# Announcements: Matlab : tutorial available at math.ufl/help/matlab-tutorial/ - PowerPoint PPT Presentation

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DTTF/NB479: Dszquphsbqiz Day 5. Announcements: Matlab : tutorial available at http://www.math.ufl.edu/help/matlab-tutorial/ Questions? Today: Wrap up Hill ciphers One-time pads and LFSR Schedule change: skip Playfair cipher, start ch 3 tomorrow. Next: one time pads.

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Announcements: Matlab : tutorial available at math.ufl/help/matlab-tutorial/

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DTTF/NB479: DszquphsbqizDay 5

• Announcements:

• Matlab: tutorial available at http://www.math.ufl.edu/help/matlab-tutorial/

• Questions?

• Today:

• Wrap up Hill ciphers

• Schedule change: skip Playfair cipher, start ch 3 tomorrow

• 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…

• 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 = ???

### Unbreakable?

• Yes, for ciphertext only…

Ciphertext:

Could be:

thephoneisringing

meetmeinthegarage

I need the whole key to decrypt.

• What’s the downside to using a one-time pad?

Variation: Maurer, Rabin, Ding et al’s satellite method

• If I’m willing to compromise some security…

### 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;

• xn+5 = xn + xn+2 (mod 2)

• What does this remind you of in math?

### 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;

• xn+5 = xn + xn+2 (mod 2)

• Another way to write is xn+5 = 1xn + 0xn+1 + 1xn+2 +0xn+3 + 0xn+4 (mod 2)

• In general,

• Generate some more terms

• How long until it repeats? (the period of the sequence)

• 10 bits (0100010100) generates ____ bits

• Demo

### 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 example:xn+31=xn + xn+3 (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, it’s maximal. Related to Mersenne primes

• See http://www.ece.cmu.edu/~koopman/lfsr/index.html for a list of maximal-period generators

• Can you devise a bad example (one with period << 2n-1)?

### LFSR Demo

• Downside: very vulnerable to known plaintext attacks.

• Determine key length by computing determinants

• Theorem: If N is the length of the shortest recurrence that generates the sequence, then det(MN) = 1 (mod 2) and det(Mn)=0 (mod 2) for all n > N.

• 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 …  length = 5

• Use the key length to solve for the recurrence.

• Verify your solution by using the solution to generate the whole key