Analysis Of Vigenère Cipher

Math Exploration

Cracking different Ciphers

Himanshu Mehta

Mr. Classey

Rationale

This mathematics exploration is focus around the topics of cryptography to be more focused this exploration will look at some of the ciphers which have been used in the past. My decision to pursue this topic was mainly because of my interest in the Enigma Code machine built by German engineer Arthur Scherbius and was used in the World War II. This device is fascinating because it was the most advanced cipher in the 1940’s and was said to be unbreakable if the Germans had implemented the cipher properly. After the war ended “It was thanks to Ultra (project associated with the cracking of the Enigma) that we won the war.” Winston Churchill to King

Sig. Giovan Battista Bellaso. This method was later misattributed to Blaise de Vigenère in the 19th century thus giving the cipher the name “Vigenère cipher”. Vigenère cipher is method that uses a series of different Caesar ciphers based on a lettered keyword. This cipher is a form of polyalphabetic substitution.

In this cipher we make a key before encrypting the message so that it is a bit more secure. In this method the key shifts the alphabets according to its position. In this encryption the key needs to be known for the person decrypting it because there are about 26n possibilities where n is the number of letters in the string.

For example let the code be: “mybirthdayisinjanuary” and let’s take the key to be “math”

Plain Text m y b i r t h d a y i s i n j a n u a r y

Key m a t h m a t h m a t h m a t h m a t h m

Encrypted Text y y u p d t a k m y b z u n c h z u t y k

Thus the encrypted text cannot be broken easily and if someone tried without a key there are 265 ≈ 1.2 X 107 possibilities unlike the Caesar cipher with just 25 possibilities.

Decrypting the Vigenère

Then the message goes to the 3 rotors these rotors combine of 3 different types of rotors a fast rotor which turns when a letter is types. Then a medium rotor which turns when the fast rotor completes a revelation and the slow rotor which rotates when the medium rotor completes a revolution. The German army had a choice of choosing 3 rotors from a set of 5 rotors and the rotors could be set in any orientation.

The number of ways the rotors can be positioned from a set of 5 are 5!/2! = 60 possibilities.

The different starting positions of the rotors as they were changed every day giving 263 = 17576 possibilities of different codes. The 1-26 numbers on the three rotors represent the alphabets from a-z. After the letter is plain text in pressed on the enigma it goes through plug board to the 3 rotors and gets ciphered. Then the cipher text goes back to the plug board usually the plug board would have been connected to the 10 wires which connected 20 alphabets together. Thus if the ciphered text is connected to a wire it would light up the alphabet it might be connected