scholarly journals New Semi-Prime Factorization and Application in Large RSA Key Attacks

2021 ◽  
Vol 1 (4) ◽  
pp. 660-674
Author(s):  
Anthony Overmars ◽  
Sitalakshmi Venkatraman
Keyword(s):  
Solution Space ◽  
Public Key Cryptography ◽  
Factorization Method ◽  
Future Research ◽  
Opposite Parity ◽  
Prime Factorization ◽  
Factorization Problem ◽  
Pythagorean Triples ◽  
Greatest Common Divisors ◽  
Pythagorean Quadruples

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

scholarly journals Linear Time Solution to Prime Factorization by Tissue P Systems with Cell Division

2011 ◽  
Vol 2 (3) ◽  
pp. 49-60 ◽  
Author(s):  
Xingyi Zhang ◽  
Yunyun Niu ◽  
Linqiang Pan ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Cell Division ◽  
Quantum Computer ◽  
Linear Time ◽  
Public Key Cryptography ◽  
Polynomial Time Algorithm ◽  
P Systems ◽  
Public Key ◽  
Prime Factorization ◽  
Factorization Problem ◽  
Time Solution

Prime factorization is useful and crucial for public-key cryptography, and its application in public-key cryptography is possible only because prime factorization has been presumed to be difficult. A polynomial-time algorithm for prime factorization on a quantum computer was given by P. W. Shor in 1997. In this work, it is considered as a function problem, and in the framework of tissue P systems with cell division, a linear-time solution to prime factorization problem is given on biochemical computational devices – tissue P systems with cell division, instead of computational devices based on the laws of quantum physical.

scholarly journals Linear Time Solution to Prime Factorization by Tissue P Systems with Cell Division

2014 ◽  
pp. 207-220
Author(s):  
Xingyi Zhang ◽  
Yunyun Niu ◽  
Linqiang Pan ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Cell Division ◽  
Quantum Computer ◽  
Linear Time ◽  
Public Key Cryptography ◽  
Polynomial Time Algorithm ◽  
P Systems ◽  
Public Key ◽  
Prime Factorization ◽  
Factorization Problem ◽  
Time Solution

Prime factorization is useful and crucial for public-key cryptography, and its application in public-key cryptography is possible only because prime factorization has been presumed to be difficult. A polynomial-time algorithm for prime factorization on a quantum computer was given by P. W. Shor in 1997. In this work, it is considered as a function problem, and in the framework of tissue P systems with cell division, a linear-time solution to prime factorization problem is given on biochemical computational devices – tissue P systems with cell division, instead of computational devices based on the laws of quantum physical.

Analysis of Various Security Issues and Challenges in Cloud Computing Environment

2016 ◽  
pp. 221-247 ◽  
Author(s):  
Zhaolong Gou ◽  
Shingo Yamaguchi ◽  
B. B. Gupta
Keyword(s):  
Cloud Computing ◽  
Comparative Analysis ◽  
Solution Space ◽  
Future Research ◽  
Security Threats ◽  
Computing Environment ◽  
Cloud Computing Environment ◽  
Security Issues ◽  
On Demand ◽  
Recent Developments

Cloud computing is a system, where the resources of a data center are shared using virtualization technology, such that it provides elastic, on demand and instant services to its customers and charges them based on the resources they use. In this chapter, we will discuss recent developments in cloud computing, various security issues and challenges associated with Cloud computing environment, various existing solutions provided for dealing with these security threats and will provide a comparative analysis these approaches. This will provide better understanding of the various security problems associated with the cloud, current solution space, and future research scope to deal with such attacks in better way.

scholarly journals Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods

2020 ◽  
Vol 32 (12) ◽  
pp. 2332-2388 ◽  
Author(s):  
Spencer J. Kent ◽  
E. Paxon Frady ◽  
Friedrich T. Sommer ◽  
Bruno A. Olshausen
Keyword(s):  
Hadamard Product ◽  
Solution Space ◽  
Local Information ◽  
Alternative Methods ◽  
High Dimensional ◽  
Factorization Problem ◽  
Gradient Based ◽  
New Type ◽  
Alternative Approaches ◽  
Theoretical Foundations

We develop theoretical foundations of resonator networks, a new type of recurrent neural network introduced in Frady, Kent, Olshausen, and Sommer ( 2020 ), a companion article in this issue, to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a resonator network can efficiently decompose the composite into these factors. We compare the performance of resonator networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that resonator networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of resonator networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing the guarantee of global convergence, resonator networks are dramatically more effective at finding factorizations than all alternative approaches considered.

scholarly journals Architectural Design Exploration Using Generative Design: Framework Development and Case Study of a Residential Block

Buildings ◽  
2020 ◽  
Vol 10 (11) ◽  
pp. 201
Author(s):  
Jani Mukkavaara ◽  
Marcus Sandberg
Keyword(s):  
Architectural Design ◽  
Solution Space ◽  
Energy Performance ◽  
Design System ◽  
Future Research ◽  
Generative Design ◽  
Design Exploration ◽  
Novel Approach ◽  
Generative Algorithms

The use of generative design has been suggested to be a novel approach that allows designers to take advantage of computers’ computational capabilities in the exploration of design alternatives. However, the field is still sparsely explored. Therefore, this study aimed to investigate the potential use of generative design in an architectural design context. A framework was iteratively developed alongside a prototype, which was eventually demonstrated in a case study to evaluate its applicability. The development of a residential block in the northern parts of Sweden served as the case. The findings of this study further highlight the potential of generative design and its promise in an architectural context. Compared to previous studies, the presented framework is open to other generative algorithms than mainly genetic algorithms and other evaluation models than, for instance, energy performance models. The paper also presents a general technical view on the functionality of the generative design system, as well as elaborating on how to explore the solution space in a top-down fashion. This paper moves the field of generative design further by presenting a generic framework for architectural design exploration. Future research needs to focus on detailing how generative design should be applied and when in the design process.

Gamification in Management Decisions: Judging Global Production Networks in a Cyber-Physical Way

2013 ◽  
Vol 769 ◽  
pp. 327-334 ◽  
Author(s):  
Till Potente ◽  
Rawina Varandani ◽  
Jan Philipp Prote
Keyword(s):  
Goal Orientation ◽  
Physical Environment ◽  
Practical Knowledge ◽  
Solution Space ◽  
Decision Makers ◽  
Future Research ◽  
Final Decision ◽  
Production Networks ◽  
Global Production Networks ◽  
Upper Management

The judgment of global production networks is facing three major challenges. The complexity, measurable through the huge solution space, as well as the time-consuming preparation of a decision in line with the limited amount of time of the final decision makers – CEOs and upper management. The experience from several industry projects showed the demand to transfer and apply the concept of gamification in a cyber-physical environment to decision-making in Global Footprint Design. Smart IT tools, which support the intuition and practical knowledge of the decision maker but do not finally make the decision for them, are needed. Based on general criteria for successful gamified IT tools three specific requirements for judging global production networks are derived: value creation, ludic goal orientation and autonomous discovery. The major challenges and these specified principles of gamification are addressed within the scope of an IT tool which simplifies and improves complex global footprint decisions by employing elements of gamification. It is analysed to which degree the criteria are already fulfilled, which benefits this prototype of a gamified IT tool can offer and what future research has to be conducted in order to fully let CEOs game their global footprint strategy.

A Survey of Fast Flux Botnet Detection With Fast Flux Cloud Computing

2020 ◽  
Vol 10 (3) ◽  
pp. 17-53
Author(s):  
Ahmad Al-Nawasrah ◽  
Ammar Ali Almomani ◽  
Samer Atawneh ◽  
Mohammad Alauthman
Keyword(s):  
Cloud Computing ◽  
Solution Space ◽  
Memory Storage ◽  
Future Research ◽  
Detection Accuracy ◽  
Security Threats ◽  
Domain Names ◽  
Future Research Directions ◽  
Fast Flux ◽  
And Control

A botnet refers to a set of compromised machines controlled distantly by an attacker. Botnets are considered the basis of numerous security threats around the world. Command and control (C&C) servers are the backbone of botnet communications, in which bots send a report to the botmaster, and the latter sends attack orders to those bots. Botnets are also categorized according to their C&C protocols, such as internet relay chat (IRC) and peer-to-peer (P2P) botnets. A domain name system (DNS) method known as fast-flux is used by bot herders to cover malicious botnet activities and increase the lifetime of malicious servers by quickly changing the IP addresses of the domain names over time. Several methods have been suggested to detect fast-flux domains. However, these methods achieve low detection accuracy, especially for zero-day domains. They also entail a significantly long detection time and consume high memory storage. In this survey, we present an overview of the various techniques used to detect fast-flux domains according to solution scopes, namely, host-based, router-based, DNS-based, and cloud computing techniques. This survey provides an understanding of the problem, its current solution space, and the future research directions expected.

Some Interesting Mathematics about Prime Factorization

1975 ◽  
Vol 68 (1) ◽  
pp. 73-74
Author(s):  
Joe Tom Rodgers
Keyword(s):  
Prime Factorization ◽  
Factoring Polynomials ◽  
Greatest Common Divisors ◽  
School Careers

Throughout their school careers, students are taught how to find the prime factorization of numbers. Most students readily learn the technique and find the process interesting. Most teachers then teach the ways in which prime factorizations can aid in factoring polynomials and finding greatest common divisors as well as least common multiples.

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