Related papers: On Shanks' Algorithm for Modular Square Roots

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a<q$ be a pair of relatively prime integers. It is proved that there is a prime primitive root $u\ll (\log p)(\log \log p)^5$ such that $u\equiv a\bmod q$ in the prime…

We provide a geometric interpretation of Brillhart's celebrated algorithm for expressing a prime $p\equiv 1\pmod 4$ as the sum of two squares.

We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial…

For any polynomial $P(x)\in\mathbb{Z}[x],$ we study arithmetic dynamical systems generated by $\displaystyle{F_P(n)=\prod_{k\le n}}P(n)(\text{mod}\ p),$ $n\ge 1.$ We apply this to improve the lower bound on the number of distinct quadratic…

Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod…

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number…

We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$…

We consider the problem of computing the monic gcd of two polynomials over a number field L = Q(alpha_1,...,alpha_n). Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over Q can be modified to work…

Algebraic matrix multiplication algorithms are designed by bounding the rank of matrix multiplication tensors, and then using a recursive method. However, designing algorithms in this way quickly leads to large constant factors: if one…

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $\gamma$$\infty$,2 $\approx$ 2.386.…

We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was…

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an…

We compute the action of the Steenrod algebra on generators of algebras of invariants of special linear group ${SL_n=SL(n,\mathbb Z/p)}$ in the polynomial algebra with $ p$ an odd prime number.

For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in\mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this…

This monograph considers a few topics in the theory of primitive roots g(p) modulo a prime p>=2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g^*(p) modulo p, a large prime, are determined. One of…

We give efficient algorithms for finding power-sum decomposition of an input polynomial $P(x)= \sum_{i\leq m} p_i(x)^d$ with component $p_i$s. The case of linear $p_i$s is equivalent to the well-studied tensor decomposition problem while…

We present a novel, highly efficient algorithm to parallelize O(N^2) direct summation method for N-body problems with individual timesteps on distributed-memory parallel machines such as Beowulf clusters. Previously known algorithms, in…

We describe several algorithms for computing $e$-th roots of elements in a number field $K$, where $e$ is an odd prime-power integer. In particular we generalize Couveignes' and Thom\'e's algorithms originally designed to compute…