Introduction to Cryptography with Coding Theory-Maths

 

        INTRODUCTION TO CRYPTOGRAPHY

“I am fairly familiar with all forms of secret writing and am myself the author of a trifling manuscript on the subject”.

                                                                                                                                        -SIR ARTHUR CONAN DOYLE

The word cryptography is derived from the Greek words Cryptos + Graphein, where cryptos mean hidden and graphein means to write. Thus, cryptography is the art of writing codes. It is the science of making communications unintelligible to all except authorized person cryptography is the only known practical means for protecting information has transmitted through public communications networks, such as those using telephone lines, microwaves, satellites, etc.

In cryptography, the codes that are used are called ciphers, the information to be the kept secret is called plain text, and the information after transformation to a secret form is called the ciphertext. The process of converting from plain text to cipher text is called encrypting (or enciphering), whereas the reverse process of changing from cipher text back to plain text is called decrypting (or deciphering).

The great roman emperor Julius Caesar used one of the earliest cryptography systems around 50 B.C. He used cryptography by replacing is a letter that occurs three places down the alphabet as below.

  Plain Text: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Cipher Text: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

For e.g.

1.      The plain text message

CAESAR SEND MONEY

 Is transformed into the ciphertext

                                    FDHVDU VHQG PRQHB

NOTE:

1.      Caesar cipher technique is not very strong technique for hiding the plain text message.

2.      It can be easily detectable.

The Caesar cipher can be described by easily using congruence theory. Any plain text is first expressed numerically by translating the characters of the text into digits by means of some correspondence such as following.

A	B	C	D	E	F	G	H	I	J	K	L	M
00	01	02	03	04	05	06	07	08	09	10	11	12

 

N	O	P	Q	R	S	T	U	V	W	X	Y	Z
13	14	15	16	17	18	19	20	21	22	23	24	25

 

If P denotes the digital equivalent of a plain text, letter, and C the digital the equivalent of the corresponding ciphertext letter, then

C  @ P+ 3 (mod 26)

For e.g.

The letter in the message “CAESAR WAS GREAT” is converted in the digital form as,

02 00 04 18 00 17 22 00 18 06 17 04 00 19

Using the congruence C @ P+3 9mod 26) i.e. P @ C-3 @ C+23 (mod 26). We get,

05 03 07 21 03 20 25 03 21 09 20 07 03 22

NOTE:

1.       There are 25 possibilities for any alphabet to be replaced by other.