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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 cryptog- raphy 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 probabil- ity 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 All Rights Reserved Cryptography: Decoding Student Learning 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 under- graduate 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.
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