The PRIM-LWE problem, introduced by Sehrawat, Yeo, and Desmedt (Theoretical Computer Science, 886 (2021)), is a variant of the Learning with Errors problem in which the secret matrix is required to have a primitive-root determinant. The dimension-uniform reduction constant is $c(p)=\inf_{n\ge 1}c_n(p)$, where $c_n(p)$ is the exact density of $n\times n$ matrices over $\mathbb{F}_p$ with primitive-root determinant. Sehrawat, Yeo, and Desmedt asked whether $\inf_{p\text{ prime}} c(p)=0$, observing that an affirmative answer would follow from the conjectural infinitude of primorial primes. We resolve this question unconditionally using only Dirichlet's theorem and Mertens' product formula, entirely bypassing the primorial-prime hypothesis. We further establish the sharp order \[ \min_{p\le x} c(p)\asymp \frac{1}{\log\log x} \qquad (x\to\infty), \] and show that the limiting distribution of $c(p)$ over the primes has support exactly $[0,1/2]$. We have not found this full-support statement in the literature. The law coincides with the classical shifted-prime distribution of $\varphi(p-1)/(p-1)$ via a transport lemma and is moreover continuous and purely singular. We also derive explicit lower bounds on $c(q)$ for primes of cryptographic interest, parameterized solely by the number of distinct prime factors of $q-1$. As a simple conservative explicit bound, for any prime $q>2^{30}$ the expected overhead $1/c(q)$ is at most $1.79\log q$. On the other hand, our results show that the worst-case overhead among primes $p\le x$ is of order $Θ(\log\log x)$, and in particular $1/c(q)=O(\log\log q)$ pointwise.

翻译：暂无翻译