Modular function in Cryptology
- One of the most important applications of modular functions involves
cryptology, which is the study of
secret messages. Here is a so-called Caesar's encryption scheme,
a process of making a message secret.
- Replace 26 letters by integers from 0 to 25
{A, B, C, ....} <--> {0, 1, 2, ...}
- Define the modular function
f(p) = (p+3) mod 26
- Then in the encrypted verion of the message, the letter
represented by p is relaced with the letter represeted by f(p)
- Example: consider the message ``MEET YOU IN THE PARK'':
First replace the letters with numbers, this produces
12, 4, 4, 19 ... 24, 14, 20 ... 8, 13... 19, 7, 4 ... 15, 0, 17, 10
Now replace each of these numbers p with f(p):
15, 7, 7, 22 ... 1, 17, 23 ... 11, 16 ... 22, 10, 7 ... 18, 3, 20, 13
The encryped message: ``PHHW BRX LQ WKH SDUN''
- To recover the original message in the example, we use the
inverse of f
f^{-1}(p) = (p-3) mod 26
This process of determining the original message from the encrypted
message is called decryption.