scholarly journals Idempotent Factorizations of Square-Free Integers

Information ◽  
2019 ◽  
Vol 10 (7) ◽  
pp. 232 ◽  
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 = p ¯ q ¯ such that λ ( n ) ∣ ( p ¯ − 1 ) ( q ¯ − 1 ) , where λ 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 Rivest–Shamir–Adleman (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.

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.

scholarly journals Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

Information ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 305
Author(s):  
Barry Fagin
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Positive Integer ◽  
Computer Science ◽  
Recent Work ◽  
Carmichael Numbers ◽  
Arbitrary Size ◽  
Explicit Formulation ◽  
Search Heuristics ◽  
Hard Problems

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.

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.

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

scholarly journals Preface for the Number-Theoretic Methods in Cryptology conferences

2020 ◽  
Vol 14 (1) ◽  
pp. 393-396
Author(s):  
Antoine Joux ◽  
Jacek Pomykała
Keyword(s):  
Number Theory ◽  
International Association ◽  
Easy Access ◽  
Annual Series ◽  
Hard Problems

AbstractNumber-Theoretic Methods in Cryptology (NutMiC) is a bi-annual series of conferences that waslaunched in 2017. Its goal is to spur collaborations between cryptographers and number-theorists and to encourage progress on the number-theoretic hard problems used in cryptology. The publishing model for the series is also mixing the traditions of the cryptography and number theory communities. Articles were accepted for presentation at the conference by a scientific commitee and werereviewed again at a slower pace for inclusion in the journal post-proceedings.In 2019, the conference took place at the Institut de Mathématiques de Jussieu, Sorbonne University,Paris. The event was organized in collaboration with the international association for cryptologic research (IACR) and supported by the European Union’s H2020 Program under grant agreement number ERC-669891. This support allowed us to have low registration costs and offer easy access to all interested researchers.We were glad to have the participation of five internationally recognized invited speakers who greatly contributed to the success of the conference.Nutmic 2019 Co-Chairs,Antoine Joux and Jacek Pomykała

scholarly journals A virtual lab model for an introductory computer science course

2005 ◽  
Vol 18 (2) ◽  
pp. 263-274 ◽  
Author(s):  
Ioanna Kantzavelou
Keyword(s):  
Computer Science ◽  
Virtual Laboratory ◽  
Abstract Concepts ◽  
Specific Topic ◽  
Preliminary Results ◽  
Introductory Computer Science ◽  
Virtual Lab ◽  
Proposed Model ◽  
Partial Implementation

This paper presents a model of a virtual laboratory for an introductory computer science course. The proposed model aims at solving a number of problems involved in the educational procedure of such a course. The model architecture consists of seven modules, each one corresponds to a specific topic of the course. Every module provides several different services in order to assist students to assimilate theory with practical exercises. Preliminary results of partial implementation of the proposed model, show the solution of some problems and better understanding of abstract concepts.

A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

2005 ◽  
Vol 01 (04) ◽  
pp. 563-581 ◽  
Author(s):  
A. KNOPFMACHER ◽  
M. E. MAYS
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Recursive Algorithms ◽  
Additive Number ◽  
Combinatorial Objects ◽  
Survey Techniques ◽  
Counting Functions ◽  
General Field ◽  
Positive Integers

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

scholarly journals A Ramsey-type property in additive number theory

1985 ◽  
Vol 27 ◽  
pp. 5-10
Author(s):  
S. A. Burr ◽  
P. Erdös
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Additive Number ◽  
Large Positive Integer ◽  
Ramsey Type ◽  
Type Property ◽  
Positive Integers

Let A be a sequence of positive integers. Define P(A) to be the set of all integers representable as a sum of distinct terms of A. Note that if A contains a repeated value, we are free to use it as many times as it occurs in A. We call A complete if every sufficiently large positive integer is in P(A), and entirely complete if every positive integer is in P(A). Completeness properties have received considerable, if somewhat sporadic, attention over the years. See Chapter 6 of [3] for a survey.

scholarly journals LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE

2018 ◽  
Vol 99 (1) ◽  
pp. 42-50
Author(s):  
FLORIAN LUCA ◽  
ANTONIN RIFFAUT
Keyword(s):  
Number Theory ◽  
Number Field ◽  
Linear Independence ◽  
Singular Moduli ◽  
Positive Integers

We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

Heat Exchanger Design for Ocean Thermal Difference Power Plants

1975 ◽  
Vol 97 (3) ◽  
pp. 1035-1045 ◽  
Author(s):  
J. G. McGowan ◽  
J. W. Connell
Keyword(s):  
Heat Exchanger ◽  
Power Plants ◽  
Thermal Power ◽  
Thermal Power Plants ◽  
Optimization Program ◽  
Preliminary Results ◽  
Heat Exchanger Design ◽  
Analytical Results ◽  
Thermal Difference

This paper discusses variations in heat exchanger design and configuration for a class of ocean thermal power plants. Details of the heat exchanger models are summarized and analytical results for component and cycle variations are presented. A heat exchanger optimization program is discussed in detail and preliminary results for this study are given.

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