Cryptography: Decoding Student Learning

ABSTRACT

PATTERSON, BLAIN ANTHONY. Cryptography: Decoding Student Learning. (Under the direction of Karen Keene.)

Cryptography is a content ﬁeld of mathematics and computer science, developed to keep information safe both when being sent between parties and stored. In addition to this, cryptography can also be used as a powerful teaching tool, as it puts mathematics in a dramatic and realistic setting. It also allows for a natural way to introduce topics such as modular arithmetic, matrix operations, and elementary group theory. This thesis reports the results of an analysis of students’ thinking and understanding of cryptography using both the SOLO taxonomy and open coding as they participated in task based interviews. Students interviewed were assessed on how they made connections to various areas of mathematics through solving cryptography problems. Analyzing these interviews showed that students have a strong foundation in number theoretic concepts such as divisibility and modular arithmetic. Also, students were able to use probability intuitively to make sense of and solve problems. Finally, participants’ SOLO levels ranged from Uni-structural to Extended Abstract, with Multi-Structural level being the most common (three participants). This gives evidence to suggest that students should be given the opportunity to make mathematical connections early and often in their academic careers. Results indicate that although cryptography should not be a required course by mathematics majors, concepts from this ﬁeld could be introduced in courses such as linear algebra, abstract algebra, number theory, and discrete mathematics. © Copyright 2016 by Blain Anthony Patterson

by Blain Anthony Patterson

A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulﬁllment of the requirements for the Degree of Master of Science

Mathematics Education

Raleigh, North Carolina

2016

APPROVED BY:

Karen Hollebrands Molly Fenn

Karen Keene Chair of Advisory Committee DEDICATION

To Sarah Elizabeth Ritchey, my best friend and the love of my life.

ii BIOGRAPHY

Blain Anthony Patterson was born on May 7, 1991 in Salem, Ohio. He grew up in the small town of Wellsville, Ohio. Blain graduated from Youngstown State University with a B.S. degree in Mathematics Education in 2014. Directly after graduating, he entered the M.S. in Mathematics Education at North Carolina State University. Blain originally intended to teach high school mathematics somewhere near his home- town in Ohio. After meeting the love of his life, Sarah Elizabeth Ritchey, Blain had a change of heart and decided that graduate school was in his future. After applying to several graduate programs in both mathematics and mathematics education, he decided that NCSU was a perfect ﬁt. Blain plans to enroll in the Ph.D. program in Mathematics Education at North Car- olina State University for the spring 2016 semester. His interests focus on how undergraduate students learn upper level mathematics, such as Linear and Abstract Algebra, Number Theory, Probability, and Real Analysis. Blain has always had a passion for teaching and mathematics. His long term goal is to be a university professor and teach mathematics at a teaching-focused college. So far, North Carolina State University has provided him with the skills to become a better educator. Blain wants to be a lifelong learner and teacher of mathematics.

iii ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Karen Keene, for her support from the time I entered the program until completion of this project. Her guidance and expertise have been an essential part of my success at North Carolina State University. I look forward to working with her in the future. Also, I would like to thank Dr. Karen Hollebrands and Dr. Molly Fenn for their suggestions and willingness to serve as committee members. I would like to thank my father, Dorman Jeﬀ Patterson II, for always believing in me and pushing me to strive for excellence. I know without a doubt I would not be where I am today without the constant love, patience, guidance, and support of my father. He has always put my needs before his own and continues to do so to this day. I would like to thank my best friend, Sarah Elizabeth Ritchey for making me a better teacher, learner, and person. She has been by my side throughout my entire graduate school journey. Sarah is the one that convinced me that graduate school was right for me. She has guided me down a path of success and happiness. We are an unstoppable team and I am looking forward to all that we can accomplish together.

iv TABLE OF CONTENTS

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

Chapter 1 Introduction ...... 1 1.1 What is Cryptography? ...... 1 1.2 Deﬁnitions and Examples ...... 2 1.3 Why Cryptography? ...... 11

Chapter 2 Literature Review ...... 13 2.1 SOLO Taxonomy ...... 13 2.2 Abstract Algebra ...... 18 2.3 Linear Algebra ...... 21 2.4 Number Theory ...... 24 2.5 Probability ...... 26 2.6 Algorithms ...... 30

Chapter 3 Methods ...... 32 3.1 Research Questions ...... 32 3.2 Setting ...... 32 3.3 Participants ...... 33 3.4 Recruitment Procedure ...... 34 3.4.1 Qualitative Studies ...... 34 3.4.2 Task-Based Interviews ...... 35 3.4.3 Interview Protocol ...... 36 3.4.4 Rationale for Protocol Questions ...... 38 3.5 Analysis ...... 38

Chapter 4 Results ...... 41 4.1 Student A ...... 41 4.1.1 Introduction ...... 41 4.1.2 Problem Solving ...... 42 4.1.3 Analysis ...... 46 4.2 Student B ...... 46 4.2.1 Introduction ...... 46 4.2.2 Problem Solving ...... 47 4.2.3 Analysis ...... 52 4.3 Student C ...... 53 4.3.1 Introduction ...... 53 4.3.2 Problem Solving ...... 53

v 4.3.3 Analysis ...... 60 4.4 Student D ...... 61 4.4.1 Introduction ...... 61 4.4.2 Problem Solving ...... 61 4.4.3 Analysis ...... 65 4.5 Student E ...... 65 4.5.1 Introduction ...... 65 4.5.2 Problem Solving ...... 66 4.5.3 Analysis ...... 70 4.6 Student F ...... 71 4.6.1 Introduction ...... 71 4.6.2 Problem Solving ...... 71 4.6.3 Analysis ...... 77 4.7 Student G ...... 77 4.7.1 Introduction ...... 77 4.7.2 Problem Solving ...... 78 4.7.3 Analysis ...... 82 4.8 Summary ...... 82

Chapter 5 Conclusion ...... 86 5.1 Interpretation of Results ...... 86 5.2 Limitations ...... 88 5.3 Implications of the Study ...... 89 5.4 Future Research ...... 89 5.5 Conclusions ...... 90

Chapter 6 References ...... 91

Appendices ...... 95 Appendix A Interview Protocol ...... 96 Appendix B Answer Sheets ...... 98 Appendix C Solutions ...... 101 Appendix D Student Work ...... 102 Appendix E Participation Email ...... 123

vi LIST OF TABLES

Table 2.1 SOLO Levels ...... 15 Table 2.2 Index of Coincidence ...... 28

Table 3.1 Student Demographics ...... 33 Table 3.2 SOLO Rubric ...... 40

Table 4.1 Student SOLO Classiﬁcation ...... 83 Table 4.2 Responses to Question 9 ...... 85

vii LIST OF FIGURES

Figure 1.1 Vigenere cipher with DOG as keyword...... 4 Figure 1.2 Caesar cipher with a shift of 4...... 4 Figure 1.3 Cipher wheel...... 5 Figure 1.4 Public-key cryptography...... 7 Figure 1.5 Elliptic curve...... 8 Figure 1.6 Elliptic curve secant addition...... 9 Figure 1.7 Elliptic curve tangent addition...... 10 Figure 1.8 Elliptic curve vertical addition...... 10

Figure 2.1 SOLO Taxonomy Pyramid...... 17 Figure 2.2 Frequency distribution of the alphabet...... 27 Figure 2.3 Euclidean algorithm for 1559 and 650...... 30

Figure 4.1 Student A assigning numbers to letters...... 42 Figure 4.2 Student A using an aﬃne cipher with a = 10 and b = 0...... 43 Figure 4.3 Student A using a Caesar cipher with 3 as the key...... 43 Figure 4.4 Student A assigning numbers to letters...... 44 Figure 4.5 Student A using a brute force method...... 45 Figure 4.6 Student A using a factor tree...... 46 Figure 4.7 Student B setting up a Hill cipher...... 47 Figure 4.8 Student B multiplying matrices modulo 26...... 48 Figure 4.9 Student B summarizes his work...... 48 Figure 4.10 Student B using 1 as the key...... 49 Figure 4.11 Student B displaying all possible keys...... 50 Figure 4.12 Student B decrypting using various keys...... 50 Figure 4.13 Student B reducing the total number of divisors...... 51 Figure 4.14 Student B factors 8911 as 8911 = 7 × 1273...... 51 Figure 4.15 Student B repeats his algorithm for 1273...... 52 Figure 4.16 Student C assigning numbers to letters...... 54 Figure 4.17 Student C encrypting use the one-time pad...... 55 Figure 4.18 Student C displaying the ciphertext...... 55 Figure 4.19 Student C using 25 as a key...... 56 Figure 4.20 Student C listing possible two letter combinations...... 57 Figure 4.21 Student C displaying the plaintext√ GOWOLFPACK...... 57 Figure 4.22 Student C approximating 8911...... 58 Figure 4.23 Student C dividing 8911 by 7...... 59 Figure 4.24 Student C checking 7 and 11 as factors ...... 59 Figure 4.25 Student C displaying the prime factorization of 8911...... 60 Figure 4.26 Student D using a Vigenere cipher with keyword SUP...... 62

viii Figure 4.27 Student D using frequency analysis...... 62 Figure 4.28 Student D decrypting using a Caesar cipher...... 63 Figure 4.29 Student D checking small prime divisors of 8911...... 64 Figure 4.30 Student D checking small prime divisors of 1273...... 64 Figure 4.31 Student D checking small prime divisors of 67...... 65 Figure 4.32 Student E using a Vigenere cipher with keyword GO...... 66 Figure 4.33 Student E assigning numbers to letters...... 67 Figure 4.34 Student E using 1 as they key...... 67 Figure 4.35 Student E decrypting using various keys...... 68 Figure 4.36 Student E using the Pollard Rho factoring algorithm with n = 8911. . 69 Figure 4.37 Student E factors 8911 as 8911 = 7 × 1273...... 69 Figure 4.38 Student E using the Pollard Rho factoring algorithm with n = 1273. . 70 Figure 4.39 Student E factors 8911 as 8911 = 7 × 19 × 67...... 70 Figure 4.40 Student F assigning numbers to letters...... 72 Figure 4.41 Student F using an Aﬃne cipher with a = 3 and b = 5...... 72 Figure 4.42 Student F attempting to decrypt with 1 as a key...... 73 Figure 4.43 Student F attempting to decrypt with various keys...... 74 Figure 4.44 Student F testing prime divisors of 8911...... 74 Figure 4.45 Student F factoring 8911 as 8911 = 7 × 1273...... 75 Figure 4.46 Student F testing prime divisors of 1273...... 76 Figure 4.47 Student F displaying the prime factorization of 8911...... 77 Figure 4.48 Student G assigning numbers to letters...... 79 Figure 4.49 Student G encrypting the message NCSU...... 79 Figure 4.50 Student G summarizes her work...... 79 Figure 4.51 Student G shifting the alphabet by 17...... 80 Figure 4.52 Student G shifting the alphabet by 17...... 80 Figure 4.53 Student G checking for small prime divisors of 8911...... 81 Figure 4.54 Student G checking for small prime divisors of 1273...... 81 Figure 4.55 Student E factors 8911 as 8911 = 7 × 19 × 67...... 82

Figure D.1 Student A problem 6...... 103 Figure D.2 Student A problem 7...... 104 Figure D.3 Student A problem 8...... 105 Figure D.4 Student B problem 6...... 106 Figure D.5 Student B problem 7...... 107 Figure D.6 Student B problem 8...... 108 Figure D.7 Student C problem 6...... 109 Figure D.8 Student C problem 7...... 110 Figure D.9 Student C problem 8...... 111 Figure D.10 Student D problem 6...... 112 Figure D.11 Student D problem 7...... 112 Figure D.12 Student D problem 8...... 113

ix Figure D.13 Student D problem 8 continued...... 113 Figure D.14 Student E problem 6...... 114 Figure D.15 Student E problem 7...... 115 Figure D.16 Student E problem 8...... 116 Figure D.17 Student F problem 6...... 117 Figure D.18 Student F problem 7...... 118 Figure D.19 Student F problem 8...... 119 Figure D.20 Student G problem 6...... 120 Figure D.21 Student G problem 7...... 121 Figure D.22 Student G problem 8...... 122

x Chapter 1

Introduction

In this chapter, I discuss the importance of cryptography for both security and pedagogical reasons and provide the background and signiﬁcance for the study. I begin by introducing some important concepts and deﬁnitions.

1.1 What is Cryptography?

Cryptography is often thought of as the practice and study of the techniques and algorithms for secure communication. It is used on a daily basis, for keeping information safe (Kahn, 1997). For example, anytime someone pays for some good or service online, their credit card information is encrypted and decrypted using advanced algorithms that are part of the cryptography. It can be described as the intersection of mathematics, computer science, and electrical engineering (Singh, 1999). Cryptography has a rich history. As a society, humans have been trying to keep messages secret from the beginning of time. In fact, historians have discovered cryptic hi- eroglyphics from around 2000 B.C (Kahn, 1997). Julius Caesar used a substitution cipher to keep messages secret (Kahn, 1997). Eventually, complex machines, like the Enigma Machine, were created to encrypt and decrypt messages (Kahn, 1997). Once the digital

1 age had surfaced, more complex methods arose. With the abundance of history related to cryptography available, an entire class could be devoted to studying this material.

1.2 Deﬁnitions and Examples

Encryption is the process of converting a message to something unintelligible (Kahn, 1997). The message we are converting is referred to as the plaintext and the message that we get after converting and should be unintelligible is the ciphertext (Kahn, 1997). The inverse of encryption is referred to as decryption, where one converts ciphertext to plaintext (Kahn, 1997). A cipher can be defined as a set of techniques and algorithms that create encryption and decryption methods (Kahn, 1997). Ciphers can be mono-alphabetic or poly-alphabetic. Mono-alphabetic can be thought of as a bijective function, where each letter is uniquely mapped to another letter (Kahn, 1997). Note that a letter can be mapped to itself. Poly-alphabetic ciphers may map the same letter to two different letters (Kahn, 1997). In order to decrypt ciphertext, one needs to use a key. Cryptosystems can be defined as the set of all possible plaintexts, ciphertexts, keys, and algorithms (Kahn, 1997). We define an algorithm as a well defined process with a finite number of steps. Finally, cryptanalysis is the study of methods for cracking encryption algorithms without a key (Kahn, 1997). In addition to jargon associated with cyrptography, one must also consider notation. The goal is to send messages securely. Keeping this in mind, the two parties who are communicating should not make an intruder’s job any easier. This may be done by removing all forms of punctuation, spaces, and capitalizing all letters (Kahn, 1997). Removing punctuation simplifies the process, by not having to assign numbers to periods, commas, etc. By removing spaces, the attacker will not be able to utilize the fact that

2 there are only a small amount of words of a given size. Similarly, capitalizing all letters will make breaking the cipher more difficult for the attacker. For example, sending the message “I need help!” instead of INEEDHELP is not wise, since there are only a small amount of words that are one letter in length. What are some examples of ciphers? The Caesar cipher is one of the oldest ciphers recorded. Julius Caesar used this particular cipher to encrypt messages of military signifi- cance (Kahn, 1997). This cipher consists of shifting each letter in the alphabet to another letter by first converting each letter to a number. A is assigned to 0, B is assigned to 1, C is assigned to 2, and so on. Once each letter is represented by a number, x, the formula for encryption is Ek(x) = x + k mod 26 and decryption is Dk(x) = x − k mod 26. Modular arithmetic uses 26 as the divisor, since there are 26 letters in the alphabet. Another elementary cipher is the Vigenere cipher. Similar to the Caesar cipher, letters are shifted by a key (number). However, instead of just a single number as the key, messages are shifted by a key word. This makes the Vigenere cipher poly-alphabetic. For example, supposed the key word is DOG. Note that D, O, and G are the fourth, fifteenth, and seventh letters of the alphabet respectively. This implies we the first letter of the plaintext will be shifted by 4, the second letter by 15, and the third letter by 7. The plaintext will most likely be longer than 3 letters or the size of the key word in general. If this is the case, the process repeats. In other words, the fourth letter will be shifted by 4, the fifth by 15, the sixth by 7 and so on. An example will make this concept more concrete. Suppose MEETATDAWN is the plaintext and DOG is the key.

3 Figure 1.1: Vigenere cipher with DOG as keyword.

This implies that M will be shifted by 4, the first E will be shifted by 15, the second E will be shifted by 7, and so on. This will result in a ciphertext of PSKWOZGOCQ. Decryption would happen in a similar manner, by using the same key. These first two examples are called shift ciphers, since each letter in the plaintext is shifted by a key to create the ciphertext. Although these methods are not used today in the practical sense, they offer a great place to start. Modular arithmetic is an elementary topic. In fact, students do modular arithmetic in grade school without realizing it. Because of the elementary knowledge needed, shift ciphers are very accessible to students. In particular, the Caesar cipher offers multiple representations. One can view this cipher using Figure 1.2.

Figure 1.2: Caesar cipher with a shift of 4.

If a student is more algebraically inclined, they may favor the following formulas for encryption and decryption.

Ek(x) = x + k mod 26 and Dk(x) = x − k mod 26

4 Students that are consider tactile learners may choose to use a cipher wheel shown in Figure 1.3.

Figure 1.3: Cipher wheel.

If students have a sufficient background in linear algebra, the Hill cipher can be discussed. Created by Lester S. Hill in 1929, the Hill cipher was the first poly-alphabetic cipher which was practical to operate on more than three symbols at once (Singh, 1999). In fact, the Hill cipher of dimension 6 was done mechanically using a system of gears and chains (Kahn, 1997). To encrypt using the Hill cipher, one must first convert letters to numbers using the standard assignment (A is assigned to 0, B is assigned to 1, C is assigned to 2, and so on). Once the plaintext is converted to numbers, the key matrix is chosen. In order for decryption to be possible, the key matrix must be invertible modulo 26. In other words, the determinant must be an integer that has an inverse modulo 26. The sender converts the plaintext to n × 1 column vectors, where n is the dimension of the key matrix. To decrypt, one simply multiplies by the inverse of the key matrix. Consider the following example. Suppose the plaintext is CAT, where C corresponds the 2, A corresponds to 0, and T corresponds to 19. Therefore the plaintext vector is the

5 following.

  2     x =  0      19

Now suppose the key matrix is below.

  1 0 0     K =  2 13 0      3 4 1

Notice that the determinant of K is 13, which has an inverse modulo 26. To encrypt, we compute the following.

      1 0 0 2 2             Kx =  2 13 0   0  =  4  = y             3 4 1 19 25

Therefore the ciphertext is CEZ. To decrypt, one must compute K−1y mod 26 ≡ x, to recover the plaintext vector, x. Because matrix theory is a part of linear algebra, a strong foundation in that content area would be extremely useful for a student studying cryptography to have. Advancing to more recent work, modern cryptography methods are primarily public- key systems. This means that the sender and receiver each have their own key, which are related to a public key through computations (Kahn, 1997). The security of public-key cryptography relies on the fact that the computations required to break the cipher is infeasible (Kahn, 1997). This means that even with the world’s fastest super computer,

6 the cipher could not be broken in a reasonable amount of time. The general scheme of public key cryptography can be seen in Figure 1.4.

Figure 1.4: Public-key cryptography.

The RSA algorithm is the most widely used public-key cryptosystem today. Created by Ron Rivest, Adi Shamir and Leonard Adleman in 1978, this encryption method takes advantage of the diﬃculty of factoring large composite integers (Kahn, 1997). With a powerful computer, multiplying two large prime numbers is computationally feasible. However, recovering the prime factors of a composite number takes the most powerful computers a great deal of time (Kahn, 1997). To encrypt a message with RSA, one must choose two “large” prime numbers p and q and compute n = pq. Then choose a number e, such that the greatest common divisor of e and (p − 1)(q − 1) is 1. The key (n, e) are made public, but p and q are kept private. In order for an adversary to break this cipher, this must factor n as n = pq, which is computationally infeasible. Other examples of public-key cryptography include Diﬃe-Hellman, Cramer-Shoup, ElGamal, and elliptic curve (Kahn, 1997). Although these methods are much more complex, they can be valuable topics to learn and teach since they utilize various areas of

7 mathematics, including algebra, number theory, probability, and geometry. One of the most accessible of these methods to students may be elliptic curve cryptography. An elliptic curve is a curve in the plane of the form y2 = x3 + Ax + B, with the condition that x3 + AX + B has distinct roots (Abraham, Kapoor, and Singh, 2008); elliptic curves are also symmetric about the x-axis. Figure 1.5 shows a graph of an elliptic curve

Figure 1.5: Elliptic curve.

For computational purposes, these curves exists over finite fields, rather than all real numbers. For example, let E be an elliptic curve over the integers mod 2. This implies that A and B must be 1 or 0 and the only possible points on the curve are (0, 0), (1, 0), (0, 1), and (1, 1). However, not all these points need be on the curve. With each elliptic curve comes an algebraic structure, where two points can be added in any order, each point has and inverse, there exists an identity point (the point at infinity), and the associative property holds. Therefore, the points on an elliptic curve form an abelian group (Abraham, Kapoor, and Singh, 2008). However to actually add points on an elliptic curve, one must use geometric properties of secant and tangent lines.

8 In order to add two points, P and Q, on an elliptic curve one most construct the line between those two points. If line PQ goes through the curve at a diﬀerent point, R, one must construct the line perpendicular to the x-axis that goes through R. The point where this line intersects the curve is P + Q (Abraham, Kapoor, and Singh, 2008). This can be seen in Figure 1.6.

Figure 1.6: Elliptic curve secant addition.

If two points, P and R, form a line that does not intersect the curve at a third point, one must construct a line perpendicular to the x-axis that goes through the curve. The point where this perpendicular line intersects the curve will be P +R (Abraham, Kapoor, and Singh, 2008). This can be seen in Figure 1.7.

9 Figure 1.7: Elliptic curve tangent addition.

Finally, if two points are symmetric about the x-axis, the line the goes through these points will not intersect the curve at a third point. Since the points on the elliptic curve form a group, the result must be another point on the curve. This point is called the point at inﬁnity, denoted by O. This point will be on any vertical line (Abraham, Kapoor, and Singh, 2008). This can be seen in Figure 1.8.

Figure 1.8: Elliptic curve vertical addition.

Clearly, to understand utilize elliptic curves for cryptography purposes, one must have a ﬁrm grasp on fundamental geometric concepts. It may be the case that students have not work with these concepts since high school. Needless to say, analyzing how students

10 understand geometric concepts can give the researcher much need insight about their ability to solve particular cryptography problems and vice versa.

1.3 Why Cryptography?

Cryptography is a field of mathematics associated with applications. Aside from security, one application is teaching. Cryptography is a very engaging subject. It puts mathematics in a dramatic setting, which can make it extremely engaging for students of all ages. Children are fascinated by intrigue and adventure. “More is at stake than a grade on a test: if you make a mistake, your agent will be betrayed” someone using cryptography to send secret codes might say (Koblitz, 1997). Additionally, cryptography enables students to discover mathematical concepts and techniques on their own. The thrill of discovery can be highly motivating for students of all ability levels. “After many hours the youngsters finally develop a method to break a cryptosystem, then they will be more likely to appreciate the power and beauty of the mathematics that they have uncovered” (Koblitz, 1997). Also, due to the uncertainty of cryptography problems, students may be less likely to maintain the notion that every problem in mathematics can be solve with a formula. Finally, cryptography allows for interdisciplinary study (Koblitz, 1997). Mathematical concepts like functions, inverses, modular arithmetic, and group theory are heavily used. In addition to these, students can cross into other fields such as statistics and linguistics. For example, a common way to break a code is to use the concept of frequency analysis. This means that students would attempt to see if certain letters always represent other letters based on their frequency (Koblitz, 1997). This strategy takes both statistical methods as well as the English language in to account.

11 Cryptography can be used to teach mathematics to students of all ages. One can learn how to send and receive secret messages at a very young age. Once the proper foundation has put down, students can start to formalize encryption and decryption with algebraic formulas. Finally, once students are exposed to mathematical theory, they can start to analyze the strengths and weaknesses of various cryptosystems. Cryptography can be a highly motivating topic for students. Students are engaged in problems where they must pull information from various areas of mathematics, as well as their personal experiences. Due to the engaging nature of cryptography, I want to investigate the following research questions.

1. How does studying cryptography enable students to use their understanding of diﬀerent advanced mathematical content?

2. What diﬀerent SOLO levels do students exhibit when they are solving cryptography problems?

In this paper, I first synthesize literature on the SOLO taxonomy and student learning of abstract algebra, linear algebra, number theory, probability, and algorithms. Then I will discuss the research methods used including the setting, participants, recruitment procedure, the interview protocol, and analysis. Next I will discuss the results of the analysis and finally, the results are interpreted and teaching implications are offered.

12 Chapter 2

Literature Review

In this chapter, I discuss how students learn various areas of mathematics used in cryptography. This includes abstract algebra, linear algebra, number theory, probability, and algorithms. Also I present the SOLO Taxonomy, which will be used to assess student understanding.

2.1 SOLO Taxonomy

One way I will study student understanding of advanced mathematics is through the structure of observed learning outcomes (SOLO) taxonomy framework. Developed by John B. Biggs and K. Collis in 1982, this framework provides a hierarchy for student understanding of a particular subject. This particular framework was chosen because of its simple, yet elegant way of assessing student understanding. According to Biggs and Collis, (1982) “The SOLO taxonomy provides a simple and robust way of describing how learning outcomes grow in complexity from surface to deep understanding.” The SOLO taxonomy is made up of ﬁve modes (Sensori-motor, Ikonic, Concrete Symbolic, Formal, Post Formal) and ﬁve stages (Pre-structural, Uni-structural, Multi-structural, Relational,

13 Extended Abstract). Modes are related to various age groups. The Sensori-motor mode occurs soon after birth, where a person reacts to the physical environment (Biggs and Collis, 1982). At age two, one reaches the Ikonic mode, which is characterized by internalization of actions in the form of images (Biggs and Collis, 1982). Concrete Symbolic occurs around age six. Here a person would think through a symbol system such as language and number systems (Biggs and Collis, 1982). In the Formal mode, which occurs around age 15, one considers more abstract concepts (Biggs and Collis, 1982). Finally, Post Formal is reached around age 22. Those in this mode are about to question or challenge the fundamental structure of theories or disciplines (Biggs and Collis, 1982). For each given mode, a person’s understanding can be categorized in one of the ﬁve stages of the SOLO taxonomy. Students functioning in Pre-structural are only requiring pieces of unconnected information, with little to no organization. Common behaviors associated with this stage includes avoiding or repeating the question being asked. If the student does engage in the problem, an incorrect process may be used, leading to irrelevant confusion (Chick, 1998). One can move into Uni-structural once a simple or obvious connection is made, however this is the only focus of the student. Students functioning in this stage apply a single process or concept, often resulting in an invalid conclusion (Chick, 1998). Once the Multi-structural stage is reached, students have made a number of connections. However, students may fail to synthesize information, which may indicate cognitive performance below that required for a successful solution (Chick, 1998). If a student reaches the Relational level, various aspects have become one and the student sees how the parts combine to make a whole. In other words, students are starting to see the “big picture.” The Relational level is completely characterized by the synthesis of information (Chick,

14 1998). Students functioning in this level have an adequate understanding of the given subject area. Finally, in the Extended Abstract level, students can generalize and transfer knowledge to a different subject area or take knowledge from a different area to solve a problem. Extended Abstract responses are structurally similar to Relational responses, but here students use concepts from outside the domain of assumed knowledge. “In this taxonomy, the structure of the learned outcome occurs within each of Piaget’s stages of cognitive development. More specifically, the three SOLO levels in the middle, namely, Uni-structural, Multi-structural and Relational, fall within the same stage whereas the extended abstract extends the level of abstraction into the next stage becoming the Uni- structural level of that next stage (Jurdak and Mouhayar, 2013).” Table 2.1 summarizes the SOLO levels.

Table 2.1: SOLO Levels

SOLO Level Description

Pre-structural No connections.

Uni-structural Single connection.

Multi-structural Multiple connections.

Relational Parts as a whole.

Extended Abstract Knowledge transfer.

It is also essential to describe how students transition from one stage to the next. When transitioning from Pre-structural to Uni-Structural, a student may attempt to answer the question, but only partially makes a connection. From Uni-structural to Multi-

15 structural, one may attempt to handle multiple connections, but no significant progress is made. During the transition from Multi-structural to Relational, a student may recognize several aspects of the problem, but fail to reconcile them. Finally, a transition from Relational to Extended Abstract can be observed by seeing students make progress towards a firm conclusion. Although participants in this study will be only be categorized in the five main categories, these transitions allow for the researcher to better analyze student understanding. A key assumption made regarding this framework is that each level incorporates the previous levels, then extends understanding. One can think of the SOLO taxonomy as a pyramid, where each level provides a foundational support for the next. Since the researcher often is asking questions that are open ended and have various entry points, this allows for students to be categorized in any of the five stages. Figure 2.1 shows the hierarchy of the SOLO taxonomy.

16 Figure 2.1: SOLO Taxonomy Pyramid.

When posing questions in task-based interviews, one must keep in mind that particular questions may limit the SOLO level of possible solutions (Biggs and Collis, 1982). For example, the question “What is public key cryptography?” would best require a pre- structural response. However, posing tasks such as “Encrypt the plaintext WOLF using an encryption method of your choice. Discuss the strengths and weaknesses of the chosen encryption method,” may allow for various SOLO levels. This is because students could do anything from not responding to using an encryption method that is outside the scope of the course. The interviewer must create questions that allow for a large range of SOLO level responses, which provides an eﬃcient tool for measuring student understanding of a particular topic.

17 2.2 Abstract Algebra

Abstract algebra is a foundational part of the field of cryptography. In fact, one can even think of cryptography as “applied abstract algebra.” If this is the case, then understanding how students think about abstract algebra is essential to the analysis of student understanding of cryptography. abstract algebra is often the first time students are faced with high levels of abstraction. Research has been done to analyze how levels of abstraction can be reduced while not sacrificing mathematical rigidity. According to Hazzan (1999), abstraction level can be interpreted three different ways:

1. Abstraction level as the quality of the relationships between the object of thought and the thinking person.

2. Abstraction level as a reﬂection of the process-object duality.

3. Abstraction level as the degree of complexity of the concept of thought.

The ﬁrst interpretation of abstraction stems from the idea that whether something is abstract or concrete is not inherent of the object, but the relationship that one has with the object (Wilensky, 1991). This implies that abstraction can vary for each person and object based on the previous connections between the two. The closer one is to an object, the more connections one will have, hence the object will become less abstract and more concrete (Hazzan, 1999). Using this perspective, one can correlate students’ mental processes with their tendency to make unfamiliar problems more familiar, i.e. making the transition from abstract to concrete (Hazzan, 1999). This process is quite common in Abstract Algebra when students use what they know about various numbers systems to solve problems involving group theory (Hazzan, 1999).

18 The abstraction level can also be view using process-object duality. To discuss this duality, it is essential that one distinguishes between process conception and object conception.“Process conception implies that one regards a mathematical object as a potential rather than an actual entity, which comes into existence upon request in a sequence of actions” (Sfard, 1991). A mathematical concept becomes an object when the concept is conceived as one entity. Therefore, for a given mathematical idea, it is ﬁrst conceived as a process, which is less abstract, then becomes an object (Hazzan, 1999). Students can reduce abstraction level by perceiving mathematical ideas as processes rather than objects. The mental process that allows students to transition from process conception to object conception was coined reﬂective abstraction by Piaget (Hazzan, 1999). Finally, level of abstraction can be interpreted as the degree of complexity of the concept of thought. This viewpoint hinges on the assumption that the more compound an entity is, the more abstract it is (Hazzan, 1999). Students can reduce abstraction this way by substituting a less complex, but related idea for a more complex one (Hazzan, 1999). For example, a student trying to decrypt a message may start by only examining one letter at a time to determine what type of cipher was used. In an Algebra course, a student may replacing an entire group with a single element to attempt to reduce the abstraction level (Hazzan, 1999). Knowing when students are attempting to reduce abstraction and how they are going about this process can give one insight about which level of understanding the student is at. Therefore, this enables a way to place students in a given stage based on their need and ability to reduce levels of abstraction. One can also consider instructional strategies for improving the understanding of various abstract concepts. According to Dubinsky et al. (1994), the following are strategies to consider when teaching abstract algebra.

19 1. Going through the Action-Process-Object-Schema Steps

2. Computer Activities and Team Work to Clear Up Misconceptions

3. Meeting Prerequisites

4. Finding Alternatives to Linear Sequencing

Dubinsky et al. (1994) asks “How can we get students to take a speciﬁc step in the development of a particular concept? In particular, what methods can be used to help students to interiorize actions to construct processes, to reverse or coordinate processes to construct other processes, to encapsulate processes, to construct objects, and to thema- tize collections of processes and objects into schemas?” Dubinsky et al. have experienced success in this regard by creating computer-based tasks to foster the constructing of processes and objects and by having students work cooperatively. Regardless of the strategies used, Dubinsky et al. suggest that students should reﬂect on their actions independently or as a class. Misconceptions may happen in any mathematics class. However, teachers must help students clear up these misconceptions or at least make them aware that they have fallen into them. The use of group work may be one solution (Dubinsky, Dautermann, Leron, and Zazkis, 1994). Students tend to be more likely to seriously consider contradictions presented by their classmates than ones presented by their teacher. When an instructor claims something is or is not true, it is convention to simply accept what is said as truth (Dubinsky, Dautermann, Leron, and Zazkis, 1994). It is essential that students have a deep understanding of sets and functions before entering an abstract algebra class (Dubinsky, Dautermann, Leron, and Zazkis, 1994). Sets and functions play an integral role in the concepts of one-to-one, onto, and isomorphism. Students should understand sets and functions both as processes and objects in order

20 to have a deeper understanding of various algebraic structures (Dubinsky, Dautermann, Leron, and Zazkis, 1994). It is natural to present mathematics in a simple linear sequence. However, Jerome Bruner prosed that a spiral curriculum is a suitable replacement (Dubinsky, Dautermann, Leron, and Zazkis, 1994). This is done under the assumption that any mathematical topic can be taught in a rigorous way at any age. Fundamental mathematical concepts can be taught a young age, then revisited over the years, adding levels of sophistication and rigor. Dubinsky et al. believes that instructors should use abstract algebra as a way to revisit concepts such as sets and functions. This way, abstract algebra can be taught through concepts students are familiar with, but have not quite mastered.

2.3 Linear Algebra

One of the ﬁrst poly-alphabetic ciphers students can learn after the Vigenere cipher is the Hill cipher. The Hill cipher encrypts messages by using an n × n matrix that has an inverse modulo 26. In order for students to encrypt, decrypt, and codebreak the Hill cipher, they must have a strong foundation in linear algebra. In this section, I provide some background on student learning of linear algebra. Although most mathematics majors take a linear algebra course, the courses often focus on computation and procedural knowledge. Although these skills are important for encryption and decryption, students often forget the procedures and algorithms (Dorier and Sierpinska, 2001) so focusing on the concepts that support the procedures and algorithms is important. For example, if a student forgets the formula for the inverse of a 2 × 2 matrix, if they had a conceptual understanding of where the formula came from, they could easily derive it.

21 In addition to its use in classical cryptography, linear algebra tends to show up in pure and applied mathematics, computer science, engineering, physics, and other sci- ences. Clearly this is an important subject for many mathematics and science majors. According to Dorier (2002) , the two main issues of teaching and learning linear algebra are Epistemological Speciﬁcity and Cognitive Flexibility (Dorier, 2002). Other diﬃculties which students are faced with when learning linear algebra is the variety of languages, semiotic registers or representation, points of view, and settings through which the objects of linear algebra can be represented (Dorier, 2002). According to Hillel (2000), there are three basic languages associated with linear algebra.

1. Abstract Language

2. Algebraic Language

3. Geometric Language

This can be an issue if the instructor switches from one representation to the next without any notice. Often students are confused the most when the language is switched from the abstract to the algebraic. With each of these languages comes a corresponding mode of thinking (Hillel, 2000).

1. Analytic-Structural

2. Analytic-Arithmetic

3. Synthetic-Geometric

It is essential that students can eﬀectively use multiple representations of objects when studying mathematics. In fact in any mathematical activity, representations are absolutely necessary, since mathematical objects cannot be perceived, therefore must be

22 represented (Dorier, 2002). For example vectors can be represented graphically by arrows, by rows or columns of coordinates, or symbolically as abstract elements of a vector space. In order to reduce levels of abstraction some have considered a more geometric approach to linear algebra. The goal is to overcome some of the abstractions by giving more concrete meaning to concepts through geometric ﬁgures. However a problem quickly ar- rises when one relies too much on the geometry of linear algebra. Geometry is limited to three dimensions, therefore various concepts have limited representations in a geometric setting. Also, if students learn linear algebra in a heavy geometric setting, they can have a diﬃcult time going back to more general and abstract cases. One example is that students may struggle to imagine a linear transformation that would not be a geometric transformation. However, some students may use the geometric ideas to their advantage when appropriate. In fact being able to decide when a geometric representation should be used or not displays a high level of understanding of the subject. “It seems that the use of geometrical representations or language is very likely to be a positive factor, but it has to be controlled and used in a context where the connection is made explicit.” Additionally, some have proposed a greater use of technology in the linear algebra classroom. In fact, according to Dikovic (2007), every linear algebra instructor should consider the following questions.

1. Why do some students learn more mathematics than other students in the same class?

2. What can linear algebra teachers do to enrich or replace traditional lecturing in order to improve meaningful learning?

23 3. What contribution of technology could be in the ﬁelds of experimenting, observa- tion, and discussing?

4. How many reasons are there, for yes or no, for example for use of graphics calculators for computing matrix inversion or for solving linear systems?

Dikovic (2007) suggests Maple, MATLAB, or Mathematica for their very powerful, numerous functions. These functions include but are not limited to instantaneous numeric and symbolic calculation, data collection and analysis, modeling, presenting two and three dimensional graphics, and application development. Hillel and et al. also suggest the use of Cabri, a dynamic geometry program, to teach and learn linear algebra. Hillel and et al. believe that by using Cabri, students can overcome the obstacle of formalism associated with vector spaces. Assuming the meaning of various mathematical objects are well represented by the computer representations, students can study vector spaces geometrically, rather than analytically (Hillel, Trgalova, and Sierpinska, 1999). One must note that these geometric representations are limited to two and three dimensions, so other representations must be considered (Dorier, 2002).

2.4 Number Theory

Number theory is often thought of as the purest form of mathematics (Cambell and Zazkis, 2002). This is a classical subject that contains several beautiful and elegant proofs. However, in recent years new light has been shed on number theory, speciﬁcally on its application to modern cryptography. Public key systems, such as RSA, rely heavily upon major theorems from number theory. For Euler’s φ function, which counts the number of integers less than n that are relatively prime to n, is an essential part of RSA encryption. The fact that this function is multiplicative (φ(pq) = φ(p)φ(q)) makes

24 RSA so effective. Additionally, foundational number theoretic concepts such as modular arithmetic, divisibility, and primality are powerful tools to the field of cryptography. What makes number theory even more important is its integration within mathematics from elementary school through college (Cambell and Zazkis, 2002). It is very common that elementary school students learn about different number systems, long division with and without remainders, and prime numbers (Cambell and Zazkis, 2002). These topics are all rooted in number theory (Cambell and Zazkis, 2002). According to The National Council of Teachers of Mathematics (NCTM), in pre-K through grade 2, all students should be able to “develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers, (NCTM, 2015).” These number theoretic concepts continue in grades 3 through 5, where NCTM (2015) claims all students should be able to “recognize equivalent representations for the same number and generate them by decomposing and composing numbers” and “describe classes of numbers according to characteristics such as the nature of their factors.” In grades 6 through 8 all students should be able to “use factors, multiples, prime factorization, and relatively prime numbers to solve problems” and “develop meaning for integers and represent and compare quantities with them,” according to NCTM (2015). NCTM (2015) then explicitly states all students should be able to use “number-theory arguments to justify relationships involving whole numbers” in grades 9 through 12. Aside from these standards, number theory can help students make the transition from arithmetic to algebra, by enabling students to develop better understandings of the abstract conceptual structure of whole numbers and integers (Wagner, 2012). Addi- tionally number theory has algebraic characteristics similar to variables (Wagner, 2012). Therefore studying concepts such as number systems, division, and primality can develop stronger mathematical reasoning.

25 Unfortunately, these concepts are forgotten after years of not engaging in them (Cam- bell and Zazkis, 2002). Although high school students know what a remainder is and how to find one, they most likely have not been introduced to the formal concept of modular arithmetic. Cambell and Zazkis (2002) found that for preservice teachers, this poses a serious problem. Their understanding of the concept of number is not where it should be. Additionally students that are required to take a number theory or modern algebra course often struggle with elementary number theory concepts (Cambell and Zazkis, 2002). Students tend to solve problems in a procedural manner rather than conceptual. In particular, students encounter difficulties when they are faced with problems that have a wide range of strategies or representations (Cambell and Zazkis, 2002). For example, students may face difficulties when attempting to link divisibility to factorization (Cambell and Zazkis, 2002).

2.5 Probability

Studying cryptography requires keen intuitions about when particular methods are going to work and when they are not. Instead of relying on intuition alone, probabilistic methods are introduced. For example, suppose that students are asked to break a code, knowing that it has been encrypted using a substitution cipher. Students can use the fact that certain letters appear more often than others. Figure 2.2 shows the frequency distribution of the letters of the alphabet.

26 Figure 2.2: Frequency distribution of the alphabet.

With this information in mind, students can make predictions about the plaintext based on the ciphertext. For example, if a student is faced with a string of ciphertext where the letter J is most common, they may assume that J represents a vowel. This is because in the English language, vowels appear more frequently than consonants, with the exception of the letter T. However, frequency analysis is only eﬀective when a substitution cipher was used. How can one decide which type if cipher was used to encrypt the plaintext? The Index of Coincidence can be an eﬃcient tool in determining whether a given cipher is mono-alphabetic or poly-alphabetic (Kahn, 1997). Index of Coincidence refers the the probability of choosing the same letter twice in a string of letters and is given by

Z P fi(fi − 1) IC = i=A N(N − 1)

27 where fi is the number of times the letter i appears for i = A, B, . . . , Z and N is the number of letters in the ciphertext. Luckily, Index of Coincidence values have been computed for various languages. If all letters are equally likely, the Index of Coincidence would be approximately 0.038 (Singh, 1999). However, in the English language certain letters appear more frequently such as the letter E, which appears approximately 13 percent of the time. This yields a higher Index of Coincidence for English, approximately 0.067 (Singh, 1999). The Index of Coincidence for various languages can be seen in Table 2.2.

Table 2.2: Index of Coincidence

Language Index of Coincidence

English 0.067

Russian 0.068

Spanish 0.075

Portuguese 0.075

Italian 0.075

French 0.078

German 0.079

Random 0.038

When a cipher is mono-alphabetic, the Index of Coincidence does not change, since the same letter frequencies exist. When the Index of Coincidence is closer to random, then one may assume that a poly-alphabetic cipher was used. Once this is known, the

28 appropriate attack can be used. The use of probability and statistics is an integral part of solving cryptography problems. However, many students have a lack of experience with probability, often only studying this subject once. This begs the question, “How can we enable students to understand probability?” To answer this question, one must consider the following three perspectives.

1. Building on the ﬁrm basis of students’ sound intuitions (Faulk, 1992).

2. Conventional teaching of probability does not establish enough connections between the intuitions of the leaner and the mathematical theory (Borovcnik, 1991).

3. Intuitions are the product of personal experience (Fischbein, 1987).

Clearly, intuitions are important in learning probability. The learning of probability should start with these intuitions, changing them as new knowledge is acquired by the student. This can be done by developing secondary intuitions to create the link between the students’ preconceptions and the theory. However, it is essential to clear up misconceptions as early as possible. Due to personal experiences, students may continue to believe something is true, even when the mathematics say otherwise. For example, a student may say that they understand that ﬂipping a coin several times has the same probability for heads each toss. However, after getting four tails in a row, students may feel that that getting heads on the next toss is now more likely. According to Fischbein (1987), “One of the fundamental tasks of mathematical education ... is to develop in students the capacity to distinguish between intuitive beliefs, intuitive feelings and formally supported convictions.”

29 2.6 Algorithms

One can think of cryptography as the intersection of mathematics and computer science. Although the complex encryption methods require rigorous mathematical theory, heavy computation must be done in real world problems. This requires both theory and application of computer science. In particular, it is essential that one has a deep understanding of algorithms, including representations, syntax, and time to run given algorithms. An algorithm is defined to be a well defined process that requires a finite amount of steps. One may view algorithms either graphically or using standard syntax. Figure 2.3 shows a graphical view of the Euclidean algorithm for 1599 and 650.

Figure 2.3: Euclidean algorithm for 1559 and 650.

This is an alternative to writing out the algorithm using symbolic notation, as seen displayed below.

30 1599 = 650 × 2 + 299

650 = 299 × 2 + 52

299 = 52 × 5 + 39

52 = 39 × 1 + 13

39 = 13 × 3 + 0

According to Byrne, Catrambone, and Stasko (1999), students benefit from seeing both symbolic and graphical representations of algorithms. This is because “Conceptual knowledge about the properties of an algorithm can help a learner to carry out the algorithm’s steps. Similarly, being able to perform the step-by-step operations of an algorithm may assist a learner in determining the veracity of a conceptual question about it. (Byrne, Catrambone, and Stasko, 1999)” This will ultimately give students a deeper understanding of algorithm and how they work. Consequently, students may gain insight on effectiveness and running time. Algorithms are an integral part of learning cryptography, since cryptanalysis is a major focus. After presented with a given cryptosystem, students may be asked to analyze its security. For example, the Caesar cipher is typically the first cipher discussed in a cryptography course. A basic algorithm to break this cipher would be to try every key from 1 to 25. Even if this was done by hand, the maximum amount of time to break this cipher would be 25 steps. As more complex ciphers are discussed, the algorithms to break them become more difficult and time consuming, allowing a perfect opportunity for graphical representations.

31 Chapter 3

Methods

In this chapter, I discuss the methods of data collection and analysis for this qualitative study. This includes the setting of the interviews, participant demographics, and a detailed description of the procedures used to collect and analyze the data from the interviews.

3.1 Research Questions

1. How does studying cryptography enable students to use their understanding of diﬀerent advanced mathematical content?

2. What diﬀerent SOLO levels do students exhibit when they are solving cryptography problems?

3.2 Setting

This experiment was conducted at the end of the spring 2015 semester at a university in the southeastern United States. Students participating in this data collection were

32 interviewed in a one-to-one setting for approximately one hour. Those participating in this study did so following the 700 level mathematics course “Applications of Algebra.” The class consisted of approximately 20 students and the material was presented using traditional lecture, twice a week for approximately 75 minutes All of the students in the course were either mathematics (pure, applied, eduction) or computer science majors. Although there are many topics that could be discussed in an applied algebra course, the professor chose to focus solely on cryptography. Topics covered in this course include shift ciphers, the Hill cipher, exponential ciphers, RSA, ElGamal, elliptic curve ciphers, Pollard’s method, and quadratic sieves.

3.3 Participants

All participants were either Mathematics (Pure or Applied) or Computer Science majors. Table 3.1 below summarizes the demographics of the participants in this research study.

Table 3.1: Student Demographics

Student Gender Age Major

A Male 22 Pure Mathematics

B Male 24 Pure Mathematics

C Female 22 Computer Science

D Female 23 Applied Mathematics

E Female 24 Pure Mathematics

F Female 21 Applied Mathematics

G Female 24 Applied Mathematics

33 A convenience sample was used for this project; the participants were selected based on their willingness to participate. All students taking the course “Applications of Alge- bra” were asked to participate by email and the ﬁrst seven to respond were chosen.

3.4 Recruitment Procedure

Prior to being interviewed students were contacted through email and asked if they would be willing to participate. Students were to sign an informed consent document, where they agreed to the use of audio recordings and their written work in this research. Participants’ work was kept and scanned for use in this research. All work can be found in Chapter 4 and Appendix C.

3.4.1 Qualitative Studies

In this study, I am analyzing student interviews and work so it lends itself to the use of qualitative research. Qualitative research emphasizes the importance of looking at variables in the natural setting in which they are found (Flick, 2009). Data can be gathered through open ended questions and tasks that provide artifacts such as sample work and direct quotations (Flick, 2009). Unlike in quantitative research, which attempts to re- move the investigator from the investigation, the researcher is an integral part of the investigation. The focus of qualitative research is to have a holistic view of what is being studied, however there exist both advantages and disadvantages to doing qualitative research. One advantage is that qualitative research provides more in depth, comprehensive information (Flick, 2009). There is a cost, however, to gaining this information. Due to subjectivity, establishing reliability and validity can be diﬃcult. Additionally, it is very diﬃcult to prevent or even detect bias induced by the researcher (Flick, 2009).

34 3.4.2 Task-Based Interviews

Task-based interviews are a particular form of clinical interviews, and date back to the time of Piaget. “A clinical task-based interview can be seen as a situation where the interviewer-interviewee interaction on a task is regulated by a system of explicit and im- plicit norms, values, are rules” (Harel and Koichu, 2007). These interviews were originally used to gain a deeper understanding of the cognitive development of children. In mathematics education, task-based interviews provide a means of gaining information about a student or group of students’ mathematical inclination. In task-based interviews, the interviewees interact with both the interviewer and a particular task environment. This implies that one of the most important aspects of the task-based interviews is the task itself. If done correctly, task-based interviews can be a very eﬀective way of analyzing mathematical behavior. Ericsson and Simon (1993) recommend the following monologue to promote think-aloud taking.

Tell me everything you are thinking from the time you ﬁrst see the question until you give an answer. I would like you to talk aloud constantly from the time I present each problem until you have given your ﬁnal answer to the question. I don’t want you to try to plan out what to say or try to explain to me what you are saying. Just act as if you are alone in the room speaking to yourself. It is most important that you keep talking. If you are silent for any long period of time I will ask you to talk.

If tasks are to be used in various settings, by various interviewees, it is essential that the procedure of interviewing can be repeated. This implies that the interview protocol needs to be explicit, so that a diﬀerent interviewee can use the same task in a diﬀerent setting. However, even if extremely explicit instructions are given, there exist

35 many nonverbal events that effect the interview process. These could include nods, smiles, and other use of body language. This issue makes completely standardizing task based interviews a difficult feat. Before the interview actually takes place, the expectations and rules must be discusses between the interviewer and interviewee. This agreement between the two parties is referred to as an experimental contract. These experimental contracts promote certain types of social behavior, which in turn acts as a mediator between the subject and knowledge of that subject. One implication of the experimental contract is that the interviewee may try to offer the correct or preferred response, instead of their actually thoughts. Therefore, it should be made clear that the interest is in the student’s thought process, not a correct or incorrect answer. In the next section, I show the protocol I used to conduct the interviews.

3.4.3 Interview Protocol

Hello, my name is Blain Patterson and I am conducting interviews for my Master’s Thesis. I am interested in how you make connections to various areas of mathematics through learning cryptography. I am going to ask you a series of questions. I am interested how you think about each problem, so I would like for you to talk to me while you work. Also, I am going to keep your work, take notes, and record audio. I am provided you with a consent form that you must sign before the interview begins. Do you have any questions before we start? Please answer the following questions (with time in between). Introductory Questions

1. Why did you decide to take this course?

2. What other mathematics courses have you taken?

36 3. What was your favorite part of the course and why?

4. What was your least favorite part of the course and why?

5. Explain the diﬀerence between private and public key cryptography. Discuss advantages and disadvantages of each.

Tasks Used

6. Consider the following plaintext: NCSU.

a. Encrypt this message using a cipher of your choice.

b. Discuss the strengths and weaknesses of the cipher your choose.

7. Consider the following ciphertext: XFNFCWGRTB.

a. Decrypt this message.

b. Discuss the strengths and weaknesses of your decryption method.

8. Consider the number 8911.

a. Is it prime? Why or why not? (If student answers yes, move on to parts b and c. Otherwise, move on to question 9.)

b. How did you determine primality?

c. Factor this number.

Closing Questions

9. What areas of mathematics did you use in this course and how were they used?

10. How did this course help you make connections to other areas of mathematics?

37 3.4.4 Rationale for Protocol Questions

The first four questions were meant to learn about the participants’ background, which can provide insight to their problem solving abilities. Questions five through eight are meant to directly categorize student understand into one of the five levels of the SOLO taxonomy. Question five asks students to encrypt using any cipher of their choice and discuss its effectiveness. This is an open ended question that allows for a wide range of SOLO levels, since students are to choose methods using as much or as little mathematics as needed. In a similar way, question six allows students to use any strategies they choose, however is more limited in the possible strategies that can be implemented. Note that students were given the opportunity to use a calculator for number eight, but were still asked to show as much work as possible. Although they were asked to factor this specific number, the factoring algorithm they use will vary. All three task questions allow for a wide range of mathematics used and SOLO classifications. Once the task portion of the interview is complete, students were asked to summarize their experience in questions nine and ten. Audio clips were recorded and student worked was kept for the analysis. Additionally, the researcher recorded notes while the students were answering questions and working through problems.

3.5 Analysis

Recall that the SOLO taxonomy is being used to classify student understanding of cryptography. For classiﬁcation purposes, I am assuming that all students are functioning at the Formal mode of the SOLO taxonomy. This is due to the fact that all students participating in this study are college students. After listening to the audio recordings and reading through the pariticpants’ work, I

38 identiﬁed when a participant used something they learned in “Applications of Algebra,” something they learned in another course, or something they learned while studying independently. This was done by keeping a list of concepts covered in the course and marking when a participant used one of these concepts to solve a problem. If a concept used was not covered in the course, I asked the student to clarify where they learned that concept and how they used it to solve a given problem. If a student refused to engage in the task, they were placed in the Pre-structural SOLO level. Students who used a single relevant aspect of cryptography to solve the problems were placed in the Uni-structural SOLO level, whereas students who used multiple relevant aspects of cryptography to solve the problems were placed in the Multi-structural SOLO level, and students who made multiple connections were placed in the Multi- structural level. If a student used multiple aspects of cryptography together to solve a problem, they were placed in the Relational SOLO level. Finally, students who used knowledge outside of the scope of cryptography were placed in the Extended Abstract SOLO level, since the Extended Abstract level of the SOLO taxonomy is associated with making connections both in and outside a given subject area. Table 3.2 summarizes how the understanding of cryptography for participants A through G will be classiﬁed using the SOLO taxonomy.

39 Table 3.2: SOLO Rubric

SOLO Level Description

Pre-structural Inability or refusal to engage in the task.

Uni-structural Uses one relevant aspect of cryptography.

Multi-structural Uses several relevant aspects of cryptography.

Relational Uses multiple relevant aspects of cryptography together.

Extended Abstract Uses knowledge outside of the scope of cryptography.

Student work is described in Chapter 4 along with the classification of each student in one of the five SOLO levels. Using student work, notes taken during the interview, and audio recordings, detailed descriptions of each interview have been compiled. This was done by transcribing each of the seven interviews and written work. I then used a list of concepts (encryption methods, decryption methods, algorithms) discussed in “Applications of Algebra” and marked when a participant used the various concepts. I also made note of when a participant used a single concept, multiple concepts disjointly, multiple concepts together, or concepts outside of cryptography. Then using the rubric in Table 3.2, the understanding of cryptography for students A through G was placed in one of the five SOLO levels.

40 Chapter 4

Results

In this chapter, results are presented including student background information (previous coursework and interest in cryptography as shared in the interviews), a description of the students’ responses and some related work samples, and SOLO level classiﬁcation. Students A through G have been labeled based on the order in which the interviews took place. Each student description includes their responses to the introductory questions, solutions to the tasks, and closing questions.

4.1 Student A

4.1.1 Introduction

Student A decided to take this this course because he felt very conﬁdent in his algebraic thinking. “I did really well in linear and abstract algebra, so I thought it would be interesting to see some applications.” Student A has taken linear algebra, abstract algebra, and real analysis. He has yet to take course in number theory, but claims he has worked with number theoretic concepts before like division and modular arithmetic. Student A’s

41 favorite part of the course was working with error correcting codes, because he liked the application of vector space theory. His least favorite part of the course was the elementary ciphers, such as the Caesar and Vigenere. He felt that these could have all been covered in the ﬁrst day, so that there was more time to focus on public key cryptography.

4.1.2 Problem Solving

Student A started encrypting NCSU by assigning letters to numbers. However, Figure 4.1 shows that instead of starting at 0 and counting to 25, he started at 1 and counted to 26

Figure 4.1: Student A assigning numbers to letters.

This came as a surprise, since the student had been using the former method the entire semester. When asked about his method, student A commented “It really doesn’t matter if I start at 1 instead of 0, I just need to make sure I do everything modulo 27.” This lead the student to convert NCSU to 14, 3, 19, 21. Student A was originally going to encrypt the plaintext NCSU using an aﬃne cipher of the form A(x) = 10x mod 27, but quickly changed his mind. He thought that 10 did not have an inverse modulo 27, therefore this encryption method did not make any sense.

42 This can be seen in Figure 4.2.

Figure 4.2: Student A using an aﬃne cipher with a = 10 and b = 0.

He then used a Caesar cipher with 3 as the key, yielding 17, 6, 22, 24. When asked why he chose the key that he did, he responded by saying “This way I don’t have to do the modular arithmetic.” Finally, 17, 6, 22, 24 was converted to QFVX. This can be seen in Figure 4.3.

Figure 4.3: Student A using a Caesar cipher with 3 as the key.

When asked about the strengths of his cipher, student A stated that “It is very easy to encrypt and decrypt, making this a convenient method for the sender and receiver. The Caesar cipher is a great cipher for recreational purposes. However, if I were sending

43 a truly secret message, I would use a more complex algorithm.” For problem six, student A instantly assumed the ciphertext was encrypted using a Caesar cipher. This lead him to again assign letters to numbers. However, this time student A started numbering at 0, which is shown in Figure 4.4. When asked why, student A responded “I just ﬁgured this is how you encrypted the message.”

Figure 4.4: Student A assigning numbers to letters.

Student A then proceeded to use a brute force method. He assumed the key was 1, so proceeded to shift each letter to the left 1. After getting WEMEBVFQSA as the plaintext, student A tried the same approach with 2 as the key. However, this time he stopped after only two letters since VD was the beginning of the would be plaintext. When asked why he stopped, student A replied “I can’t think of any word that starts with VD.” For the next few attempts, he stopped as soon as the word did not make any sense. Student A then came to the conclusion that the second letter, F, must be a vowel. With this in mind, he assumed F shifted from A, E, I, O, or U. After failed attempts with A and U, he tried O, yielding the correct plaintext shown in Figure 4.5.

44 Figure 4.5: Student A using a brute force method.

Once student A was faced with the task of determining primality of 8911, he was quickly able to eliminate possible divisors of 8911. Student A claimed “I know two doesn’t go it, since it is odd. So that means no even numbers will go into it. The sum of its digits is nineteen, which is not divisible by three, so 8911 is not divisible by three. Also, it doesn’t end in a ﬁve or zero, so ﬁve doesn’t go into it either.” Without ever writing anything down, student A was nearly convinced 8911 was prime. He then tried 7 as a divisor and found that 8911 = 7 × 1273. Student A repeated the same process for 1273, once again almost coming to the conclusion that it was in fact a prime number. To be sure, he started dividing primes into 1273 and noticed that 1273 = 19 × 67. Without hesitation student A claimed that 67 was prime and wrote the prime factorization as 8911 = 7 × 19 × 67 shown in Figure 4.6.

45 Figure 4.6: Student A using a factor tree.

4.1.3 Analysis

Student A falls under the Uni-structural SOLO level. He applied the concept of modular arithmetic to encrypt NCSU using an Aﬃne cipher. However, his initially attempt to encrypt with E(x) = 10x mod 27, where he incorrectly stated that 10 does not have an inverse modulo 27. His vague understanding of modular inverses ultimately led him to a basic cipher in the form E(x) = x + 3. Besides the use of modular arithmetic, student A was unable to make any other connections to the class or to previous coursework.

4.2 Student B

4.2.1 Introduction

When asked why he took this course, student B replied “Abstract algebra and number theory have been my top two courses so far, so seeing applications would be very interesting.” Student B has taken abstract and linear algebra, real analysis, number theory, and a few geometry courses. His favorite part of the course was elliptic curve cryptography.

46 Student B said “I have always enjoyed geometry, it was amazing to geometry and algebra used together in this manner.” When asked about his least favorite part of the course, he commented “I like when mathematics is clean and well, aesthetically pleasing. Some of exponential ciphers had some heavy computation, which took away from the beauty of the theory.”

4.2.2 Problem Solving

Student B decided to encrypt the message NCSU, using a Hill cipher. The Hill cipher is done by choosing an n × n matrix K as the key and then converting the plaintext into column vectors of length n. To encrypt, one must compute Kx for each vector x. Note that K must be invertible modulo 26. This means that the determinant must not only be nonzero, but invertible modulo 26. Before any computation can be done, Student B had to convert letters to numbers and create column vectors. This can be seen in Figure 4.7.

Figure 4.7: Student B setting up a Hill cipher.

47 In Figure 4.8, student B then multiplied each vector by K modulo 26. The resulting vectors are 2 × 1, with entries from the integers modulo 26.

Figure 4.8: Student B multiplying matrices modulo 26.

Finally, student B converts all numbers back to letters resulting in the ciphertext NBSW. Student B summarized his work in Figure 4.9.

Figure 4.9: Student B summarizes his work.

When asked about the choice for entries in K, student B replied “The determinant of K will be 7, which has an inverse mod 26. In fact, that inverse is 15.” Student B felt that the Hill cipher is more diﬃcult to break than a simple shift or substitution cipher.

48 He followed by saying “The Hill cipher is a linear transformation, meaning an adversary could break it by setting up a system of equations.” Student B assumed the cipher used was substitution since the letter F was repeated. Moreover, he assumed that the message was encrypted using a Caesar cipher. Student B started problem seven by assuming that F corresponded to E, since E is the most commonly used letter in the english alphabet. This turned out to be a dead end, as decryption left him with an unintelligible word. Student B’s initial decryption attempt can be seen in Figure 4.10.

Figure 4.10: Student B using 1 as the key.

In Figure 4.11, student B then assumed that F had to correspond to a vowel, since vowels are very common.

49 Figure 4.11: Student B displaying all possible keys.

Figure 4.12 shows that after trial and error, student B came to the conclusion that the key was 17, yielding the message GOWOLFPACK.

Figure 4.12: Student B decrypting using various keys.

Student B was quickly able to signiﬁcantly shorten the list of possible divisors by taking the square root of 8911, shown in Figure 4.13. When asked why he did this, student B responded “By taking the square root, I know that if there is a divisor, it will be less than the square root of 8911.”

50 Figure 4.13: Student B reducing the total number of divisors.

Then he used a brute force algorithm to determine the ﬁrst divisor, 7. To determine whether or not 2, 3, or 5 were divisors, student B used simple divisibility tricks he recalled from number theory.

Figure 4.14: Student B factors 8911 as 8911 = 7 × 1273.

From Figure 4.14, student B now knew that 8911 = 7×1273. He paused for a moment before saying “I suppose I can just repeat this algorithm for 1273.” After repeating this for 1273 and 67, student B was able to determine that the prime factorization was 8911 = 7 × 19 × 67. Figure 4.15 shows student B looking for a prime divisor of 1273.

51 Figure 4.15: Student B repeats his algorithm for 1273.

4.2.3 Analysis

Student B falls under the Relational SOLO level. He used what knew about matrix theory and number theory to encrypt NCSU. Using these concepts together, student B was able to implement the Hill cipher by choosing a matrix with entries from the integers modulo 26. Moreover, student B had to ensure that this matrix had an inverse modulo 26. Just choosing the encryption matrix took several aspects from both number theory and matrix theory. Once a valid matrix was chosen, student B then had to use more knowledge from matrix theory to multiply each column vector by the encryption matrix. By synthesizing what he learned from the class, number theory, and matrix theory, student B was able to eﬀectively encrypt the message NCSU.

52 4.3 Student C

4.3.1 Introduction

Student C was very excited about taking this course since she had previous experience with both advanced mathematics and computer science. She had seen cryptography from a computer science stand point before and said “I was very interested in the theory behind all of the algorithms.” Student C has previously taken mathematics courses in abstract algebra, real analysis, number theory, and diﬀerential equations. She also mentioned her experience in computer science course such as programming (Java, Python, C), algorithms, computer architecture, and data structures. When asked about her favorite part of the course, student C responded, “I really enjoyed learning the theory behind all of the algorithms I already knew.” She mentioned that she struggled with the elliptic curve cryptography, claiming “I have not taken a geometry courses since high school.” Addi- tionally, student C claimed she did not care for any of the block ciphers. “I understand why they are used, they just seem so random, I like when algorithms are well deﬁned and easy to follow.”

4.3.2 Problem Solving

Student C decided that she was going to encrypt the message NCSU using a one-time pad. This was interesting, since this particular cipher was not covered in the course. The one-time pad is theoretically unbreakable if done correctly (Kahn, 1997). This is because the plaintext is encrypted using a random secret key. This key is often referred to as the one-time pad. Then each letter of the plaintext is encrypted using the corresponding letter of the key using modular addition. One can think of the one-time pad as a Vigenere cipher, where the key word is random instead of an actual English (or perhaps another

53 language) word. To begin encrypting using this method, student C did only wrote down the alphabet and attempted to just count. She quickly realized this may cause errors, leading her to perform the standard number to letter conversion shown below in Figure 4.16.

Figure 4.16: Student C assigning numbers to letters.

The next step was to generate a random key. Student C has a strong background in computer science, so she is aware of the diﬀerence between true random and pseudo random. When it came time to generate a random key, student C decided that she was just going to come up with a string of four letters, such that these letters did not create an English word or acronym. “This is more diﬃcult than I thought. When I try to come up with random letters for the key, I just end up using patterns to come up with those letters.” This led her to close her eyes and point to numbers to create the key 13, 12, 21, 11. Student C then simply performed modular addition to yield the ciphertext AONF, shown in Figure 4.17 and Figure 4.18.

54 Figure 4.17: Student C encrypting use the one-time pad.

When asked why she used numbers rather than letters for the key, student C responded “Generating random numbers seems easier than generating random letters. Anytime the string of letters was too close to a word, I wouldn’t use it. If all letters are equally likely , then ABCD is just as likely as HELP.”

Figure 4.18: Student C displaying the ciphertext.

Student C began decrypting XFBFCWGRTB by assuming this was shifted by 25. She claimed that E was the most common letter in the alphabet, so assuming F (which appears twice) came from an E is reasonable. However, when student C decrypted the entire message with 25 as the key, the resulting plaintext was WEMEBVFQSA. This can be seen in Figure 4.19. She assumed this was not what the original message said, so she moved on.

55 Figure 4.19: Student C using 25 as a key.

Before continuing, student C paused for a moment and said, “If I do this by brute force, this is going to take a while. I think I can make this easier on myself.” She then proceeded to look at letter pairs, instead of single letters. “There are deﬁnitely a lot less two letter combinations to try, since only so many of them will make sense.” Student C then listed all two letter combinations that made sense to her, shown below in Figure 4.20.

56 Figure 4.20: Student C listing possible two letter combinations.

Student C began checking if the rest of the message made sense with various shifts. “Actually, I doubt that a word will begin with CK, so I’m going to skip that one. Also, I think that GO is a more likely beginning to a word or phrase than EM, so I will start with that.” This resulted in the plaintext, GOWOLFPACK, shown below in Figure 4.21.

Figure 4.21: Student C displaying the plaintext GOWOLFPACK.

After writing down that the key was 9, student C said “Well actually the key is -9, which is really 17 mod 26. Counting back 9 was just easier than counting forward 17.”

57 When given the number 8911, student C stated “That sure looks prime, but since its four digits, I should check some smaller prime factors just to be safe. The square root of 8911 is less than 95, so I guess I could check those factors.” Shown below in Figure 4.22 √ is student C approximating 8911.

√ Figure 4.22: Student C approximating 8911.

Student C then began to check the ﬁrst several prime numbers. “I remember some basic divisibility tricks from number theory like to be divisible by three the sum of a its digits must be divisible by three.” The ﬁrst prime divisor student C discovered was 7. This left her with 8911 = 7 × 1273, shown in Figure 4.23.

58 Figure 4.23: Student C dividing 8911 by 7.

“I suppose I can just run the same algorithm with 1273, except this time, I can just start with 7.” When asked why, she claimed “If 3 didn’t go into 8911 then it can’t go into any of its factors. 7 may go into it twice, so I need to check that again.” She proceeded to check the next several prime numbers, shown in Figure 4.24.

Figure 4.24: Student C checking 7 and 11 as factors .

After testing factors 7, 11, 13, 17, and 19, student C determined that 19 was a divisor of 1273, which resulted in the factorization 8911 = 7 × 19 × 67. She was certain that 67 was prime, thus claimed that she had completely factored 8911. Student C’s ﬁnal

59 factorization can be seen in Figure 4.25 below.

Figure 4.25: Student C displaying the prime factorization of 8911.

4.3.3 Analysis

Student C falls under the Extended Abstract SOLO level. Not only was she able to apply what she learned from the class, she applied concepts from her undergraduate coursework in computer science. Student C used the one-time pad encryption method, an extremely secure encryption method she learned from a computer science based cryptography course. Additionally, she was aware they the one-time pad was nothing more than a Vigenere cipher, with a random key. By synthesising what she learned both in and outside of class, student C was able to successful encrypt the message NCSU and discuss the eﬀectiveness of her encryption method.

60 4.4 Student D

4.4.1 Introduction

Student D had taken courses with this professor before and though that this course sounded interesting. Additionally, she knew the course would have a light workload so it would improve her grade point average. Prior to this course, student D had taken courses in linear and abstract algebra, topology, diﬀerential equations, numerical analysis, and matrix theory. This explains her favorite part of the course. Student D claimed “I guess applying Markov chains to cryptography. I never saw that connection before.” Her only complaint about the class was its early time slot. Since student D focuses on applied mathematics, she stated “I have a really hard time seeing how pure areas of math like abstract algebra or analysis can be used in applications, so doing that allowed me to make some connections with modeling that I wouldnt have otherwise.”

4.4.2 Problem Solving

Student D decide to encrypt the message NCSU with a Vigenere cipher using the keyword SUP, shown below in Figure 4.26.

61 Figure 4.26: Student D using a Vigenere cipher with keyword SUP.

When asked about her motivation for using the Vigenere cipher, she replied “I like this cipher a lot. It is simple enough that one could encrypt using it without any advanced mathematical knowledge. Yet, it it is poly-alphabetic, so it is much safer than a simple substitution cipher.” When faced with the ciphertext XFNFCWGRTB, Figure 4.27 shows student D instantly thought about frequency analysis. “If this message was encrypt using a substitution cipher, the F that appears twice would correspond to the same letter.”

Figure 4.27: Student D using frequency analysis.

On her ﬁrst attempt, student D assigned F to O and paused to think about what to do with the rest of the ciphertext. She then stated “I have a feeling this was encrypted with

62 a Caesar cipher.” When asked why, she replied “It just seems like the most systematic substitution cipher, decrypted a random substitution cipher would take a great deal of trial and error.” She then realized that to get from O to F, one would need to shift the alphabet left by 17. Once student D implemented this shift, she deduced that the plaintext was GOWOLFPACK. Student D’s ﬁnal decryption can be seen in Figure 4.28.

Figure 4.28: Student D decrypting using a Caesar cipher.

When faced with the number 8911, student D thought that it initially looked like a prime number. “In my experience, it seems like a lot of numbers that end in one are prime. To be safe, I guess I’ll start by checking some small primes.” Student D checked prime numbers 2, 3, and 5 without doing any arithmetic. When asked how she knew those were not divisors, she replied “Well 8911 is odd so obviously two doesn’t go into it. If you sum the digits of 8911, 8 + 9 + 1 + 1, you get 19, which is not divisible by three, so 8911 isn’t either. Then 8911 doesn’t end in a zero or ﬁve, so it isn’t divisible by ﬁve.” Figure 4.29 below shows student D checking small prime divisors of 8911.

63 Figure 4.29: Student D checking small prime divisors of 8911.

The ﬁrst prime divisor student D discovered was 7, allowing her to factor 8911 as 8911 = 7 × 1273. She then said “I guess I can just use the same brute force procedure on 1273.” Student D then proceeded to check small prime numbers using the same divisibility tricks until she concluded that 19 was a divisor of 1273, shown in Figure 4.30 below.

Figure 4.30: Student D checking small prime divisors of 1273.

Student D then had 8911 expressed as 8911 = 7 × 19 × 67. She paused for a moment and was unsure if 67 was prime or not. “It seems like a weird number, one I don’t run into that often. It looks prime, but I’ll check just to be sure.” Student D only wrote down primes 2, 3, 5, and 7. She claimed “The square root of 67 is less than nine, so I really only need to check primes less than nine.” She was quickly able to deduce that 67 was in fact

64 prime. This led the prime factorization 8911 = 7 × 19 × 67, shown in Figure 4.31.

Figure 4.31: Student D checking small prime divisors of 67.

4.4.3 Analysis

Student D falls under the Multi-structural SOLO level. She was able to eﬀectively use the Vigenere cipher, a cipher she learned in class, to encrypt the message NCSU. Ad- ditionally, student D used frequency analysis to aid in the decryption of the message XFNFCWGRTB. Finally, she used elementary number theory to factor 8911 into prime components. Although student D was able to make multiple connections, they were all disjoint. In other words, no synthesis of information occurs. This did not, however, hinder her ability to successful solve the given problems.

4.5 Student E

4.5.1 Introduction

Student E’s motivation for taking this course was primarily to increase her grade point average. She said “MA 792 was a special topics course, which should be an easy A.” Her previous coursework includes real and numerical analysis, abstract, linear, and Lie

65 algebra, matrix theory, and topology. When asked about her favorite part of the course, student E replied “The basic ciphers where we assign letters to numbers, because it was simple and easy.” Student E deﬁnitely did not care for any algorithmic process. In fact she stated “I always struggle with ﬁnding the inverses of numbers mod something with the algorithm that I always forget. Also, it takes forever.” Student E studies pure mathematics, which is evident in her resistance to “messy” algorithms.

4.5.2 Problem Solving

Student E chose to encrypt the message NCSU using a Vigenere cipher with GO as the keyword, as seen in Figure 4.32.

Figure 4.32: Student E using a Vigenere cipher with keyword GO.

She was able to encrypt without converting letters to numbers and shifting. She chose to encrypt with a Vigenere cipher because “Its poly-alphabetic, yet is easy to do. In fact

66 encrypting using a Vigenere cipher is much strong than an Aﬃne cipher, yet is easier to encrypt.” To decrypt the message XFNFCWGRTB, student E started assigning numbers to letters. However, instead of starting at 0 as done in class, she started by assigning 1 to A, shown below in Figure 4.33.

Figure 4.33: Student E assigning numbers to letters.

She quickly realized that if the message had been encrypted using a substitution cipher, the numbers she assigned to the letters are arbitrary. When asked why she assumed a substitution cipher, she said ”The second and fourth letters are repeated, which makes me think they are both vowels, maybe E’s.” Student E not only assumed a substitution cipher, but a Caesar cipher. With this in mind, she attempted to shift the F back to an E and found that the resulting message was invalid. This can be seen in Figure 4.34.

Figure 4.34: Student E using 1 as they key.

After a few more attempts of trial and error, student E was able to successfully decrypt

67 the message XFNFCWGRTB to GOWOLFPACK, as seen in Figure 4.35. She felt that her method of decryption was rather eﬃcient, since it only took her three attempts. She stated “I knew that E occurs most frequently in the english language, so that was a good place to start. Also, T’s seem to be very common as well. After just a few letters, the word didn’t make sense, so I knew T wasn’t correct.”

Figure 4.35: Student E decrypting using various keys.

Student E decided to factor 8911 using the Pollard Rho method. To perform this algorithm one must take an integer, x, then compute y = x2 + 1 mod n, where n is the number to be factored. Now compute the greatest common divisor of |x − y| and n. If these numbers are relatively prime, try again. Otherwise the greatest common divisor is a factor of n. Student E’s algorithm can be seen in Figure 4.36.

68 Figure 4.36: Student E using the Pollard Rho factoring algorithm with n = 8911.

Once student E knew 7 was a factor, she divided 8911 by 7 to ﬁnd 8911 = 7 × 1273, as seen in Figure 4.37.

Figure 4.37: Student E factors 8911 as 8911 = 7 × 1273.

She then ran through the algorithm again, shown below in Figure 4.38, letting n = 1273. Using 2, 5, and 26 as values for x all yielded no results. Student E then chose 677 as the value for x, which resulted in 19 as a factor of 1273.

69 Figure 4.38: Student E using the Pollard Rho factoring algorithm with n = 1273.

After computing 1273 ÷ 19, she found the prime factorization of 8911 to be 8911 = 7 × 19 × 67, shown below in Figure 4.39.

Figure 4.39: Student E factors 8911 as 8911 = 7 × 19 × 67.

4.5.3 Analysis

Student E falls under the Multi-structural SOLO level. She was able to eﬀectively use the Vigenere cipher, a cipher she learned in class, to encrypt the message NCSU. Additionally, student E used frequency analysis to aid in the decryption of the message XFNFCW-

70 GRTB. Finally, she used the Pollard Rho factoring algorithm to factor 8911 into its prime components. Although she made strong connections within individual questions, student E was unable to use these connections together to solve a given problem.

4.6 Student F

4.6.1 Introduction

Student F’s background varies quite drastically from all other participants. Since her major of study as an undergraduate was engineering, she has yet to be exposed to rigorous number theory or abstract algebra. However, she also claimed to have a strong background in linear algebra. She has always been interested in cryptography. In fact, she has experience as a teaching assistant for a cryptography course. Although she does not have the formal background in number theory, she has a strong grasp on modular arithmetic, one of the core foundations of this class. Student F claims that her favorite part of the course was learning about RSA. “Breaking RSA relies on solving such a simple problem, yet actually doing the computation takes too long. Public key cryptography is very intriguing.” Her least favorite part of the course had to be all of the group theory. She felt it was boring and unnecessary. “This may have to do with my lack of experience with pure math.”

4.6.2 Problem Solving

Student F decided to encrypt the message NCSU using an Affine cipher. “The affine cipher is still a substitution cipher, so people could do frequency analysis to break it. But the Affine cipher is more secure than the Caesar cipher, since the Caesar cipher only has 25 possible keys.” Using the encrypting formula E(x) = ax + b mod 26, she chose a = 3

71 and b = 5. Figure 4.40 shows that without writing down the entire alphabet, student F knew which numbers the letters N, C, S, and U corresponded to.

Figure 4.40: Student F assigning numbers to letters.

She then proceeded to compute E(x) = 3x + 5 mod 26 for x = 13, x = 2, x = 18, and x = 20. Converting these numbers back to letters, student F was left with the ciphertex SLHN, as seen in Figure 4.41.

Figure 4.41: Student F using an Aﬃne cipher with a = 3 and b = 5.

Student F quickly assumed the message was encrypted using a substitution cipher. “There are two F’s, so I think there is a good chance that F corresponds to the same letter. I’m guessing that letter is an E.” She then shifted all letters in the ciphertext

72 under the assumption that F came from an E. This left her with the unintelligible text WEMEBVFQSA, shown below in Figure 4.42.

Figure 4.42: Student F attempting to decrypt with 1 as a key.

Still focused on discovering what letter F came from, student F attempted a few more shifts. She thought that F may been shifted from an A, T, or I. However these shifts all left student F with plaintext that did not make any sense. A ﬁnal shift from the letter O, resulted in the plaintext GOWOLFPACK. This can be seen in Figure 4.43.

73 Figure 4.43: Student F attempting to decrypt with various keys.

When presented with the number 8911, student F seemed unsure about the primality. “For some reason this number just looks prime. I feel like a lot of numbers that end in 1 are prime like 11, 31 and 41.” She started testing prime divisors and found that 7 was a factor, as seen in Figure 4.44 and Figure 4.45.

Figure 4.44: Student F testing prime divisors of 8911.

74 Figure 4.45: Student F factoring 8911 as 8911 = 7 × 1273.

Student F ﬁgured the best way to check the primality of 1273 would be to repeat this process. “I already tried divisors up to 7, so I don’t need to check anything smaller.” She proceeded to check prime divisors until she found that 1273 = 19 × 67, shown in Figure 4.46.

75 Figure 4.46: Student F testing prime divisors of 1273.

Without hesitation, student F was sure that 67 was prime. She then displayed the complete factorization of 8911 using a factor tree and by also listing the factors. This can be seen in Figure 4.47.

76 Figure 4.47: Student F displaying the prime factorization of 8911.

4.6.3 Analysis

Student F falls under the Uni-structural SOLO level. She used her prerequisite knowledge of modular arithmetic to encrypt the message NCSU. Before using an Aﬃne cipher (E(x) = ax + b mod 26), student F had to determine appropriate values for a and b. She knew that b could be any integer between 0 and 25 and that a had to have an inverse modulo 26. Using these number theoretic concepts, student F was able to successfully encrypt the message NCSU. However, besides the use of modular arithmetic, student G was unable to make any connections to the class or to previous coursework.

4.7 Student G

4.7.1 Introduction

Student G has taken courses in Algebra and Number Theory, which she feels are very important to know when studying cryptography. Additionally, she has taken a cryptography course in the past, but claims that this course was much more rigorous and theoretical. When asked about her interest in cryptography, student G responded with the following.

77 Codes are essential in this technological age. We can make purchases online and transfer money safely because the codes are secure and protect our information. Without secure codes, it would be difficult to keep important information from getting in the wrong hands. Strong codes are important for both the individual and the country’s well-being. These codes are so important and powerful, and they work all because of properties of numbers and their factorizations. I am interested in the power of these mathematical properties and what they enable us to do. There are a lot of interesting research questions that result from these properties in relation to cryptography, like trying to write an unbreakable code. I find cryptography interesting because the mathematical ideas behind it are simple (ex factorization), yet actually very difficult to carry out if a number is large enough. Cryptography reminds me of a puzzle and you have to be clever to mathematically encrypt or decrypt.

4.7.2 Problem Solving

Student G began encrypting the message NCSU by assigning numbers to letters. However, instead of starting with 0 and numbering to 25 as done in class, she began at 1 and numbered to 26. “I won’t be using modular arithmetic, so the correspondence between the numbers and letters is doesn’t really matter.” She also assigned 0 to a space between words. Student G’s encryption scheme can be seen in Figure 4.48 below.

78 Figure 4.48: Student G assigning numbers to letters.

She then partitioned NCSU into two groups of two letters NC and SU, where NC corresponded to 14, 3 and SU corresponded to 19, 21. These letters were represented by column vectors as shown below in Figure 4.49.

Figure 4.49: Student G encrypting the message NCSU.

It appeared that student G was going to use the Hill cipher to encrypt NCSU, but at this point she stopped. She said “It’s really just a simple substitution cipher, where each vector corresponds to a two letter pair. I guess it’s really only secure if the adversary is unaware of the cipher used. If someone knows the scheme, it would be extremely easy to decrypt any ciphertext.” Student G’s cipher is summarized in Figure 4.50.

Figure 4.50: Student G summarizes her work.

79 Assuming the plaintext was encrypted using a Caesar cipher, student G noticed that the letter F appeared twice. This lead her to conclude that F corresponded to a vowel in the plaintext. “To minimize the number of shifts I have to try to decode the message, I tried to strategically guess what certain letters correspond to. For instance, since there are two F’s close to each other in the code, I assumed F would likely correspond to a vowel.” Student G attempted the corresponding shifts for A, E, and found the plaintext to be unintelligible. Once she shifted F to O along with the remainder of the alphabet, she was left with the following transformation in Figure 4.51.

Figure 4.51: Student G shifting the alphabet by 17.

By shifting every letter in the alphabet by 17, student G was left with the plaintext GOWOLFPACK as seen in Figure 4.52.

Figure 4.52: Student G shifting the alphabet by 17.

Student G began problem number 8 by checking for small prime factors. “I can’t really think of a better way than brute force. I really hope that I don’t have to check that many. I’m going to use some basic number theory tricks to check small primes.” Figure

80 4.53 below shows student G checking for small prime divisors of 8911.

Figure 4.53: Student G checking for small prime divisors of 8911.

This lead to the factorization 8911 = 7 × 1273. Repeating her algorithm, student G began to factor 1273 by checking small prime numbers, starting with 2 once again. Once student G reached 19, she had factored 1273 as 1273 = 19 × 67. This can be seen below in Figure 4.54.

Figure 4.54: Student G checking for small prime divisors of 1273.

Without hesitation, student G claimed that 67 was prime and wrote her ﬁnal factorization as 8911 = 7 × 19 × 67, as seen in Figure 4.55.

81 Figure 4.55: Student E factors 8911 as 8911 = 7 × 19 × 67.

4.7.3 Analysis

Student G falls under the Multi-structural SOLO level. She made connections to matrix theory to encrypt the message NCSU by equating groups of letters to column vectors. Student G also made connections to number theoretic concepts, which helped her eﬃ- ciently factor 8911 into its prime components. However, student G was unable to make signiﬁcant connections to topics that were not directly covered in “Applications of Alge- bra.” Also, the connections that student G did make were completely disjoint and where not used coherently to solve a given problem. Although her work suggests that student G understands individual concepts from the course, she is unable to see how these parts form a whole.

4.8 Summary

With the exception of the Pre-structural stage of the SOLO taxonomy, there were students functioning at all SOLO levels. Out of seven students interviewed, two fell in Uni-structural, three fell in the Multi-structural, one fell in Relational, and one fell in Extended Abstract. Students who where categorized in the Uni-structural state made a single connection to the class. Those who fell in Multi-structural where able to two or more connections to the class when solving the given problems. The students who were

82 categorized in the Relational and Extended Abstract levels both synthesized information to problem solve. The diﬀerence is that the student who fell in the Extended Abstract SOLO level was able to information outside the scope of the course and connect it to they learned in the course. Table 4.1 summarizes the SOLO classiﬁcations for students A through G.

Table 4.1: Student SOLO Classiﬁcation

Student SOLO Level

A Uni-structural

B Relational

C Extended Abstract

D Multi-structural

E Multi-structural

F Uni-structural

G Multi-structural

It was most common that students were functioning in the Multi-structural level of the SOLO taxonomy. There were no instances of participants not responding or refusing to engage in the task, therefore every participant was at least at the Uni-structural level. It was surprising, however, that two of the seven participants did fall in the Uni-structural SOLO level, since this level requires very little understanding of the subject matter. A variety of strategies were used including methods from number theory, abstract algebra, linear algebra, and probability. There exists a relationship between the SOLO

83 level a student fell in and the areas of mathematics that were used to problem solve. Students functioning at the Uni-structural level typically used number theoretic concepts to solve the given problems. Those functioning at the Multi-structural level used strategies from number theory and probability separately. Since only one student was placed in the Relational and Extended Abstract levels, a relationship between the SOLO level and strategies used cannot be established. Since I assumed that all participants in this study were all in the formal mode of the SOLO taxonomy because of their age, they were quite aware of exactly what what areas of mathematics they used to problem solve. Following the task questions, all students were asked questions nine and ten below.

9. What areas of mathematics did you use in this course and how were they used?

10. How did this course help you make connections to other areas of mathematics?

Question nine was used to conﬁrm their SOLO classiﬁcation. All students described exactly what areas of mathematics they used to solve the problems. All participants used number theory when solving the cryptography problems. This may be due to the fact that several of the elementary ciphers discussed in “Applications of Algebra” were based on number theoretic concepts. The two participants who were functioning at the Uni-structural level both used number theory alone to problem solve. The three participants who were functioning at the Multi-structural level used both number theory and probability, but used them disjointly rather than together. The one participant who was functioning at the Relational level used number theory, linear algebra, and probability. What distinguished this participant from the others using multiple aspect of cryptography, was the fact that he used number theory and linear algebra together to implement the Hill cipher. Finally, the one participant functioning at the Extended Ab-

84 stract level used and applied knowledge from computer science by encrypting a message using the one-time pad method. Question ten was not used for research purposes, rather to prompt the student to think about how they might apply what they learned in this course to future coursework. Table 4.2 summarizes student responses to question nine.

Table 4.2: Responses to Question 9

Student Mathematics Used

A Number Theory

B Number Theory, Linear Algebra, and Probability

C Number Theory and Computer Science

D Number Theory and Probability

E Number Theory and Probability

F Number Theory

G Number Theory and Probability

85 Chapter 5

Conclusion

In this chapter, I interpret the results of the research and discuss pedagogical implications for of the teaching and learning of cryptography. I also discuss the possibility of assessing the use of technology to solve cryptography problems.

5.1 Interpretation of Results

Recall the following research questions.

1. How does studying cryptography enable students to use their understanding of diﬀerent advanced mathematical content?

2. What diﬀerent SOLO levels do students exhibit when they are solving cryptography problems?

Analyzing these interviews showed that students have a strong foundation in number theoretic concepts, such as divisibility, primality testing, prime factorization, and modular arithmetic. In general, deep connections were not made between abstract algebra, linear algebra, and probability. Several students claimed that number theory is closely related

86 to abstract algebra. Under this assumption, students who made connections to number theory also made connections to abstract algebra. The knowledge of number theoretic concepts enabled students to effective encrypt using Caesar and Affine ciphers and to determine primality of integers. These concepts can be natural introduced to students using cryptography. Instead of just presenting students with the idea of modular arithmetic, teachers can first introduce students to Caesar ciphers. The act of wrapping the alphabet around to the beginning is an intuitive model for modular arithmetic. If presented in the context of cryptography, topics such as modular arithmetic can be accessible to even middle school students. Students had tendencies to focus on more concrete topics from abstract algebra. For example, instead of encrypting messages using the algebraic structure of elliptic curves or finite polynomial fields, students choose to use the integers modulo 26. This provided students with a more concrete structure to encrypt with. Students only used probability as a way to build on their intuition. There was no instance where students used formal probability theory, but used intuitive ideas about the frequency of the English alphabet. This is most likely because none of the problems required more formal reasoning. According to Singh (1999), student intuitions can be extremely helpful in developing sound conceptual understanding of probability. Stu- dents used these preconceptions about the English alphabet to develop algorithms for decryption. There was also a tendency for students to display their algorithms in both a graphical and algebraic manner. For example, several students used both a factor tree and the division algorithm for integers to factor 8911. This enabled students to display the complete factorization of 8911 in a way that that is clear to both the student and the interviewer. The results from the task-based interviews showed that most common SOLO classifi-

87 cation was the Multi-structural level. Students in this level have a clear understanding of cryptography, but may fail to see how various areas of mathematics can be used together to solve cryptography problems. Since every student engaged in the task, there was no instance of students functioning at the Uni-structural level. More students functioned at the Uni-structural level (two students) than the Relational (one student) or Extended Abstract (one student) levels.

5.2 Limitations

There are several limitations of this study. The experiment was conducted with students in a mathematics elective course, meaning students taking this course did so because of their interests and strengths in abstract algebra and number theory. The results may vary if a diﬀerent demographic of students was chosen, such as major or age group. For example, a similar experiment could be done with students in elementary, middle, or high school. The researcher would have vary the questions asked and keep in mind that these students have been exposed to less mathematics. This experiment was done on a small scale. Because of the small sample of students, this could be a limitation. Although the majority of students in this study were functioning at the Multi-structural SOLO level, this does not imply the same for a larger group of students. It would be beneﬁcial to perform a similar experiment with a much larger sample size to observe the distribution of SOLO levels.

5.3 Implications of the Study

This research aﬃrms that cryptography promotes problem solving and critical thinking. A central focus of cryptography is that we do not know how to solve certain problems

88 efficiently (Koblitz, 1997). Often students think that any mathematical problem can be solved with a formula (Koblitz, 1997). This was clearly not the case for those participating in this study. When asked to decrypt ciphertext, participants would often try one method, reflect and realize that it was not feasible, then attempt another method. In order to solve problems in cryptography, students have to think critically and be engaged in a productive struggle. The SOLO taxonomy provided an efficient means for categorizing student understanding of cryptography. Ideally more students would be functioning in the Relational and Extended Abstract levels. This implies that students should see connections between mathematical ideas early and often in their academic careers. One way to ensure that this happens is to introduce concepts through cryptography. Studying ciphers for example, is a natural way to introduce functions and discuss concepts such as domain, range, inverses, composition, one-to-one, and onto. Additionally, cryptography lends itself to the use of intuitive probability. These intuitive concepts students have, such as the distribution of the alphabet, can be used as a foundation to build mathematical knowledge. Cryptography is a beautiful subject that should be used as facilitator of mathematics from kindergarten through college.

5.4 Future Research

Future research includes looking at how students solve cryptography problems with various types of technology. Since cryptography encompasses so many ﬁelds of mathematics, one could research how students interact with Computer Algebra Systems (CAS), Sta- tistical Software, Dynamic Geometry Software (DGS), and programming languages such as Python, C++, Java, etc. Another task-based interview could be conducted, where stu-

89 dents are asked to write short programs to solve various problems. This would require both the interviewer and interviewee to have the prerequisite knowledge of coding. Also, more research would have to be done in the ﬁeld of technology education.

5.5 Conclusions

Through this experiment I have come to the conclusion that students are able to make connections to various areas of mathematics through studying cryptography. However, the connections students made were mostly from the domain of number theory. In this particular study, students were able to use various number theoretic concepts to solve cryptography problems. I conclude that students see a strong connection between cryptography and number theory, but fail to see deep connections in other areas such as abstract algebra, linear algebra, and probability.

90 Chapter 6

References

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94 APPENDICES

95 Appendix A

Interview Protocol

1. Why did you decide to take this course?

2. What other mathematics courses have you taken?

3. What was your favorite part of the course and why?

4. What was your least favorite part of the course and why?

5. Explain the diﬀerence between private and public key cryptography. Discuss advantages and disadvantages of each.

96 6. Consider the following plaintext: NCSU.

a. Encrypt this message using a cipher of your choice.

b. Discuss the strengths and weaknesses of the cipher your choose.

7. Consider the following ciphertext: XFNFCWGRTB.

a. Decrypt this message.

b. Discuss the strengths and weaknesses of your decryption method.

8. Consider the number 8911.

a. Is it prime? Why or why not? (If student answers yes, move on to parts b and c. Otherwise, move on to question 9.)

b. How did you determine primality?

c. Factor this number.

9. What areas of mathematics did you use in this course and how were they used?

10. How did this course help you make connections to other areas of mathematics?

97 Appendix B

Answer Sheets

NCSU

98 XFNFCWGRTB

99 8911

100 Appendix C

Solutions

6. NCSU can be encrypted any way student chooses.

7. XFNFCWGRTB should be decrypted as GOWOLFPACK.

8. 8911 = 7 × 19 × 67.

101 Appendix D

Student Work

102 Figure D.1: Student A problem 6.

103 Figure D.2: Student A problem 7.

104 Figure D.3: Student A problem 8.

105 Figure D.4: Student B problem 6.

106 Figure D.5: Student B problem 7.

107 Figure D.6: Student B problem 8.

108 Figure D.7: Student C problem 6.

109 Figure D.8: Student C problem 7.

110 Figure D.9: Student C problem 8.

111 Figure D.10: Student D problem 6.

Figure D.11: Student D problem 7.

112 Figure D.12: Student D problem 8.

Figure D.13: Student D problem 8 continued.

113 Figure D.14: Student E problem 6.

114 Figure D.15: Student E problem 7.

115 Figure D.16: Student E problem 8.

116 Figure D.17: Student F problem 6.

117 Figure D.18: Student F problem 7.

118 Figure D.19: Student F problem 8.

119 Figure D.20: Student G problem 6.

120 Figure D.21: Student G problem 7.

121 Figure D.22: Student G problem 8.

122 Appendix E

Participation Email

Hello, My name is Blain Patterson and I am conducting interviews for my Masters Thesis. I am interested in how you make connections to various areas of mathematics through learning cryptography. I would like to ask you some questions and have you do some problems related to what you have learned this semester. The interviews will take approximately 45 minutes. Would you consider being interviewed for this purpose? If so, you will need to sign a consent form before the interview begins. Best, Blain Patterson, B.S. Graduate Student Mathematics Education North Carolina State University Raleigh, NC 27695

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