Euclid's Number Theory Research Paper

1. Abstract 2. Acknowledgements
Table of Contents
1. Abstract ………………………………………………………………………………………… 2
2. Acknowledgement ………………………………………………………………………….. 3
3. Introduction …………………………………………………………………………………… 5
4. Number Theory and Modular Arithmetic ………………………………………..
4.1. Modular Arithmetic
4.2. Congruence Modulo 6

5. Number Theory and Prime Numbers ………………………………………………
5.1. Properties of Primes
5.1.1. Euclid’s Proof (Infinity of Primes)
5.1.2. Fermat’s Little Theorem
5.1.3. Euler’s φ Function 8

6. Standard RSA Cryptography ……………………………………………………………
6.1. Key Generation
6.2. Encryption
6.3. Decryption: Using Euler’s Theorem
6.4. Worked Example 11
7. RSA: Safe and Secure …………………………...…………………………………………. 14
8. Multi-prime RSA ……………………………………………………………………………... 15
9.

A public key is generated so that a sender can use it to encrypt a message and the receiver uses a private key, which is kept secret, to decrypt the message only by him. For this section, we will use Alex as a sender and Emily as a receiver.
4.1 Key Generation
This is the first process of the RSA cryptography. In this process, the receiver, Alex, generates two keys for encoding and decoding. There are 5 main steps involved in this process.
The first step is to choose two distinct prime numbers p and q. These numbers should be very big and chosen at random because firstly, it is very difficult to identify really big prime numbers and secondly, choosing them at random makes it really hard for an adversary to try to decrypt a secret message.
The next step is to compute the number n by simply multiplying the two prime numbers p and q; n = pq. Here n is used as a modulus for both the public key and the private key.
The third step is to compute ϕ(n) = ϕ(p)ϕ(q). From the Euler’s ϕ Function, we know that ϕ(p) = (p – 1), thus we can say that ϕ(p)ϕ(q) = (p – 1)(q –