Cryptography and Matrices
Linear Application and Hill Cipher.
Cryptography has played an important role in information and communication security for thousand years. It was first invented due to the need to maintain the secrecy of information transmitted over public lines. The word cryptography came from the Greek words kryptos and graphein, which respectively mean hidden and writing (Damico). Since the ancient days, many forms of cryptography have been created. And in 1929, Lester S. Hill, an American mathematician and educator, introduced a method of cryptography, named Hill cipher, which was based on linear algebra applications. (Anton Rorres 719) Like other forms’, Hill cipher’s basic idea is that by using matrix multiplication, an original message – plaintext – will be converted into a coded message, called ciphertext. This converting process has three main steps: translating the message into a matrix, then multiplying with a cipher matrix to create the encrypted one, and finally converting it into letters.
The very first step of transforming the given message into numbers is to assign every alphabet letter a numerical value. There are many ways to do that, and one of those is that each letter except Z is assigned the numerical value that specifies its position in the alphabet. For example, the letter A is assigned the value of 1, B is 2 and so on. The letter Z, in order to use modular arithmetic later, will be assigned the value of 0. Therefore, if the plaintext is DEPAUW UNIVERSITY, then the corresponding series of numbers is 4, 5, 16, 1, 21, 23, 21, 14, 9, 22, 5, 18, 19, 9, 20 and 25 (the space is omitted).
In order to create a plaintext matrix equivalent to the message, the message is divided into groups of two, three, or more (for a Hill n-cipher, the letters will be separated into groups of n elements). Because the number of letters in a group decides the size of the cipher matrix used in the next step, to make the encoding process less complicated, Hill 2- and 3- cipher
Cryptography has played an important role in information and communication security for thousand years. It was first invented due to the need to maintain the secrecy of information transmitted over public lines. The word cryptography came from the Greek words kryptos and graphein, which respectively mean hidden and writing (Damico). Since the ancient days, many forms of cryptography have been created. And in 1929, Lester S. Hill, an American mathematician and educator, introduced a method of cryptography, named Hill cipher, which was based on linear algebra applications. (Anton Rorres 719) Like other forms’, Hill cipher’s basic idea is that by using matrix multiplication, an original message – plaintext – will be converted into a coded message, called ciphertext. This converting process has three main steps: translating the message into a matrix, then multiplying with a cipher matrix to create the encrypted one, and finally converting it into letters.
The very first step of transforming the given message into numbers is to assign every alphabet letter a numerical value. There are many ways to do that, and one of those is that each letter except Z is assigned the numerical value that specifies its position in the alphabet. For example, the letter A is assigned the value of 1, B is 2 and so on. The letter Z, in order to use modular arithmetic later, will be assigned the value of 0. Therefore, if the plaintext is DEPAUW UNIVERSITY, then the corresponding series of numbers is 4, 5, 16, 1, 21, 23, 21, 14, 9, 22, 5, 18, 19, 9, 20 and 25 (the space is omitted).
In order to create a plaintext matrix equivalent to the message, the message is divided into groups of two, three, or more (for a Hill n-cipher, the letters will be separated into groups of n elements). Because the number of letters in a group decides the size of the cipher matrix used in the next step, to make the encoding process less complicated, Hill 2- and 3- cipher