scholarly journals On a restricted linear congruence

2016 ◽  
Vol 12 (08) ◽  
pp. 2167-2171 ◽  
Author(s):  
Khodakhast Bibak ◽  
Bruce M. Kapron ◽  
Venkatesh Srinivasan
Keyword(s):  
Number Theory ◽  
Fourier Transform ◽  
Computer Science ◽  
Short Proof ◽  
Arithmetic Functions ◽  
Number Of Solutions ◽  
Finite Fourier Transform ◽  
Linear Congruence ◽  
Special Cases ◽  
Ramanujan Sums

Let [Formula: see text], [Formula: see text], and [Formula: see text] be all positive divisors of [Formula: see text]. For [Formula: see text], define [Formula: see text]. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence [Formula: see text], with [Formula: see text], [Formula: see text], is [Formula: see text] where [Formula: see text] is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory 133 (2013) 705–718], one of the main results of the paper by Sander [J. Number Theory 129 (2009) 2260–2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory 10 (2014) 1355–1363].

scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

Some comments on inverse arithmetic functions

2004 ◽  
Vol 89 (516) ◽  
pp. 403-408
Author(s):  
P. G. Brown
Keyword(s):  
Number Theory ◽  
Prime Factor ◽  
Arithmetic Functions ◽  
Natural Numbers ◽  
Finite Mathematics ◽  
Factor Decomposition

In many of the basic courses in Number Theory, Finite Mathematics and Cryptography we come across the so-called arithmetic functions such as ϕn), σ(n), τ(n), μ(n), etc, whose domain is the set of natural numbers. These functions are well known and evaluated through the prime factor decomposition of n. It is less well known that these functions possess inverses (with respect to Dirichlet multiplication) which have interesting properties and applications.

scholarly journals Idempotent Factorizations of Square-free Integers

2019 ◽  
Author(s):  
Barry Fagin
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Infinite Number ◽  
Preliminary Results ◽  
Analytical Results ◽  
Hard Problems ◽  
Positive Integers

We explore the class of positive integers n that admit idempotent factorizations n=pq such that lambda(n) divides (p-1)(q-1), where lambda(n) is the Carmichael lambda function. Idempotent factorizations with p and q prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p and/or q. Idempotent factorizations are exactly those p and q that generate correctly functioning keys in the RSA 2-prime protocol with n as the modulus. While the resulting p and q have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution.

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