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Non-malleable codes (NMCs) protect sensitive data against degrees ofcorruption that prohibit error detection, ensuring instead that a corruptedcodeword decodes correctly or to something that bears little relation to theoriginal message. The split-state model, in which codewords consist of twoblocks, considers adversaries who tamper with either block arbitrarily butindependently of the other. The simplest construction in this model, due toAggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bitmessages to $O(k^7)$-bit codewords. It is conjectured, however, that theconstruction allows linear-length codewords. Towards resolving this conjecture,we show that the construction allows for code-length $O(k^5)$. This is achievedby analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for generalAbelian groups. Closely following the excellent exposition of this result forthe group $\mathbb{F}_2^n$ by Lovett, we expose its dependence on $p$ for thegroup $\mathbb{F}_p^n$, where $p$ is a prime.

Revisiting the Sanders-Freiman-Ruzsa Theorem in $\mathbb{F}_p^n$ and its Application to Non-malleable Codes