Related papers: On Shanks' Algorithm for Modular Square Roots
We investigate logarithmic and square-root types of bounds for the general difference of two primes, $P_{k+q}-P_k$, $k, q\in\mathbb{N}$.
Let $g>1$ be an integer and $f(X)\in{\mathbb Z}[X]$ a polynomial of positive degree with no multiple roots, and put $u(n)=f(g^n)$. In this note, we study the sequence of quadratic fields ${\mathbb Q}(\sqrt{u(n)}\,)$ as $n$ varies over the…
It is known that greedy methods perform well for maximizing monotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show - arguably, surprisingly - that invoking the…
We give an O(sqrt n log n)-query quantum algorithm for evaluating size-n AND-OR formulas. Its running time is poly-logarithmically greater after efficient preprocessing. Unlike previous approaches, the algorithm is based on a quantum walk…
In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski.…
These are pedagogical notes on Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the…
Let $p$ be an odd prime number and let $X_0^+(p)$ be the quotient of the classical modular curve $X_0(p)$ by the action of the Atkin-Lehner operator $w_p$. In this paper we show how to compute explicit equations for the canonical model of…
Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work…
We study the shared processor scheduling problem with a single shared processor where a unit time saving (weight) obtained by processing a job on the shared processor depends on the job. A polynomial-time optimization algorithm has been…
In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop…
We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree $n$ over a finite field $\F_q$, the average-case complexity of our algorithm is an expected $O(n^{1 + o(1)} \log^{2 +…
An algorithm to extract the square root in radicals from a multivector (MV) in real Clifford algebras Cl(p,q) for n=p+q <=3 is presented. We show that in the algebras Cl(3,0), Cl(1,2) and Cl(0,3) there are up to four isolated roots in a…
The aim of this paper is to present an algorithm the complexity of which is polynomial to compute the semi-simplified modulo $p$ of a semi-stable $\Q_p$-representation of the absolute Galois group of a $p$-adic field (\emph{i.e.} a finite…
We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map…
We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo $p$ of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if $p-1$ has a $(\ln…
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times…
We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…
The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example \cite{863046,863031} and the references therein). In this paper we develop a simple differential compensation (much…
Using Watson's terminating $_8\phi_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019),…
We present a new GCD algorithm of two integers or polynomials. The algorithm is iterative and its time complexity is still $O(n \\log^2 n ~ log \\log n)$ for $n$-bit inputs.