This document discusses cryptography and linear algebra. It defines cryptography as the study of secure communication techniques that allow only the sender and intended recipient to view message content. Encryption transforms information into an unreadable form using a secret key, while decryption transforms the encrypted code back into the original using the same key. An example demonstrates encrypting a message by multiplying it by an encoding matrix, transmitting the encrypted result, and decrypting it by multiplying by the inverse matrix.
1. L I N E A R A L G E B R A I N
C RY P T O G R A P H Y
2. C R Y P T O G R A P H Y
• Cryptography term is derived from the Greek
word kryptos, which means hidden and
graphy is writing.
• Cryptography is the study of secure
communications techniques that allow
only the sender and intended recipient of
a message to view its content.
3. E N C R Y P T I O N
A N D
D E C R Y P T I O N
• Encryption means the process of
transformation of an information (plain
text) into an unreadable form (cipher).
• Decryption means the transformation of
the coded message (cipher text) back
into original from (plain text).
• Encryption and decryption require a
secret technique which is known only to
the sender and the receiver.
4. • The message to be sent is called the plaintext message. The
disguised message is called the ciphertext. The plaintext and the ciphertext
are both written in an alphabet, consisting of letters or characters. Characters
can include not only the familiar alphabetic characters A, ,…, Z and a, ,…, z but
also digits, punctuation marks, and blanks.
5. • The secret is called a key. One way of generating a
key is by using a non-singular matrix to encrypt a
message by the sender. The receiver decodes
(decrypts) the message to retrieve the original message
by using the inverse of the matrix. The matrix used for
encryption is called encryption matrix (encoding matrix)
and that used for decoding is called decryption matrix
(decoding matrix).
6. E X A M P L E
Let the message to sent by the sender be "WELCOME".
Let the encoding matrix be
Since the key is taken as the operation of post multiplication by a square matrix of order 3, the
message is cut into pieces (WEL), (COM), (E), each of length 3 and converted into a sequence
of row matrices of numbers:
[ 23 5 12], [3 15 13], [5 0 0]
7. [ 23 5 12], [3 15 13], [5 0 0]
Now that, we have included two zeros in the last row matrix. The reason is to get a row
matrix with 5 as the first entry.
Next, we encode the message by post-multiplying each row matrix as given below:
8. So the encoded message is [45 –28 23] [[46 –18 3] [5 -5 5]
The receiver will decode the message by the reverse key, post- multiplying by thee inverse of A.
The reciever decodes the coded message as follows:
9. So the sequence of
decoded row matrices is
[23 5 12], [3 15 13], [5 0 0].
Thus the receiver reads the
message as "WELCOME".
