PLC XOZ OVM QLI LDV PBB IPI FHB FQF PDL FKD QLY BYL QEC RKX KAB XPV. Oops! I'm getting ahead of myself. We'll get back to this sentence in a very short time, but first, lets discuss what roles mathematics plays in cryptology.
The simplest type of cipher is the monoalphabetic substitution cipher (Lewand 1). As its name suggests, this enciphering scheme is based on the idea of replacing each character of a message (the plaintext message) with a unique alternate character to obtain an encryption (the ciphertext message). For example, you may choose to replace each occurrence of the letter b in the message with the letter y. One of the rules of using monoalphabetic substitution for a given letter, you have to stick with that same substitution throughout the entire message. That's what makes this system monoalphabetic. Here's the example of Julius Caesar's form of substitution:
Using this scheme, the plaintext message, "So far cryptology seems like it is going to be both fun and easy" becomes "PL CXO ZOVMQLILDV PBBJP IFHB FQ FP DLFKD QL YB YLQECRK XKA BXPV."
Ordinarily, punctuation and the use of lower case letters to indicate plaintext, and upper case letters for ciphertext are neglected. The ciphertext is also grouped into clusters of an agreed upon number of characters to deprive would-be interceptors of any clues they can derive about our message based on the lengths of words. Thinking back...does the above ciphertext look similar to that of the one I used in the beginning?
Most cryptographic techniques rely on properties of the set of integers. One such important property is known as an axiom (Way 13). Perhaps you recall from geometry that an axiom is a statement that appears to be self-evident and that is accepted as...