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：PRIM-LWE问题由Sehrawat、Yeo和Desmedt提出（《理论计算机科学》886卷，2021年），是带误差学习问题的一个变体，其秘密矩阵要求具有本原根行列式。维度一致归约常数为$c(p)=\inf_{n\ge 1}c_n(p)$，其中$c_n(p)$是$\mathbb{F}_p$上具有本原根行列式的$n\times n$矩阵的精确密度。Sehrawat、Yeo和Desmedt询问是否$\inf_{p\text{ 为素数}} c(p)=0$，并指出若本原素数猜想成立则可给出肯定回答。我们仅使用狄利克雷定理和梅滕斯乘积公式无条件解决了这一问题，完全绕开了本原素数假设。我们进一步建立了精确阶\[\min_{p\le x} c(p)\asymp \frac{1}{\log\log x} \qquad (x\to\infty),\]并证明$c(p)$在素数上的极限分布支撑集恰好为$[0,1/2]$。据我们所知，这一全支撑结论在文献中尚未出现。该分布通过传输引理与经典的$\varphi(p-1)/(p-1)$移位素数分布重合，且具有连续性和纯奇异性。我们还推导了密码学相关素数$q$上$c(q)$的显式下界，该下界仅由$q-1$的不同素因子个数参数化。作为一个简单的保守显式界，对任意素数$q>2^{30}$，期望开销$1/c(q)$至多为$1.79\log q$。另一方面，我们的结果表明，素数$p\le x$中的最坏情形开销阶为$Θ(\log\log x)$，且特别地，$1/c(q)=O(\log\log q)$逐点成立。