Non-malleable Codes against Constant Split-State Tampering

Abstract

Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs[1] as an elegant generalization of the classical notions of error detection, where the corruption of a codeword is viewed as a tampering function acting on it. Informally, a non-malleable code with respect to a family of tampering functions F consists of a randomized encoding function Enc and a deterministic decoding function Dec such that for any m, Dec(Enc(m)) = m. Further, for any tampering function f ∈ F and any message m, Dec(f(Enc(m))) is either m or is e-close to a distribution Df independent of m, where e is called the error. Of particular importance are non-malleable codes in the C-split-state model. In this model, the codeword is partitioned into C equal sized blocks and the tampering function family consists of functions (f1, ,fC) such that fi acts on the ith block. For C = 1 there cannot exist non-malleable codes. For C = 2, the best known explicit construction is by Aggarwal, Dodis and Lovett [2] who achieve rate = Ω(n -- 6/7) and error =2-Ω(n-1/7), where n is the block length of the code. In our main result, we construct efficient non-malleable codes in the C-split-state model for C = 10 that achieve constant rate and error =2-Ω(n). These are the first explicit codes of constant rate in the C-split-state model for any C = o(n), that do not rely on any unproven assumptions. We also improve the error in the explicit non-malleable codes constructed in the bit tampering model by Cheraghchi and Guruswami [3]. Our constructions use an elegant connection found between seedless non-malleable extractors and non-malleable codes by Cheraghchi and Guruswami [4]. We explicitly construct such seedless non-malleable extractors for 10 independent sources and deduce our results on non-malleable codes based on this connection. Our constructions of extractors use encodings and a new variant of the sum-product theorem.