# Introduction system of manipulating integers, where the integers are

Introduction

The essence of cryptography has been around for thousands of years. The first recorded uses of a written form of cryptography goes back to ancient Egypt and Mesopotamia. In Egypt, non-standard hieroglyphic symbols were utilized in an inscription, while in Mesopotamia, a tablet was found that contained a formula for the making of glazes for pottery. Special codes have been long utilized for military purposes as well. In modern society and with much more advanced technology, cryptography and digital encryption have been integral to the protection of information.

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There is some important terminology to understand regarding cryptography. A cipher is a method or series of steps, or algorithm, that is used to encrypt information or create a code. A key is a specific value that is applied using the cipher to that produces the encrypted text. Plaintext describes a set of text or another message that has not been encrypted. Ciphertext is the text information that is produced as a result of encryption. An important mathematical concept to understand is modular arithmetic. Modular arithmetic is a system of manipulating integers, where the integers are said to “wrap around” a fixed quantity that is used, called the modulus. Much of advanced cryptography utilizes modular arithmetic, as it has a way of being able to scramble the results of different calculations so patterns are not as obvious or existent at all, which is quite useful when it comes to cryptography.

With increasing frequency in modern society, hackers have been able to breach and leak significant personal information of millions of people around the world. The more I hear about these breaches, it only furthers my interests in increasing and improving cybersecurity and encryption in general, hence the topic of this paper.

This aim of this investigation will be to better understand and comprehend the overall concept behind ciphers, as well as the overall history and progression of ciphers and complexity. The mathematics behind certain ciphers and the techniques used to decipher and crack such ciphers and codes will be analyzed and investigated in order to more completely comprehend the concepts of cryptography. The ciphers that will be mentioned in this paper will be mostly investigated in their chronology within history.

Monoalphabetic ciphers

Some of the earliest but simplest ciphers utilized are categorized as monoalphabetic ciphers. One of the first such ciphers often used was the ATBASH cipher. The name derives from the first, last, second, and second to last Hebrew letters (Aleph-Tav-Beth-Shin). It consists of substituting the first letter aleph for the last letter tav so on for the rest of the alphabet. In essence, this simply reverses the alphabet. The original purpose of this cipher involved encoding the Hebrew alphabet, specifically seen with several examples within the book of Jeremiah within the Hebrew Bible. This can be seen perhaps as one of the first example of an affine cipher as well, which can be represented as E(x) = (ax + b) mod 26.

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One of the first ciphers utilized for major military purposes was in Rome, under the rule of Julius Caesar. This is also a simple monoalphabetic substitution cipher, where every letter in the code is shifted to the right a certain value, traditionally 3 letters.

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EXPERIENCEISTHETEACHEROFALLTHINGS

HASHULHFHLVWKHWHDFLHURIDOOWKLQJV

This code can be represented as a change from the position of the letter in the alphabet n, to n+3. Every letter in the alphabet is shifted over to the right 3 positions. For example, the letter J in the 11th position is now coded for M, which is in the 14th position. The letters at the end wrap around and become the letters at the beginning of the alphabet, such as X in the 24th position now has a value of A, which is in the 1st position. There is very similar to how the premise of modular arithmetic, in that once the position has gone too far, it resets at the beginning. In practice, this type of cipher is profusely difficult to break, but it was certainly effective during a time when not as many people could read, and only Caesar or his generals knew what it said.

Transposition ciphers

A simple yet relatively more secure type of substitution cipher is the transposition cipher. The earliest uses of this type of cipher can be seen in ancient Egypt and Greece, especially used in the ancient Greek city-state of Sparta. In 4th century B.C., a device named a scytale was used to encrypt Spartan government and military messages. A scytale was composed simply of a wooden stick with a strip of parchment wrapped around it. This technique had tremendous use in ancient battles. Each general was given a stick of a uniform diameter so that he could quickly understand messages that had received from other generals. The premise of a transposition cipher is that a message is written within a matrix of a fixed size (the following example uses a 10×6 matrix).

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Initially, when read straight across, the message certainly makes sense. However, the code is encrypted by writing down the message as words straight down in each column.

Therefore, the following message will be read down the columns in the matrix used.

TROOPS HAVE BEEN STRANDED AT THERMOPYLAE. REINFORCEMENTS ARE NEEDED.

This message is instead written as:

TBEPOR REDTRA OEALCN ONTAEE PSTEME STHRED HREENE AARITE VNMNSD EDOFA

However, as most ciphers developed around this, a transposition cipher such as this was fairly easy to decipher. In a battle situation, the most likely way to crack this would be to steal a general’s scytale. Then, each message could be read easily. However, it can be cracked even without the scytale itself. The way the simplest of these works is by picking a matrix of a fixed size and then writing your message across the rows. The encipherment step consists of writing down the letters in the matrix by following the columns.