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\beginproof (Sketch) The proof combines three ingredients: \beginenumerate \item The LFSR's linear span ensures no low-degree polynomial approximation (Massey's theorem). \item Borell's inequality bounds the probability that $\Phi$ deviates from its mean. \item A union bound over all $2^L$ possible initial states shows the total distinguishing advantage decays exponentially in $L$ and $m$. \endenumerate The full derivation follows the Fourier–Gaussian approach of \citeborrellus2024. \endproof
\bibitemborell1975brunn C. Borell, ``The Brunn–Minkowski inequality in Gauss space,'' \textitInvent. Math., vol. 30, no. 2, pp. 207–216, 1975.
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The search for pseudorandom sequences with provable resistance to correlation attacks dates back to the work of Siegenthaler \citesiegenthaler1984correlation and Meier & Staffelbach \citemeier1989fast. Recent advances in Gaussian analysis—particularly Borell's inequality \citeborell1975brunn—have remained largely disconnected from cryptographic practice.