Cryptography: ways to keep a secret

One of the World’s Oldest Professions: The Guarding of Secrets

Linguists, Mathematicians, Engineers, Physicists, and puzzle enthusiasts collectively have contributed their diversity of skills to encode and decode the secrets of a lover, a hotel chef, a tribe, an army and a nation. More recently, a plethora of Computer Scientists have joined the ranks of the Guardians of Secrets.

Natural language alphanumeric shifting, shuffling, substitutions and chicanery were typical methods used to prevent the unlocking of coded messages for many millennia.

Secret messages have been communicated using runners, pigeons, drop boxes, radio signals, Morse code dots and dashes, telegraph, telephone, Bomb boxes, Enigma boxes, subliminal television images, and the Internet.

Enter the world of deception, encryption and decryption.

In the reams of data filled with dizzying mixes of letters, numbers, and symbols, one wonders:
Where is the Mathematics? This cloud of obfuscation overshadows the theoretical roots of the underlying mathematical foundations of how and why it all works.

An Overview of Asymmetric Cryptography

By Craig Belair (Graduate of the Athabasca University B.Sc. major in Applied Mathematics program)

This research paper is a fascinating read which outlines the role of pure mathematical theory in the structure and the security mechanisms of our modern, public key, encryption systems. The foundational concepts arise out of Number Theory, in particular the existence of very large prime numbers, and how to find them using Euler’s Totient Function. To date, the largest known prime number has 22,338,618 decimal digits. From that upper bound, the following concepts and theories play a significant role in the development of our modern encryption systems:

1. the Euclidean Algorithm for Factoring with Remainders,
2. the Theory of Modular Arithmetic,
3. the Chinese Remainder Theorem,
4. the Miller-Rabin Primality Test,
5. the Theory of Cyclic Groups and Automorphisms,
6. the Theory of Elliptic Curves,
7. the Repeated Squares Algorithm, and
8. some Probability Theory.

To find out more about the threads of Number Theoretic results which lace the mathematical backdrop behind modern encryption and decryption systems, see An Overview of Asymmetric Cryptography.

• Amazing History of Secret Codes and Cryptography
  (published on May 19, 2020 by Documentary Archive)

“Cryptography, the use of codes and ciphers to protect secrets, began thousands of years ago. Until recent decades, it has been the story of what might be called classic cryptography — that is, of methods of encryption that use pen and paper, or perhaps simple mechanical aids. In the early 20th century, the invention of complex mechanical and electromechanical machines, such as the Enigma rotor machine, provided more sophisticated and efficient means of encryption; and the subsequent introduction of electronics and computing has allowed elaborate schemes of still greater complexity, most of which are entirely unsuited to pen and paper. “

• A Classical Assumption in Cryptology:  the Riemann Hypothesis
  (published on January 4, 2021 by Quanta Magazine)

“The Riemann hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer.”

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A Classical Problem in Mathematics: The Difficulty of Factoring Very Large Numbers

It has been claimed by experts in the field that the security of the RSA encryption system is dependent on the relative difficulty of factoring large integers. The two are correlated positively – the more difficult the factoring of the numbers central to the code, the higher the security of the code.

It is widely believed that the mathematical procedure of factoring (you know, that lowly algebraic method taught in high school) is the most efficient mathematical attack on RSA to break the encryption code.

• Factoring large numbers into primes | Famous Math Problems 1
  (published on January 12, 2013 by Insights into Mathematics)

This is the first video of a series by N.J. Wildberger, a pure mathematician at the University of New South Wales, Australia. The series will discuss a wide variety of famous (and perhaps not so famous) mathematical problems, ranging from antiquity to modern times.

• Encryption and HUGE numbers
  (published on December 9, 2012 by Numberphile)

“Banks, Facebook, Twitter and Google use epic numbers - based on prime factors - to keep our Internet secrets. This is RSA public-key encryption.”

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Elliptic Curve Theory Leading to More Recent Encryption Systems

The theory of finite Cyclic Groups in Modular Arithmetic (in which the numbers in the group consecutively cycle back to the first number using the group operation) has played a role in the coding of several encryption systems. The fundamental idea is that one has to find a mathematically coherent way back to the original encrypted message. Because elliptic curves, graphically, look a lot like ellipses, points on these curves provide another rubric for cycling back to an original point. This abstraction opened up a new way to view encryption.

• What is … an Elliptic Curve? 
  (published on January 14, 2021 by Alvaro Lozano-Robledo)

“In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond.”

• Elliptic Curve Diffie-Hellman
  (published on December 10, 2014 by Robert Pierce)

This video describes the basics of the Elliptic Curve Diffie-Hellman protocol for key exchanges.

• What is BITCOIN? The maths behind blockchain and cryptocurrencies    
  (published on June 4, 2019 by Discovermaths)

This video explains, in a remarkably transparent way, the key to cryptocurrencies.