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 $$\mathcal {F}$$ 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 $$f \in \mathcal {F}$$ 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 $$t = \mathcal {O}(n)$$ t = O ( n ) with n being the codeword length and, in (ITCS 2014), show an upper bound of $$1-1/t$$ 1 - 1 / t on the best achievable rate for any t -split state NMC. For $$t=10$$ 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 $$t= o(n)$$ 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 $$t=4$$ t = 4 , that achieves a constant rate of $$\frac{1}{3+\zeta }$$ 1 3 + ζ , for any constant $$\zeta > 0$$ ζ > 0 , and error $$2^{-\varOmega (\ell / log^{c+1} \ell )}$$ 2 - Ω ( ℓ / l o g c + 1 ℓ ) , where $$\ell $$ ℓ is the length of the message and $$c > 0$$ c > 0 is a constant.

中文翻译：

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具有显式恒定速率的四态不可延展代码

由 Dziembowski、Pietrzak 和 Wichs (ITCS 2010) 引入的不可延展码 (NMC) 在经典纠错码概念无法提供任何保证的情况下提供了强有力的保证：解码后的消息要么相同要么完全独立无论引入到代码字中的错误数量如何。非正式地，NMC 是根据一系列篡改函数 $$\mathcal {F}$$ F 定义的，并保证任何被篡改的代码字都解码为相同的消息或独立的消息，只要它使用函数进行篡改$$f \in \mathcal {F}$$ f ∈ F 。对 NMC 进行充分研究的篡改系列之一是 t 分裂状态系列，其中对手任意但独立地篡改代码字的每个 t“状态”。Cheraghchi 和 Guruswami (TCC 2014) 在 $$t = \mathcal {O}(n)$$ t = O ( n ) 其中 n 是码字长度并且在 ( ITCS 2014)，显示任何 t 分裂状态 NMC 的最佳可实现速率的上限为 $1-1/t$$ 1 - 1 / t。对于 $$t=10$$ t = 10 ，Chattopadhyay 和 Zuckerman (FOCS 2014) 实现了常数未知的恒定速率构造。总之，对于任何 $$t= o(n)$$ t = o ( n ) 都没有已知的 NMC 构造，更不用说接近匹配 Cheraghchi 和 Guruswami 的下界的了！在这项工作中，我们在 t -split-state 模型中构建了一个高效的不可延展代码，对于 $$t=4$$ t = 4 ，它实现了 $$\frac{1}{3+\ zeta }$$ 1 3 + ζ ，对于任何常数 $$\zeta > 0$$ ζ > 0 ，