Primitive-Root Ratio over Prime Fields: A Shifted-Prime Distribution of Hausdorff Dimension Zero and Implications for PRIM-LWE

TL;DR

This paper investigates the distribution of primitive-root ratios over prime fields, revealing a singular, Hausdorff dimension zero measure with implications for lattice-based cryptography and LWE problem reductions.

Contribution

It establishes the limiting distribution's properties, including singularity and Hausdorff dimension zero, and connects these findings to cryptographic overhead estimates in PRIM-LWE.

Findings

The infimum of c(p) over primes p is zero, with order roughly 1/ log log p.

The limiting distribution is purely singular with Hausdorff dimension zero.

Explicit bounds for cryptographic overhead are derived for current NIST primes.

Abstract

For a prime $p$, let $c(p)$ denote the limiting fraction of $n\times n$ matrices over $\mathbb{F}_p$ whose determinant is a primitive root modulo $p$. The quantity $c(p)$ is a natural multiplicative deformation of the totient ratio $\varphi(p-1)/(p-1)$ and inherits its distributional behaviour over the primes. Existence and continuity of the limiting law follow from the shifted-prime Erd\H{o}s--Wintner--Hildebrand framework. We prove the following new results: unconditionally, $\inf_p c(p)=0$ and the sharp order is $\min_{p\le x}c(p)\asymp 1/\log\log x$; the reciprocal satisfies $\limsup_{p\to\infty, \, p\text{ prime}} 1/(c(p)\log\log p)=e^{\gamma}$, and no smaller constant suffices. We give a complete proof, combining an adaptation of Erd\H{o}s's argument with the Jessen--Wintner pure-type dichotomy, that the limiting distribution is purely singular, and strengthen this to $\dim_H(\mu_G)=0$, i.e. the limiting measure is carried by a Borel set of Hausdorff dimension zero. The distribution has full topological support $[0,\tfrac12]$ and admits a Bernoulli-product representation indexed by the odd primes. The moments are given by convergent Euler products, and the Mellin transform $\mathbb{E}[X^s]$ extends to an entire function of $s$, non-vanishing on $\operatorname{Re}(s)>0$. Near the right endpoint, $1-G(\tfrac12-\varepsilon)\sim\kappa/\log(1/\varepsilon)$ with an explicit constant $\kappa$. The quantity $1/c(p)$ equals the dimension-uniform expected rejection-sampling overhead in the reduction from learning with errors (LWE) to PRIM-LWE in lattice-based cryptography. The explicit bounds yield concrete overhead estimates for all primes appearing in current NIST post-quantum standards and representative NTT-friendly moduli.