Announcements:

1
DTTF/NB479 Dszquphsbqiz Day 7

• Announcements
• Assignment 2 finalized
• Questions?
• Today
• Wrap up Hill ciphers
• One-time pads and LFSR

2
Hill Ciphers

• Lester Hill, 1929. Not used much, but first time
  linear algebra used in crypto
• Use an n x n matrix M. Encrypt by breaking
  plaintext into blocks of length n (padding with
  xs if needed) and multiplying each by M.
• Example Encrypt hereissomeonetoencrypt using M
• her eis som eon eto enc ryp txx
• (7, 4, 17) (4, 8, 18)
  (19, 23, 23)
• (2, 5, 25) (0, 2, 22) (0, 22, 15)
• cfz acw yga vns ave anc sdd awp
• CFZACWYGAVNSAVEANCSDDAWP

3
Hill Cipher Demo

• Encryption
• Easy to do in Matlab.
• (Otherwise, youll need to find/write a matrix
  library for language X.)
• Decryption
• Uses matrix inverse.
• How do we determine if a matrix is invertible mod
  26?
• Does this cipher exhibit diffusion?

4
Next one time pads

• Back to Vigenere if the codeword were really
  long, say 25 as long as the entire plaintext,
  how many characters would contribute to each dot
  product? ____
• What does this say about our ability to do a
  frequency analysis?
• Now consider the extreme case, the one-time pad

5
One-time pads

• Represent the plaintext in binary, length n
• Works for text (from ASCII), images, music, etc
• The key is a random vector of length n
• Ciphertext plaintext XOR key
• Do
• message 1000011, key 1110010
• Cipher ???
• ciphertext XOR key ???

6
Unbreakable?

• Yes, for ciphertext only
• Ciphertext
• EOFMCKSSDKIVPSSAD
• Could be
• thephoneisringing
• meetmeinthegarage
• I need the whole key to decrypt.
• Whats the downside to using a one-time pad?
• Variation Maurer, Rabin, Ding et als satellite
  method
• If Im willing to compromise some security

7
Linear Feedback Shift Register (LFSR) Sequences

• Name comes from hardware implementation

Generated bit stream
Shift register
b1 b2 b3 b4 bm-1 bm
To encrypt plaintext of length n, generate an
n-bit sequence and XOR with the plaintext.
Feedback function

• Need initial conditions (bits in register) and a
  function to generate more terms.
• Example
• x1 0, x2 1, x3 0, x4 0, x5 0
• xn5 xn xn2 (mod 2)
• What does this remind you of in math?

8
Linear Feedback Shift Register (LFSR) Sequences

• A recurrence relation!
• Specify initial conditions and coefficients, for
  example
• x1 0, x2 1, x3 0, x4 0, x5 0
• xn5 xn xn2 (mod 2)
• Another way to write is xn5 1xn 0xn1
  1xn2 0xn3 0xn4 (mod 2)
• In general,
• Generate some more terms
• How long until it repeats? (the period of the
  sequence)
• 10 bits generates ____ bits
• Demo

9
Long periods

• LFSR can generate sequences with long periods
• Like Vigenere with long key hard to decrypt!
• Lots of bang for the buck!
• But it depends on the key
• Good examplexn31xn xn3 (mod 2)
• How many bits do we need to represent this
  recurrence?
• 62 bits
• How long is the period?
• Over 2 billion! Why?
• There exist (231 1) 31-bit words
• Why -1?
• If it cycles through all of these, its maximal.
  Related to Mersenne primes
• See http//www.ece.cmu.edu/koopman/lfsr/index.htm
  l for a list of maximal-period generators
• Can you devise a bad example (one with period ltlt
  2n-1)?

10
Linear Feedback Shift Register (LFSR) Sequences

• Downside very vulnerable to known plaintext
  attack. Why?
• Discuss with a partner
• If time, my example