Four-State Non-malleable Codes with Explicit Constant Rate

Non-malleable codes (NMCs), introduced by Dziembowski, Pietrzak and Wichs (ITCS 2010), generalize the classical notion of error correcting codes by providing a powerful guarantee even in scenarios where error correcting codes cannot provide any guarantee: a decoded message is either the same or completely independent of the underlying message, regardless of the number of errors introduced into the codeword. Informally, NMCs are defined with respect to a family of tampering functions F and guarantee that any tampered codeword either decodes to the same message or to an independent message, so long as it is tampered using a function f is an element of F. Nearly all known constructions of NMCs are for the t-split-state family, where the adversary tampers each of the t "states" of a codeword, arbitrarily but independently. Cheraghchi and Guruswami (TCC 2014) obtain a Rate-1 non-malleable code for the case where t = O(n) with n being the codeword length and, in (ITCS 2014), show an upper bound of 1 - 1/t on the best achievable rate for any t-split state NMC. For t = 10, Chattopadhyay and Zuckerman (FOCS 2014) achieve a constant rate construction where the constant is unknown. In summary, there is no known construction of an NMC with an explicit constant rate for any t = o(n), let alone one that comes close to matching Cheraghchi and Guruswami's lowerbound! In this work, we construct an efficient non-malleable code in the t-split-state model, for t = 4, that achieves a constant rate of 1/3+zeta, for any constant zeta > 0, and error 2(-Omega(l/logc+1l)), where l is the length of the message and c > 0 is a constant.