Announcements:

1
DTTF/NB479 Dszquphsbqiz Day 6

• Announcements
• Pass in Assignment 1 now
• Meet my assistant Kevin Reed
• Assignment 2 (tentative) posted
• Questions?
• Today Finish Vigenere, start Hill ciphers

2
Vigenere Ciphers

• Idea the key is a vector of shifts
• Ex. Use a word like hidden (7 8 3 3 4 13).
• Example
• The recent development of various methods of
• 7 8 3 3 413 7 8 3 3 413 7 8 3 3 413 7 8
  3 3 413 7 8 3 3 4 13 7 8 3 3 413 7 8
• 015 7 20 815112122 6 8
  811191718161720 1 17 8 25132416172322 2511
  11017 7 5 2113
• aph uiplvw giiltrsqrub ri znyqrxw zlbkrhf vn
• Encryption
• Repeat the vector as many times as needed to get
  the same length as the plaintext
• Add this repeated vector to the plaintext.
• Demo

3
Summary from yesterday

• What makes a Vigenere cipher more secure than a
  shift cipher?
• How do we find the key length?
• Why does the max of dot(A0,Ai) occur when i0?
• What are the advantages and disadvantages of
  using the dot product method (method 2) vs.
  method 1 to decrypt the key?

4
English letter frequencies

• A 0.082
• B 0.015
• C 0.028
• D 0.043
• E 0.127
• F 0.022
• G 0.020

H 0.061 I 0.070 J 0.002 K 0.008 L 0.040 M
0.024 N 0.067
O 0.075 P 0.019 Q 0.001 R 0.060 S 0.063 T
0.091
U 0.028 V 0.010 W 0.023 X 0.001 Y 0.020 Z
0.001
Graph
5
Finding the key length

• What if the frequency of letters in the plaintext
  approximates A?
• Then for each k, frequency of each group of
  letters in position p k (mod L) in the
  ciphertext approximates A.
• Then loop, displacing the ciphertext by i, and
  counting the number of coincidences.
• Get max when displace by correct key length
• So just look for the max!
• shift
• APHUIPLVWGIILTRSQRUBRIZNYQRXWZLBKRHFVN (0)
• NAPHUIPLVWGIILTRSQRUBRIZNYQRXWZLBKRHFV (1)
• VNAPHUIPLVWGIILTRSQRUBRIZNYQRXWZLBKRHF (2)
• KRHFVNAPHUIPLVWGIILTRSQRUBRIZNYQRXWZLB (6) 5
  matches

6
Visualization

• http//www.usafa.af.mil/df/dfcs/acis/applets/Vigen
  ere.html
• Play with now and for homework
• Thanks to Dr. Dino Schweitzer, USAFA, who I met
  last Friday, for pointing me to his demo!
• Arent you glad I was at SIGCSE?

7
Vigenere Closing Thought

• What if we modified the Vigenere cipher so that
  each individual letter was not simply shifted,
  but the result of an affine function?

8
Block Ciphers

• So far, changing 1 character in the plaintext
  changes ___ characters in the ciphertext.
• Shannon outlined qualities of good ciphers
• Diffusion Changing one character of the
  plaintext changes _____ characters in the
  ciphertext
• Makes frequency analysis much tougher!
• Confusion Each character of the ciphertext
  interacts with several parts of the key
• Block ciphers have both qualities
• DES (64 bits), AES (128 bits), Hill ciphers
  (smaller today)

9
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

10
Decrypting

• Reverse the process, multiplying each block by M
  inverse
• Theorem If a matrix M is invertible mod n, then
  gcd(det(M), n) 1
• Proof on board (uses a lemma from Ch 3)

11
How to break via known plaintext?

• Think about this

12
Does this cipher exhibit diffusion?