On Grosswald's conjecture on primitive roots

Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409<p< 2.5\times 10^{15}$ and for all $p>3.67\times 10^{71}$.Comment: 7 page

Computation of the least primitive root

Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems as a consequence. In particular, we show that $g(p)<p^{5/8}$ for all primes $p>3$ and that $h(p)<p^{2/3}$ for all primes $p$

The Counting function for Elkies primes

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ where $q$ is a prime power. The Schoof--Elkies--Atkin (SEA) algorithm is a standard method for counting the number of $\mathbb{F}_q$-points on $E$. The asymptotic complexity of the SEA algorithm depends on the distribution of the so-called Elkies primes. Assuming GRH, we prove that the least Elkies prime is bounded by $(2\log 4q+4)^2$ when $q\geq 10^9$. This is the first such explicit bound in the literature. Previously, Satoh and Galbraith established an upper bound of $O((\log q)^{2+\varepsilon})$. Let $N_E(X)$ denote the number of Elkies primes less than $X$. Assuming GRH, we also show $$ N_E(X)=\frac{\pi(X)}{2}+O\left(\frac{\sqrt{X}(\log qX)^2}{\log X}\right)\,. $

Deterministic root finding over finite fields using Graeffe transforms

We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field F q . Our algorithms were designed to be particularly efficient in the case when the cardinality q − 1 of the multiplicative group of F q is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders