Polyalphabetc Substitution Ciphers

Notes

The first two messages in the previous set of notes were not too bad to break
- Shorter is harder
- Different words/ letter frequency is harder
- Removing spaces, transmitting as a block is harder.
- But in general, this can be done.
The last message in the previous notes could not be broken with frequency analysis.
- This is because it was not encrypted with a monoalphabetic substitution cipher.
It was encrypted with the Vigenere cipher (vi-j-ne-are)
This is a polyalphabetic cipher.
- Or a cipher where each letter is encrypted with a potentially different alphabet.
- Or shift of the alphabet in this case.
Vigenere takes a keyword and a plaintext message
- In the simple form, each letter is encrypted with shift amount determined by a different letter in the key.
- The message : send more money.
- The key: edinboro
- s -> 's' + 'e'-'a' = 'w'
- e -> 'e' + 'd'-'a' = 'h'
- n -> 'n' + 'i'-'a' = 'v'
- d -> 'd' + 'n'-'a' = 'q'
- space -> space
- m -> 'm' + 'b'-'a' = 'n'
- o -> 'o' + 'o'-'a' = 'c'
- r -> 'r' + 'r'-'a' = 'i'
- e -> 'e' + 'o'-'a' = 's'
- space -> space
- m -> 'm' + 'e'-'a' = 'q'
- o -> 'o' + 'd'-'a' = 'r'
- n -> 'n' + 'i'-'a' = 'v'
- e -> 'e' + 'n'-'a' = 'r'
- y -> 'y' + 'b'-'a' = 'z'
Note that multiple letters map to q, r, ...
There are advanced methods for breaking this, but they are much more complex.
And mostly involve finding the key.

An interesting polyalphabetic cypher involves the random number generator

Since this will generate the same random stream

// encryption
initialize the random number generator to a known quantity
for each plainLetter in the message
    offset = plainLetter -'a'
    cryptLetter = 'a' + (offset + rand() ) %26

// decryption
initialize the random number generator to a known quantity
for each plainLetter in the message
    offset = plainLetter -'a'
    cryptLetter = 'a' + (offset - rand() ) %26

In this case, the key is the seed to the random number generator.

The Enigma Machine

(Bennett)
Used by Germany in WWII
Polyalphabetic Cipher
Message (plain or crypt) is typed in on the kyboard.
The encryption/decryption appears in the lights above the keyboard
Each day, a key is entered
- Select the roters
  - Actually, first select n from a set of M roters
  - 3/5, M depended on the version of the machine and branch of service
  - Put the roters in the proper order.
- These were usually stored in a codebook, that was required for operation.
- And the wires on the plugboard are changed
- Finally, for each message , and indicator group was set, ie put the roters in the proper position.
The roters served as a simple letter mapping
- All roters of a type (I) were the same, and would map, E->T (example, might not be right)
- Ie a simple substitution cipher
- But each time a letter was coded/decoded the roter stepped providing a different alphabet for the next letter.
- Depending on a roter, this would then step the next neighbor by o or more steps.
  - The middle roter might step once every 26 letters
  - Some roters were designed to step more than one step
- The plugboard provided an additional letter swapping.
All of these led to 158,962,555,217,826,360,000 different settings
- Chose 3 roters from 5 : 5 * 4 * 3 = 60
- 26 different positions per roter: 26³ = 17,576
- 26! different plugboard combinations : 150,738,274,937,250
(Wikipedia, MesserWoland, CC BY-SA 3.0)
But the Allies broke it
- Bad design: a letter could not map its image a->a is forbidden
  - This gave cryptoanalyists a way to remove a number of rotor order combinations
  - Any combo where letter->letter was discarded
- Bad use: the initial letter settings were sent before the message
  - This gave cryptoanalysts a way to remove some possible initial rotor positions
  - And a clue into some possible initial positions
- Not all roters incremented by the same amount
  - This also helped determine rotor location
- Only resetting the machine once a day
  - Provided more messages for decryption
- Some letters were re-transmitted as themselves (special characters)
  - Provided insight into the messages
- And some spy work (1931-1932)