# Four-State Non-malleable Codes with Explicit Constant Rate

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## Abstract

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 \(\mathcal {F}\) 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 \(f \in \mathcal {F}\). 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 \(t = \mathcal {O}(n)\) 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 \(\frac{1}{3+\zeta }\), for any constant \(\zeta > 0\), and error \(2^{-\varOmega (\ell / log^{c+1} \ell )}\), where \(\ell \) is the length of the message and \(c > 0\) is a constant.

## Notes

### Acknowledgements

We thank Yevgeniy Dodis for insightful comments related to the generalization in Sect. 5. We also thank the anonymous referees for several helpful comments. Research of the first author was supported, in part, by Department of Science and Technology Inspire Faculty Award.

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