Intro

Throughout the years society has had an interest in keeping information secure. Encryption has evolved from being a simple method of keeping secrets to methods that were thought to be unbreakable for centuries and finally to modern advanced methods of computational algorithms. Cryptology has been used by society in many contexts such as for private use, art, religion, military, and diplomatic use; in all fields in order to keep information indecipherable by those not authorized. As we can see encryption has been part of human life for many years. Encryption either used for business, military, or personal used it is a branch of mathematics is being used every day. Although we have computers to aid is with algorithms for encryption, we can also use the methods used in earlier years. Lately, the way we use cryptology has grown and adapted from simple ciphers more complicated algorithms. It is clear that encryption is an art just blooming.

Cryp-Tol-O-Gy

Cryptology is the union of two domains: Cryptography and Cryptanalysis. The word cryptography comes from ancient Greek literally meaning "hidden writing" thus "communicating confidential [information] through an insecure channel".

Cryptanalysis is literally breaking down a code that is "deciphering those communications when one is not the legitimate receiver".

Brief Outline: History of Cryptology

* Cryptology has been used since ancient times.

* There is not much known about the mathematical aspects of cryptology before the twentieth century (Koblitz).

* Around 50BC- Julius Caesar would send encrypted messages using the 3 letter shift method (Tapson).

* Leon Battista Albeti is considered the father of cryptology. Around the 1400s he wrote the treatise ‘De componendis cifris’: in it he embodied the example of polyalphabetic substitution with mixed alphabets and variable period. He is also attributed for being the inventor of “the earliest Western exposition of cryptanalysis, the invention of polyalphabetic substitution, and the invention of enciphered code” ( Barr).

* In 1586 a French diplomat Blaise de Vigenere published his description of a polyalphabetic cipher which was "considered practically impossible to be broken for almost four centuries"(History of Cryptography). During the nineteenth century, the Kaisiski's method was first published in "Die Geheimschriften und die Dechiffrierkunst”.

* During wars- Cryptanalysts needed to decode telegraph messages in order to prepared and know the plan of attack of the enemy (Morain).

* During the last fifty years, the development of computers have extended the link between computer science and mathematics, some new applications of cryptology have appear in electronic trade, money, and notarial deeds (Morian).

Encrypt: Caesar's Method

The following is an example of how to mathematically encrypt a message.

We will be using the following table to encode our message:

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

T

S

U

V

W

X

Y

Z

X

Y

Z

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

Caesar's cipher is denoted as f(p) =(p+3) mod 26 where p is the integer number of the plain-text.

In this case the key is a shift of 3.

Message to be encoded: "I LOVE MATHEMATICS"

Caesar's cipher: "F ILYB JXPEBJXPFZQ"

Now try to encrypt your own message by clicking on the link below:

http://www.crypo.com/eng_caesar.php

What's so Math-ie?

From a mathematical view, cryptography consists of two basic principles: substitution and transposition. Substitution consists of permuting the letters of the alphabet, in other words the letters in plaintext are replaced by other letters or symbols. Transposition consists of jumbling the letters in plaintext; their normal order is disarranged (Kahn). Modular arithmetic is a mathematical tool used for cryptography. The concept of congruency is essential since it is a way to mathematically describe simple rotations of ciphers. The alphabet is viewed as a loop, such that when it reaches the end, it starts all over.