Optimal Marginal Mixing for Transvection-Based Trapdoored Matrices over Large Fields
Mark Braverman, Aayush Jain, Stephen Newman
\emph{Trapdoored matrices} are pseudorandom matrices admitting a fast matrix-vector multiplication algorithm given the secret key. We study a candidate family of such matrices: matrices generated by short random \emph{transvection walks} on \(\mathrm{SL}_n(\mathbb{F}_q)\), whose steps are given by adding a random multiple of a random row of the current state to another. We show that the worst-case time for any nonempty set \(\mathcal{S}\subseteq [n]\times[n]\) of matrix entries to come within TV distance \(\varepsilon\leq \frac{1}{4}\) of its stationary marginal under this walk is \(Θ(|\mathcal{S}|+n\log(n/\varepsilon))\), uniformly over all \(\mathcal{S}\) and \(q\geq Q_\mathrm{all}(n)\). This is asymptotically optimal, and resolves open questions about the mixing times of transvection walks. We derive concrete upper and lower bounds for \(n\geq 256\). Surprisingly, our bounds also hold conditioned on the row-pair structure of the walk, except on a fraction at most \(\varepsilon\) of such structures. This implies that these walks may be pseudorandom even after conditioning on the row-pair structure, and therefore that trapdoored matrices built with these walks may be universal. Assuming the conjectured structure-conditioned pseudorandomness of short walks, we exhibit highly efficient, highly parallelizable, and universal trapdoored matrices over \(\mathrm{SL}_n(\mathbb{F}_q)\), \(\mathrm{GL}_n(\mathbb{F}_q)\), and \(\mathbb{F}_q^{n\times n}\). The families over \(\mathrm{SL}_n(\mathbb{F}_q)\) and \(\mathrm{GL}_n(\mathbb{F}_q)\) admit equally efficient trapdoors for their inverses, making them the first to do so for finite-field trapdoored matrices. We discuss attacks, analyzing one in detail, and lay out directions for further investigation.
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