Codes, Ciphers, and Cryptography-Ch 3.2

1
Codes, Ciphers, and Cryptography-Ch 3.2

• Michael A. Karls
• Ball State University

2
Main Ingredients of the Enigma

• Keyboard (with A-Z)
• Light bulbs (with A-Z)
• Plugboard (with A-Z)
• Rotors (with 0-25)
• Reflector (with 0-25)

3
User Interface

• The keyboard and light bulbs were the user
  interface for entering and reading messages.
• The plugboard was used to increase the number of
  keys (settings).

4
Rotors

• Each rotor was connected with a series of wires
  that created a permutation cipher.
• An electric signal would enter the rotor at one
  position and the permutation would cause the
  signal to leave at a different position on the
  far side of the rotor.

5
Reflector

• The reflector rotor at the end was a fixed
  product of disjoint two-cycles (i.e. a
  permutation), such as
• ? (0, 12)(1, 9)(2, 3) (4, 24)(5, 18)(6,
  23)(7, 8) (10, 25)(11, 13)(14, 21) (15, 17)(16,
  19)(20, 22)
• Note that ?-1 ?, so x?? x for all x.

6
Mathematics of the Enigma

• If we ignore the plugboard settings, the Enigma
  is a product of permutations!

x plain
y
z
w
d cipher
b
a
c
reflector
rotor 3
rotor 2
rotor 1
7
Mathematics of the Enigma (cont.)
8
Mathematics of the Enigma (cont.)

• Equation (1) is a model for the Enigma if we
  encrypt one letter!
• Recall that the rotors move like an odometer,
  with rotor 2 changing when rotor 1 changes from
  25 to 0 and rotor 3 changing in a similar fashion.

9
Mathematics of the Enigma (cont.)

• To account for the rotation of the rotors, we
  introduce a shift variable for each rotor,
  which keeps track of how far a rotor has shifted
  from its initial position!
• Shift variables
• s1 ? rotor 1
• s2 ? rotor 2
• s3 ? rotor 3
• Note For s2 to increase, we need s1 to change
  from 25 to 0 and for s3 to increase, we need s2
  to change from 25 to 0.

10
Mathematics of the Enigma (cont.)

• To account for the shift s1, instead of computing
• we compute
• Instead of computing the inverse of
• to get x, i.e.
• we need to solve (2) for x in terms of y.

11
Mathematics of the Enigma (cont.)

• Applying ?1-1 to both sides of (2) yields
• Adding - s1 mod 26 to both sides of (3),
• We will use functions of form (2) and (4) to
  account for the moving rotors!
• To encrypt, we actually compute the following

12
Mathematics of the Enigma (cont.)
13
Mathematics of the Enigma (cont.)

• After each letter, check shift variables (mod
  26)
• s1 (s1 1) mod 26
• If s1 0, then s2 (s2 1) mod 26
• If s2 0, then s3 (s3 1) mod 26

14
Example 1

• Suppose we are given ?1, ?2, ?3, and ? as
  follows
• ?1 (0, 15, 7, 8, 19, 23) (1, 4, 17, 2) (3,
  13, 25, 5, 21, 12) (6, 9, 10, 22, 11, 14, 24, 16,
  18, 20)
• ?2 (0, 5, 9) (1, 2, 3, 7, 6, 8, 4) (10, 11, 12,
  15, 18, 13, 14, 16, 19, 17) (20, 25, 22, 21, 24,
  23)
• ?3 (0, 10, 20, 3, 13, 23, 7, 17, 5, 15, 25, 1,
  11, 21) (2, 12, 22, 4, 14, 24, 8, 18, 9, 19, 6,
  16)
• ? (0, 12) (1, 9) (2, 3) (4, 24) (5, 18) (6, 23)
  (7, 8) (10, 25) (11, 13) (14, 21) (15, 17) (16,
  19) (20, 22),
• with s1 24, s2 10, and s3 5.
• Use these initial settings to encipher the
  message Get your forces ready.

15
Example 1 (cont.)

• Solution Take x 6 for G and compute with (5).

16
Example 1 (cont.)
17
Example 1 (cont.)
18
Example 1 (cont.)
19
Example 1 (cont.)

• Thus, d 15, which implies G is encrypted as P.
• Finally, update shift variables.
• s1 (s1 1) mod 26 (241) mod 26 25
• s2 10
• s3 5
• Repeat with x 4 for E,
• Homework Finish enciphering message!

20
References for Enigma Notes and Photos

• D. W. Hardy and C. L. Walker, Applied Algebra,
  Codes, Ciphers, and Discrete Algorithms, Prentice
  Hall, New Jersey, 2003
• http//jproc.ca/crypto/enigma.html
• http//math.arizona.edu/dsl/enigma.htm