Introduction to the Special Issue: Resources for Undergraduate Cryptology

Abstract

This editorial introduces the special issue, Resources for Undergraduate Cryptology. We begin by describing possible roles for cryptology in the undergraduate mathematics curriculum together with a brief overview of the subject. We conclude with a brief preview of each paper included in this issue.

KEYWORDS:

• Cryptology
• cryptography
• cryptanalysis
• discrete mathematics
• number theory
• computer science

1. CRYPTOLOGY IN THE UNDERGRADUATE MATHEMATICS CURRICULUM

Cryptology consists of two separate, but intertwined, components: cryptography, which is the construction of codes and ciphers, and cryptanalysis, which is the breaking of codes and ciphers. The inherent competition between these two components makes cryptology exciting and thought-provoking.

Additionally, cryptology is a rich field that uses ideas, tools, and skills that are present in many undergraduate mathematics courses and is a great setting for students to work on engaging and meaningful applications of mathematics. While these applications attract many mathematics students to cryptology, the inherent puzzle-solving nature of cryptology often appeals to students from outside of mathematics as well. In increasing numbers, cryptology courses are offered in mathematics departments to support programs in mathematics, applied mathematics, or computer science. Sometimes these courses even target non-mathematics majors to fulfill broader interdisciplinary goals. With this special issue on Resources for Undergraduate Cryptology, we invite you to explore the myriad ways cryptology may enrich the lives of your students.

Cryptographers construct encryption functions. Encryption functions take as input plaintext language, which has patterns, and produce output, called ciphertext, which is “patternless.” The ciphertext is then transmitted over an open (insecure) channel, and the lack of pattern in the ciphertext is one aspect of the security of the encryption. But, when the authorized receiver obtains the ciphertext, that person must be able to read the message; therefore, the encryption function must be invertible (called the decryption function). Undergraduate mathematics is filled with the study of functions, processes, operations, and their inverses. Cryptography in undergraduate mathematics courses can reinforce these mathematical ideas and explore them in a unique setting. For example, shift and Vigenère ciphers involve understanding addition modulo 26 (and its inverse), the Hill cipher involves working with invertible matrices modulo 26; being able to invert permutations (and compositions of permutations) are important in a variety of other cryptographic settings.

Encryption and decryption also rely on a key, which “sets” the encryption and decryption processes. Often the same key is used for both encryption and decryption. Such a system is called “symmetric-key cryptography” because sender and receiver use the same key. The security of the encryption depends on the secrecy of the key. For a long time, it was necessary to have a secure channel over which the key could be exchanged. But in 1976, Whitfield Diffie and Martin Hellman introduced “public-key cryptography” by describing a method that allowed two parties to securely exchange a key over an insecure channel. Exponentiation modulo a prime number was at the heart of this seemingly impossible task. It was at this point in time that academic mathematicians became attracted to cryptology and the encryption/decryption processes became more explicitly mathematical as cryptography began to use ideas from abstract algebra, number theory, and discrete mathematics. Cryptography in undergraduate mathematics courses can build on these mathematical ideas and allow students to use these skills in the rich environment of codemaking and codebreaking.

While the language of mathematics can be used to explicitly define encryption and decryption functions, it is the process of cryptanalysis that allows students to flex their problem-solving muscles. Cryptanalysts attempt to recover the original plaintext message from the seemingly incomprehensible ciphertext, without any knowledge of the secret key! Plaintext language is full of patterns and predictability. Yet, ghosts of these patterns from the plaintext can remain in the ciphertext. Cryptanalysts hunt for and exploit those ghosts. Initially the search for patterns employed analysis of frequencies, understanding of the structures of languages, and problem-solving skills. During World War II, a number of electromechanical cryptanalytic machines were developed to search for patterns. Today, formal statistical testing and digital computers are cryptanalytic tools. In the near future, it is expected that those tools will include quantum computers. In cryptanalysis, there are no “rules” to follow – creativity, ingenuity, and perseverance are integral components to this engaging practice and students are often excited to jump in and try to solve these puzzles. Certainly, mathematics students have experience working with patterns. Students in undergraduate statistics courses learn to search for patterns in data. Students in undergraduate mathematics courses learn how to exploit patterns by creating functions, operations, algorithms, and other ways to abstractly represent patterns. Cryptanalysis provides a great problem-solving environment for students throughout the mathematical sciences to use their mathematical knowledge.

In what follows you will find articles written by authors who have successfully engaged undergraduate students in cryptologic ideas in courses that have cryptology or cryptography as a primary topic or in courses that use cryptology as a hook to interest students in exploring the mathematical ideas on which cryptology is based.

2. PREVIEW OF PAPERS

In this PRIMUS Special Issue on Resources for Undergraduate Cryptology, you will find lively and engaging articles describing successful pedagogies and course structures, carefully designed classroom activities, ideas for undergraduate projects in cryptology, and innovative course content. There are articles to support a wide range of expertise, ranging from instructors looking to teach a short module on cryptology for the first time to experienced instructors looking to enhance an upper-level mathematics or computer science course.

This issue opens with three papers that focus on special topics in cryptology that could be used to enhance an existing cryptology or mathematics course. In “Smudge Attack or: How to Secure Your Dirty Keypad” [5], Ulrich A. Hoensch explores how identifying fingerprint smudges on certain keys can affect the security of a keypad. This activity introduces multinomial coefficients and uses similar counting techniques. As the author notes, “The content is appropriate for undergraduate combinatorics, discrete math, probability and statistics, and cryptography courses.” Cheryl Beaver's “Adventures in Cryptology: Exploration-Worthy Project Topics” [1] provides a plethora of cryptology-related topics for students to explore with accompanying descriptions and resources. This paper is organized based on background prerequisites and has topics appropriate for a wide scope of interests, ranging from World War II cryptology to homomorphic cryptography to the security of public key ciphers. Markus Reitenbach's “Using Secret Sharing to Store Cryptocurrency” [8] provides an application of Shamir's secret sharing scheme to cryptocurrency storage along with sample activities appropriate for an audience comfortable with modular arithmetic. All three of these articles provide rich opportunities for student exploration, with Beaver's work supplying a wide range of projects, while both Hoensch and Reitenbach present projects specific to their topics and discuss opportunities for further investigation.

The next set of three articles all provide detailed descriptions of undergraduate cryptology courses along with innovative assignments from those courses. In “Cryptology as a way to Teach Advanced Discrete Mathematics” [3], David P. Bunde and John F. Dooley share the inner workings of their Cryptography and Computer Security course for students with a background in computer programming and discrete mathematics. In this paper, they share cryptanalysis and encryption algorithm implementation assignments that involve programming. Aaron Wootton's “Cryptography as a Bridge to Abstract Mathematics” [9] describes a variety of learning modules that can be incorporated into a mathematics major cryptology course or as special topics in abstract algebra, linear algebra, or number theory courses. He also provides a carefully detailed learning module on linear feedback shift registers with motivating questions and discussion points. The third course-focused paper, “An Inquiry-Based Learning Approach to Teaching Undergraduate Cryptology” [2], by Stephanie Blanda, describes an upper-level (computer science) cryptology course with several in-class activities and a discussion on how they are implemented in an inquiry-based learning setting.

A unique entry in our special edition, and more broadly, in the literature on undergraduate cryptology, is Melinda Lanius's “On Conceptual Metaphor in Cryptology Education” [7]. This article describes the impact of culture and metaphor on cryptology education and describes best practices in facilitating discussions about important conceptual metaphors in cryptology, such as “cryptology is information privacy” and “encryption is communication.”

The final two articles in this special edition discuss topics in cryptology for a more advanced mathematical audience. Jeffrey Ehme's “Variations on a Miller-Rabin Theme” [4] analyzes the Miller-Rabin test for finding large primes along with variations of this test that were developed in conjunction with undergraduate students. Depending on which version of the test is used, background requirements could range from introductory number theory to more advanced abstract algebra. The last paper in our issue, “Alkaline: A Simplified Post-Quantum Encryption Algorithm for Classroom Use” [6] by Joshua Holden, gives a simplified version of a recently developed encryption algorithm designed to be resistant to attacks by quantum computers. This article brings students to the forefront of recent and ongoing research in cryptology. While this topic is the most technically challenging in this issue, readers will see some unique applications of ideas and tools from linear algebra and abstract algebra.

By sharing this wide range of resources coming from expert instructors, this special issue aspires to promote further implementation of cryptology-based topics into the undergraduate mathematics curriculum. We know these articles will be helpful to both veterans of cryptology instruction and novices looking to develop their first cryptology course.

ACKNOWLEDGMENTS

The authors are thankful to the editors of PRIMUS for giving us the opportunity to put together this special issue on Cryptology.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Stuart Boersma

Stuart Boersma is a professor and former chair of the Department of Mathematics at Central Washington University. He is co-creator of the undergraduate cryptology competition Kryptos, enjoys writing expository mathematics papers, and received the 2005 Trevor Evans Award from the Mathematics Association of America.

Chris Christensen

Chris Christensen is a professor of mathematics and former chair of the Department of Mathematics and Statistics at Northern Kentucky University. He received a BS in mathematics from Michigan Technological University and a PhD from Purdue University while studying algebraic geometry with Professor S. S. Abhyankar. He is a member of the editorial board of Cryptologia.

Christian Millichap

Christian Millichap received his BS in mathematics and philosophy from Dickinson College and his PhD in mathematics from Temple University, with a focus on geometric topology. He enjoys engaging his students in cryptology-related topics through teaching, book clubs, scavenger hunts, and study-away programs. When he is not playing with shapes or ciphers, he enjoys running, hiking, and reading science fiction.