Introduction

Euler’s number was first introduced to me when I was investigating the methods of encryption to passkeys and RSA algorithms. I was researching the methods of encrypting keys and discovered how Euler’s totient can be used to scramble the public key in a private-public key combination in order to unlock and grant access to digital spaces and logins. This interested me due to me immense fascination in online security and hacking and so I decided to discover the presence of math in encryption algorithms. Other details about Euler suggested that he was a highly regarded mathematician in the 1700s and is responsible for many of the first proofs for conjectures and other mathematic problems. His discovery of the totient theorem and also the use of modular arithmetic will be used to discover its involvement with encryption.

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Euler’s Totient Theorem

In number theory, Euler’s totient1 theorem states that if n and a are coprime positive integers, then

a?n ? 1 (mod n)

Where ?n is Euler’s totient function

Defining Modular Arithmetic

Modular arithmetic deals with the relationship between a repeating mathematical function, exampled popularly as a clock, where the hour hand will move around all the way to 12 and then reset to 0. This is how modular functions work as they always suggest a pattern of recurring numbers. If we have A mod B and we increase A by a multiple of B, its results in the same spot ie.

A mod B = (A+K x B) mod B for any integer K.

Example,

3 mod 10=3

13 mod 10=3

23 mod 10=3

33 mod 10=3

A?B (mod C)

Modular arithmetic can be described mathematically by suggesting a congruence relation between integers that are compatible with the operations on integers: addition, subtraction, and multiplication. For positive integer n, two numbers a and b are said to be congruent modulo n, if the difference (a – b) is an integer multiple of n (that is, if there is an integer k such that a ? b = kn).

Euler’s Totient Function

Euler’s totient function,2 pictured as ?n, is a collection and analysis of the numbers that are smaller than n and relatively prime to n.

For example, ?10 is 4 because there are 4 numbers that are smaller than 10 and that are relatively prime to 10. (1, 3, 7, 9). ?6 is 2 as 1 and 5 are relatively prime to 6 and 2, 3, and 4 are not.

The following operation of totients of numbers 2 through 20 it listed in the chart.

n

?n

2

1

3

2

4

2

5

4

6

2

7

6

8

4

9

6

10

4

11

10

12

4

13

12

14

6

15

8

16

8

17

16

18

6

19

18

20

8

Examples pulled from the table,

n = 10 and a = 3. Both 10 and 3 are relatively prime to each other. ?10 = 4.

If so, 34 = 81 = 1(mod 10)

Another listed example might be n = 15 and a = 2, we can see that 28 = 256 = 1(mod 15)

1 https://en.wikipedia.org/wiki/Euler’s_theorem

2 http://mathworld.wolfram.com/TotientFunction.html