Non-malleable Codes for Decision Trees

We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by decision trees of depth$$d= n^{1/4-o(1)}$$. In particular, each bit of the tampered codeword is set arbitrarily after adaptively reading up to d arbitrary locations within the original codeword. Prior to this work, no efficient unconditional non-malleable codes were known for decision trees beyond depth$$O(\log ^2 n)$$. Our result also yields efficient, unconditional non-malleable codes that are$$\exp (-n^{\varOmega (1)})$$ -secure against constant-depth circuits of$$\exp (n^{\varOmega (1)})$$ -size. Prior work of Chattopadhyay and Li (STOC 2017) and Ball et al. (FOCS 2018) only provide protection against$$\exp (O(\log ^2n))$$ -size circuits with$$\exp (-O(\log ^2n))$$ -security. We achieve our result through simple non-malleable reductions of decision tree tampering to split-state tampering. As an intermediary, we give a simple and generic reduction of leakage-resilient split-state tampering to split-state tampering with improved parameters. Prior work of Aggarwal et al. (TCC 2015) only provides a reduction to split-state non-malleable codes with decoders that exhibit particular properties.