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，其要求秘密矩阵具有素根行列式。维度一致归约常数为$c(p)=\inf_{n\ge 1}c_n(p)$，其中$c_n(p)$是$\mathbb{F}_p$上具有素根行列式的$n\times n$矩阵的精确密度。Sehrawat、Yeo和Desmedt提出是否$\inf_{p\text{ prime}} c(p)=0$，并指出若素数猜想成立则可得肯定结论。我们仅利用狄利克雷定理和梅滕斯乘积公式无条件解决了该问题，完全绕开了素素数假设。我们进一步建立了锐阶估计 \[ \min_{p\le x} c(p)\asymp \frac{1}{\log\log x} \qquad (x\to\infty)， \] 并证明$c(p)$在素数上的极限分布支撑集恰为$[0,1/2]$。该全支撑结论在现有文献中尚未见报道。该分布律通过一个传递引理与经典的移位素数分布$\varphi(p-1)/(p-1)$一致，且具有连续性和纯奇异性质。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。作为简明的保守显式界，对于任意素数$q>2^{30}$，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$p\le x$中的最坏情况开销为$Θ(\log\log x)$量级，特别地，$1/c(q)=O(\log\log q)$逐点成立。