Cryptographic Randomness

1
Cryptographic Randomness

• Haile Eyob
• CS 240 Software Project
• May 13, 2003

2
Cryptographic Randomness

• Cryptography
• Classic Cryptography
• Symmetric Cryptography
• Public-Key Cryptosystem
• Pseudorandom Numbers
• Rule 30 Randomness
• Conclusion

3
What is Cyptography?

• Cryptography is science of hiding data
• Plaintext - Data that can be read
• Ciphertext - in unreadable text.
• Encryption C E(M)
• Decryption D(C) M.

4
Caesar Cipher
Encryption C (M 3) MOD(26) SOFTWAREPROJECTCO
URSE VRIEWDUHSURMHFWFRXUVH Decryption M (C -
3) MOD(26) VRIEWDUHSURMHFWFRXUVH SOFTWAREPROJECT
COURSE
5
Caesar Cipher cont.

• Relative Frequency of Letters in English Text 8

6
Vigenère Cipher

• Polyalphabetic substitution cipher
• Suppresses the normal frequency data
• Example
• Plaintext SOFTWAREPROJECTCOURSE
• Key PRIVATEKEYPRIVATEKEYPRIV
• Cipher HFNOWTVOTPDAMXTVSEVQT

7
Vigenère Cipher cont.

• The modern Vigenère tableau 8

8
Rotor Machine

• Rotor machine 8

9
Rotor Machines cont.
Rotor1 "KNYIHTCSQGFBWZPMAXULVOJRDE" Rotor2
"VDBFPLNMJISAZEXWOHYTUQGKRC" Rotor3
"OWTVSKYPJIFMLUAHRQECNDBZGX" Example Plaintext
SOFTWAREPROJECTCOURSE Cipher
JWZOYSGUANMRQWWNCIHNM
10
Data Encryption Standard (DES)

• 64-bit text and 56 bit-key
• Secret key shared
• Key for encryption and decryption
• Three operations XOR, substitution, and
  permutation.

11
Data Encryption Standard cont
Simplified DES scheme 8
12
Data Encryption Standard cont.

• Key Generation for Simplified DES (Schaeffer97

13
Public Key Cryptography

• No shared key
• Pair of keys public and private key
• Public key for encryption
• Private key for decryption
• Difficulty of factoring
• RSA
• Choose primes, p and q
• n p ? q
• C ? Me mod n
• M ? Cd mod n

14
Public Key Cryptography cont
Example p 2503 q 3011 n p q
7536533 f(n) (p 1)(q 1) 7531020 e
753103 d ? e-1 mod(f(n)) 2259307 M
373737 C ? Me mod n 6486063
15
Pseudorandom Numbers

• Generated by software functions.
• Sequence is deterministic.
• Random numbers - No correlations
• Difficult to predict the next number

16
Random Number Generation

• Physical sources - pulse detectors of ionizing
  radiation events, gas discharge tubes,
  radioactive source.
• Hardware device keystrokes, hard disks
• Random numbers contain correlations.
• Provide fewer real bits of unpredictability

17
Linear Congruential Generator (LCG)

• Linear recurrence              xn axn-1 b
  (mod m)
• RANDU
• m 231, a 65539, b 0, x0 1
• Better
• m 231 - 1, a 16807, b 0
• Period m 1

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Linear Congruential Generator - RANDU
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Linear Congruential Generator - RANDU

• Distribution of points obtained from RANDU

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Linear Congruential Generator (LCG)

• Fast and easy
• Partial output is needed to get a, b, m
• Seed recovered in polynomial time.

21
Linear Feedback Shift Register (LFSR)

• LFSR generates binary bits
• Bits shifted right by one position
• Function - XOR of some bits
• Feedback put in leftmost cell
• Oututput LSB

22
Linear Feedback Shift Register cont.

• Linear shift feedback register with 4 bit register

23
Linear Feedback Shift Register cont.

• Maximum period 2n 1
• States represented by polynomial function.
• LFSRs are based on running rule 60 in registers
  with a limited number of cells and with a certain
  type of spiral boundary conditions.

24
Linear Feedback Shift Register cont

• Example of LSFR with 4 bits
• Initial state 1010
• 1010 1001 1000 0111
• 1101 0100 1100 1011
• 0110 0010 1110 0101
• 0011 0001 1111 1010
• Sequence 0101100100011110
• Maximum period 24 1 15.

25
Blum-Blum-Shub (BBS)

• Cryptographically strong generator
• p ? q ? 3 mod 4,
• n p?q.
• xi x2i-1 mod n.
• p ? q ? 3 mod 4,
• To determine xi is QR is intractable.
• Difficulty of factoring n
• Computation intensive

26
Blum-Blum-Shub cont.
p 366571831 q 652367675 n
239139613109962925 x0 986960469269329 Bits
generated 1101101001110010001000100000100101101
110011010010
27
Rule 30 Randomness

• Intrinsic randomness
• Physical processes
• Physical laws are deterministic
• LCG
• LSFR
• BBS

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Rule 30 Randomness cont.

• Pattern produced by rule 30

29
Simple Encryption

• A simple encryption scheme 9.

30
Repeated Encryption Sequence

• Repeated encryption key 9.

31
Cellule Automata Encryption

• Encryption using rule 60 9.

32
Evolution of Rule 60

• Evolution of rule 60 both downward and sideways
  9.

33
Cellule Automata Additive Property

• Rule 60 is additive
• Given some segment of the encrypting sequence,
  find a row
• Rule 30 is not additive
• Determine the color of a cell means enumerating
  all initial conditions.

34
Cellule Automata Encryption cont.

• Encryption using rule 30 9.

35
Randomness using RULE 30 (Wolfram)

• One can find the form of a row in the cellular
  automaton, if given some segment of the
  encrypting sequence, corresponding to a
  particular column. Rule 30 is not additive and to
  determine the color of a cell from the colors of
  its neighbor columns is the same as enumerating
  all possible initial conditions.

36
Evolution of Rule 30

• Evolution of rule 30 both downward and sideways
  9.