Related papers: Deterministic Primality Testing - understanding th…
We present a variant of the Agrawal-Biswas algorithm, a Monte Carlo algorithm which tests the primality of an integer $N$ by checking whether or not $(x+a)^N$ and $x^N + a$ are equivalent in a residue ring of $\mathbb{Z}/N\mathbb{Z}[x]$.…
We prove the correctness of the AKS algorithm \cite{AKS} within the bounded arithmetic theory $T^{count}_2$ or, equivalently, the first-order consequences of the theory $VTC^0$ expanded by the smash function, which we denote by $VTC^0_2$.…
A locally testable semigroup S is a semigroup with the property that for some nonnegative integer k, called the order or level of local testability, two words u and v in some set of generators for semigroup S are equal in the semigroup if…
Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two…
In recent years, several powerful techniques have been developed to design {\em randomized} polynomial-space parameterized algorithms. In this paper, we introduce an enhancement of color coding to design deterministic polynomial-space…
The purpose of this article is to delve into the properties of invariants. The properties, explained in [2], reveal new ways to develop algorithms that allow us to test the primality of a number. In this article, some of these are shown,…
This review is aimed to evaluate the importance of Symmetric Key Cryptography for Security in Distributed Systems. Businesses around the world as well as research and other such areas rely heavily on distributed systems these days. Hence,…
Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by…
Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy. Developing new algorithms or improving upon existing ones could…
Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among…
Using ideas from automata theory we design a new efficient (deterministic) identity test for the \emph{noncommutative} polynomial identity testing problem (first introduced and studied in \cite{RS05,BW05}). We also apply this idea to the…
This note considers reciprocal of primes in binary representation and shows that the conjecture that 0s exceed 1s in most cases continues to hold for primes less one million. The conjecture has also been tested for ternary representation…
In order to avoid unnecessary applications of Miller-Rabin algorithm to the number in question, we resort to trial division by a few initial prime numbers, since such a division take less time. How far we should go with such a division is…
We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices $(B_1, \dots, B_m)$ and $(C_1, \dots, C_m)$, whether there exists an…
This article presents a validation of a recently proposed strongly polynomial-time algorithm for the general linear programming problem. The proposed algorithm is an implicit reduction procedure that combines primal and dual linear…
Current asymmetric cryptography is based on the principle that while classical computers can efficiently multiply large integers, the inverse operation, factorization, is significantly more complex. For sufficiently large integers, this…
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of…
The assumed computationally difficulty of factoring large integers forms the basis of security for RSA public-key cryptography, which specifically relies on products of two large primes or semi-primes. The best-known factoring algorithms…
We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the…
The strong probable primality test is an important practical tool for discovering prime numbers. Its effectiveness derives from the following fact: for any odd composite number $n$, if a base $a$ is chosen at random, the algorithm is…