science Foundation, or the U.S. Government. [email protected], University of Maryland. Supported in part by an NSF CAREER Award #CNS-1453045, by a research partnership award from Cisco and by financial assistance award 70NANB15H328 from the U.S. Department of Commerce, National Institute of Standards and Technology. ‡ [email protected], Northeastern University. Supported by NSF grants CNS1314722 and CNS-1413964 § [email protected], Columbia University. Supported in part by the Defense Advanced Research Project Agency (DARPA) and Army Research Office (ARO) under Contract W911NF-15-C-0236, NSF grants CNS1445424 and CCF-1423306, and the Leona M. & Harry B. Helmsley Charitable Trust. ¶ [email protected], Toyota Technological Institute. Supported by NSF grant CCF 1563122.