Four-State Non-malleable Codes with Explicit Constant Rate

Non-malleable codes (NMCs), introduced by Dziembowski, Pietrzak and Wichs (ITCS 2010), provide a powerful guarantee in scenarios where the classical notion of 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 decodes either to the same message or to an independent message, so long as it is tampered using a function f ∈ F . One of the well-studied tampering families for NMCs is 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 + ζ , for any constant ζ > 0 , and error 2 - Ω ( ℓ / l o g c + 1 ℓ ) , where ℓ is the length of the message and c > 0 is a constant.