Group: Spencer Fajardo

Title: Linear Algebra and some applications to Cryptography.

Abstract: The Hill Cipher is a somewhat mathematically involved encryption
method that uses matrices to encrypt plain text. You can use a four letter
word "key", put it in a matrix using the letter's numerical value that
comes from its place in the alphabet, and then put a message into as many
2x1 matrices as is needed to hold all the characters you wish to encrypt.
Then, you multiply the message columns by the key matrix, and then take the
resulting values mod26 to keep the values within the alphabet. Now you have
an encrypted message, and you can use the key's inverse to decipher the
message via the same process.

The WWII Enigma machine was a complex device used to create an encryption
that became one of the main focuses of the allies to decipher, since it was
used to transmit encrypted messages by the axis. The enigma machine used
two key ideas to create its encryption. The first is that it used a 26x26
identity matrix that represented the alphabet. That is, a corresponded to
the 1 in the first column, b was the 1 in the second column, and so on.
Then, the machine would swap these rows, mixing the message up into
unreadable text. In addition to this, every time a letter was typed into
the enigma machine, a rotor would turn that would "push" the alphabet up
one letter. So, a would go to b, b would go to c, and z would cycle back to
a. In this way, an encrypted message could be deciphered if someone knew
the settings of the machine when it was typed since you could "undo" all
the matrix operations performed on the original identity.