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

TL;DR

This paper improves the understanding of non-malleable codes in the split-state model by analyzing a special case of Sanders's Bogolyubov-Ruzsa theorem for Abelian groups, leading to a better code length bound.

Contribution

It demonstrates that a known construction for non-malleable codes can be extended to allow code lengths of O(k^5) by analyzing a specific case of Sanders's theorem for 5 groups.

Findings

Code length bound improved to O(k^5)

Analysis of Sanders's theorem dependence on prime p

Application to non-malleable code construction

Abstract

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