Recall Techniques for Cipher design

• Substitution– permutations on texts

• Transposition– permutations on places

We have discussed substitution-based Ciphers

– Mono-Alphabetic Ciphers

• Caesar/shift

– Poly-Graphic Ciphers

• Playfair Cipher (keyed-table lookup)

• Hill Cipher (matrix-multiplication operation)

– Poly-Alphabetic Ciphers

• Vigenere Cipher

Transposition Methods

• Permutations on places.

• Key: the indication of the transposition mapping.

Rail-Fence Cipher

• Method:– Plaintext is written down as a sequence of

diagonals and then read of as a sequence of rows

• Plaintext:we will meet on monday

• Example:w w l m e o m n a

e i l e t n o d y

• Ciphertext:wwlmeomnaeiletnody

Repeated Row-Column Cipher

Plaintext is written in a rectangle, row by row, and ciphertext is read from the rectangle, column by column, in a permuted column order.

Key : 2 5 4 1 3

Plaintext: we will meet on monday

Table w e w i l

l m e e t

o n m o n

d a y X X

Ciphertext: ieoXwlodltnxwemyemna

Repeat the row-column transposition several times

Weaknesses in Transposition Methods

• Transposition cipher can be easily recognized, because it has the same letter appearance frequencies as the original text.

• Appearance frequencies of digrams or trigrams may also be useful in breaking the code.

• Frequent plaintext words may repeat at same locations in many places, which will result in repetition of certain letter combinations in the ciphertext.

• Key can be determined by placing the ciphertext in a rectangle and playing with the rows and the columns.

Rotor Machines

• Rotor Machines combine principles of Substitution and Transposition Methods.

• Rotor Machines generate ciphers that are difficult to break.

• Rotor Machines in World War II:– “Enigma” used by the German.– “Purple” used by the Japanese.

• The breaking of both Rotor Machines by the Allies was a significant factor in the outcome of the war.

Rotor Machines

• Rotor Machine is a set of L independent rotating cylinders, each having 26 input pins and 26 output pins, with internal wiring that connects each input pin to a unique output pin.

• Each cylinder implements a fixed permutation between its 26 input pins and its 26 output pins.

• The 26 rotations of a cylinder implement 26 permutations.

• The set of L cylinders implements 26L permutations.

Rotor Machine (3 cylinders)

A

B

C

.

.

P

.

.

Y

Z

A

B

C

.

.

.

.

.

Y

Z

A

B

C

.

.

.

.

X

Y

Z

Maps: AX, BY, …, ZB

Enigma Machine

• Uses 3 cylinders, each of which has 26 input pins and 26 output pins, totaling 263 = 17,576 permutations and has a reflector at the end.

• A plaintext letter typed at the keyboard goes through 3 cylinders.

• Decryption is mirror image of Encryption. If letter X is mapped to letter Y then letter Y returns back to letter X.

Enigma Machine (Initial Setting : Key)

– The key of the Rotor Machine consists of the order of the L cylinders.

– Since number of permutations (17,576) was not considered high enough – the design of the Enigma machines allowed swapping the 3 cylinders. This multiplied the number of initial settings by 3! = 6

– In addition, there were 6 plug-board cables that enabled swapping pairs of letters at the keyboard. This multiplied the number of initial options by 1011

– Total number of initial settings of the Enigma Machine is about ~ 1016

Enigma Machine

• German Military Version of Enigma Machine:– In 1939, the Germans increased the Enigma

security – Use of 3 out of 5 cylinders increased number of

options from 6 to 60– Use of 10 instead of 6 plug-board cables increased

number of options by another factor of about 103

– Total number of initial settings of the German Military Enigma Machine is about 1.59 x 1020

– Polish cryptanalysts weren’t able to crack the Enigma anymore

– In the early 1940’s, the British were able to crack the Enigma

Source : http://ed-thelen.org/comp-hist/NSA-Comb.html

Recommended Readings

• Enigma : http://ed-thelen.org/comp-hist/NSA-Comb.html