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Non-malleable Reductions and Applications, by Divesh Aggarwal and Yevgeniy Dodis and Tomasz Kazana and Maciej Obremski

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Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs~\cite{DPW10}, provide a useful message integrity guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely ``unrelated value''. Although such codes do not exist if the family of ``tampering functions'' $\cF$ allowed to modify the original codeword is completely unrestricted, they are known to exist for many broad tampering families $\cF$. The family which received the most attention~\cite{DPW10,LL12,DKO13,ADL14,CG14a,CG14b} is the family of tampering functions in the so called {\em split-state} model: here the message $x$ is encoded into two shares $L$ and $R$, and the attacker is allowed to {\em arbitrarily} tamper with each $L$ and $R$ {\em individually}. Despite this attention, the following problem remained open: \begin{center} {\em Build efficient, information-theoretically secure non-malleable codes in the split-state model with constant encoding rate}: $|L|=|R|=O(|x|)$. \end{center} In this work, we resolve this open problem. Our technique for getting our main result is of independent interest. We \begin{itemize} \item[(a)] develop a generalization of non-malleable codes, called {\em non-malleable reductions}; \item[(b)] show simple composition theorem for non-malleable reductions; \item[(c)] build a variety of such reductions connecting various (independently interesting) tampering families $\cF$ to each other; and \item[(d)] construct our final, constant-rate, non-malleable code in the split-state model by applying the composition theorem to a series of easy to understand reductions. \end{itemize}

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