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## Dr Jonathan Kenigson, FRSA: Applied Cryptography from Pure Number Theory – The Calculus of Recondite Things

Cryptography is a fascinating field of study, and a powerful tool for protecting sensitive information. At its core, cryptography is the science of making and breaking codes. It uses mathematical techniques such as Public Key Cryptography, Number Theory, and various algorithms to transform information into an unintelligible form, known as ciphertext. This ciphertext can then be transmitted over a communication channel and decrypted to its original form at the other end. This process keeps the data secure and prevents it from being stolen or misused. Cryptography is an essential part of our digital world, and it’s used in a variety of contexts, from securing financial transactions to protecting the privacy of our online communications. It’s a powerful and important technology, and one that we rely on every day. With the right understanding and implementation, cryptography can help ensure the safety and security of our data and communications.

What is Number Theory? Number Theory is a branch of mathematics that deals with the properties of whole numbers, integers, and their relationships with each other. It looks at the patterns and properties of numbers, as well as how to apply them in various fields such as computer science, cryptography, engineering, and physics. Number theory also explores the concept of prime numbers, which are numbers divisible only by themselves and one. It also investigates irrational numbers, which are numbers that cannot be expressed as a fraction, and the concept of infinity. Number theory is an intriguing and deep field of study that has been studied by mathematicians for centuries.

It has applications in many areas, such as cryptography, computer science, engineering, and physics. If you are interested in exploring the fascinating and complex world of number theory, there are many educational resources available to help you get started. Modular Arithmetic, also known as clock arithmetic, is an important tool in mathematics. It is used to solve problems in number theory, cryptography and computer science. In its most basic form, Modular Arithmetic is based on the idea of a “modulus” or divisor. This is an integer greater than zero which defines the size of the “clock” and the range of numbers that can be used. An example of this is the “mod 12” clock, which has the numbers 1-12 on its face. Modular Arithmetic works by performing calculations between numbers on this clock, and the results are always within the range of 1-12. For example, if you add 7 and 8, the result will be 15, but since 15 is not within the range of 1-12, you must use the modulus to bring it back into the range, in this case 3. The Euclidean Algorithm is another tool used in conjunction with Modular Arithmetic, and it can be used to calculate the greatest common divisor of two numbers.

By combining these two powerful mathematical techniques, it is possible to solve a wide range of problems. Algebraic Number Theory is a branch of mathematics that focuses on the study of algebraic structures related to numbers and operations. It is a very important field of mathematics, as it is used to provide the theoretical basis for many areas of mathematics, such as cryptography, number theory, algebraic geometry, and algebraic topology. Algebraic Number Theory is used to study the properties of algebraic objects, such as fields, rings, and groups, and to determine the properties of the numbers represented by these objects. It also provides a framework for understanding the properties of more complex algebraic objects, such as elliptic curves and abelian varieties. Algebraic Number Theory is a vast field, and is composed of many smaller topics, such as algebraic geometry, algebraic topology, algebraic K-theory, and Iwasawa theory.

By studying Algebraic Number Theory, researchers can gain insights into the structure of numbers and how they interact with other algebraic objects. It is an incredibly useful tool for studying the mathematics of numbers. Integral domains are an important concept in algebra. Simply put, an integral domain is a type of algebraic structure in which every element is divisible by 0 and 1. This means that multiplication and addition operations are allowed and well-defined. Integral domains are important because they form the basis for more complex algebraic structures. For example, they are used in fields like algebraic geometry and number theory. An important property of integral domains is that they are closed under addition and multiplication, meaning that the result of a multiplication or addition operation will always be an element of the domain. In other words, the operations of addition and multiplication will always produce a valid result. This property is extremely useful for solving complex equations and understanding the structure of algebraic systems. Integral domains are an invaluable tool for understanding and working with algebraic systems.

Finally, Analytic Number Theory is the study of the distribution and properties of prime numbers. It is an area of mathematics that is both beautiful and deep and has many applications in cryptography and other areas of computer science. Analytic Number Theory uses techniques from calculus and other branches of mathematics to study the properties of prime numbers and other number-theoretic objects. It involves the use of complex analysis, zeta functions, and Fourier analysis to study questions like the distribution of prime numbers or the existence of certain patterns in prime numbers. Analytic Number Theory is a fascinating and challenging field of mathematics, and it is the foundation of many modern applications, such as cryptography and data compression. So, if you’re looking for a challenging and rewarding field of mathematics, Analytic Number Theory is worth considering!

Anderson, D. D., David F. Anderson, and Muhammad Zafrullah. “Factorization in integral domains.” Journal of pure and applied algebra 69.1 (1990): 1-19.

Anderson, Daniel. Factorization in integral domains. Vol. 189. CRC Press, 1997.

Baldoni, Maria Welleda, Ciro Ciliberto, and Giulia Maria Piacentini Cattaneo. Elementary number theory, cryptography and codes. Berlin: Springer, 2009.

Buchmann, Johannes. Introduction to cryptography. Vol. 335. New York: Springer, 2004.

Coutinho, Severino Collier. The mathematics of ciphers: number theory and RSA cryptography. AK Peters/CRC Press, 1999.

Ferguson, Niels, and Bruce Schneier. Practical cryptography. Vol. 141. New York: Wiley, 2003.

Katz, Jonathan, and Yehuda Lindell. Introduction to modern cryptography. CRC press, 2020.

Koblitz, Neal. A course in number theory and cryptography. Vol. 114. Springer Science & Business Media, 1994.

Kovács, Béla, and Attila Petho. “Number systems in integral domains, especially in orders of  algebraic number fields.” Acta Sci. Math.(Szeged) 55.3-4 (1991): 287-299.

Lefton, Phyllis. “Number theory and public-key cryptography.” The Mathematics Teacher 84.1 (1991): 54-63.

Menezes, Alfred J., Paul C. Van Oorschot, and Scott A. Vanstone. Handbook of applied  cryptography. CRC press, 2018.

Miller, Steven J., and Ramin Takloo-Bighash. “An invitation to modern number theory.” An  Invitation to Modern Number Theory. Princeton University Press, 2021.

Mollin, Richard A. An introduction to cryptography. Chapman and Hall/CRC, 2006.

Rivest, Ronald L. “Cryptography.” Algorithms and complexity. Elsevier, 1990. 717-755.

Shparlinski, Igor. Finite Fields: Theory and Computation: The meeting point of number theory,   computer science, coding theory and cryptography. Vol. 477. Springer Science & Business Media, 2013.

Yan, Song Y. Computational number theory and modern cryptography. John Wiley & Sons, 2013.

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