Number Theory Applications in Cryptography

This thesis provides a unique cryptosystem comprised of different number theory applications. We first consider the well-known Knapsack Problem and the resulting Knapsack Cryptosystem. It is known that when the Knapsack Problem involves a superincreasing sequence, the solution is easy to find. Two cryptosystems are designed and displayed in this thesis that allow two parties often called Alice and Bob use a common superincreasing sequence in the encryption and decryption process. They use this sequence and a variation of the Knapsack Cryptosystem to send and receive binary messages. The first cryptosystem assumes that Alice and Bob agree on a shared superincreasing sequence prior to beginning encryption. The second cryptosystem involves Alice and Bob constructing a common, secret, superincreasing sequence built from subsequences of the Fibonacci sequence during the encryption process. Elliptic curves were explored on a smaller scale as they are also applied in cryptography. For a fixed prime number p and a special class of elliptic curves over Zp, we investigate how many of them intercept the y-axis. Additionally, the research presented in this paper was successfully implemented into a middle school classroom. Chapter 1 includes introductory material about cryptography. Chapter 2 discusses superincreasing sequences and their appearance in Fibonacci subsequences. It also includes important properties of the Fibonacci sequence. The two cryptosystems are presented in Chapter 3 followed by the brief findings of the intersection of elliptic curves with y-axis in Chapter 4. Finally, Chapter 5 introduces a middle school lesson plan that provided students the experience of cryptography and increased their appreciation of mathematics. A few other lesson plans are provided in the appendix