Random Numbers

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Random Numbers
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Random Numbers

• Random number
• Pseudo random number
• Uses for random number in cryptography

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Random Numbers

• Random number should not be predictable
• The numbers should not be uniform
• Measure of randomness is entropy
• A 32-bit key that is generated completely
  randomly has entropy of 32 bits
• A 32-bit key that has only 4 different values has
  entropy of 2 bits ( Entropy comes from the
  exponent value of 22 4)

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Random Numbers

• Many programming languages support random number
  generators. User provides the seed value
• Random numbers are in the range 0 to 1. To get a
  random number between 1 and 100 the user
  multiplies the random value by 100, takes the
  integer value and then add the value 1.
• Rule is to multiply the random value by the upper
  limit of the range, get the integer value of the
  product and then add the lower limit to the result

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Random Numbers

• E.g., Find a random number between 1 and 100.
  Your Java or C program produces a random value
  using the random number generator as 0.520647
• INT(0.520647 100) 52
• Add 1
• So the random number is 1 52 53

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Random Numbers

• Randomness is measured in a statistical sense
• Criteria for randomness
• Uniform distribution, i.e., the frequency of
  occurrence of the numbers is approximately the
  same
• Independence, i.e., one value in the sequence has
  no relation to any other value in the sequence

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Pseudorandom Numbers

• Pseudorandom numbers are predictable
• Generated by deterministic methods
• Common abbreviation for Pseudorandom Number
  Generator is PRNG
• John Kelsey developed a PRNG called Yarrow in
  1999
• Yarrow has an internal state that can be updated
  using a block cipher

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Pseudorandom Numbers

• Ferguson and Schneier have developed another PRNG
  method called Fortuna
• Fortuna has three parts
• Fixed size seed
• An accumulator collects and pools the entropy
• Seed file control ensures randomness
• Fortunas generator is a block cipher in CTR
  mode. Recall that CTR generates a stream of data.

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Pseudorandom Numbers

• PRNGs are generated using a method known as
  linear congruence
• m 0 is the modulus
• n is the multiplier
• c is the increment
• s0 is the seed
• Linear congruence methods generates random
  numbers iteratively

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Pseudorandom Numbers

• si1 (nsi c) mod m
• 0 ? si
• Choice of n, c and m are critical
• n c s0 1 and m 8 produces random numbers
  2, 3, 4, 5, 6, 7, 0, 1. This set is not random
  at all.
• n 7, c 0, m 32 and s0 1 produces the set
  7, 17, 23, 1, 7, 17, 23, 1, . This set has a
  period 4, meaning that after the 4th number there
  is repetition in the sequence

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Pseudorandom Numbers

• Suggested rules to follow for a PRNG
• The function should generate all numbers between
  0 and m-1 (both inclusive) before repeating
• The sequence generated must be random
• The function should be efficient when implemented
  with 32-bit arithmetic (Recall that CTR method
  was more efficient than CBC method)

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Pseudorandom Numbers

• For a 56-bit DES key, a counter with period 256
  is used
• Another popular PRNG is ANSI X9.17
• X9.17 is a very strong PRNG
• It is used in PGP and financial security
  applications
• X9.17 uses two inputs one is the 64-bit
  representation of current date and time and the
  other is a 64-bit arbitrary seed value

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Pseudorandom Numbers

• Keys are generated using three 3DES modules.
  Each of the three modules use the same pair of
  56-bit keys. Recall that in 3DES (all three
  keys could be distinct or two of the keys could
  be the same)
• first key encrypts plaintext producing cipher1
• second key decrypts cipher1 producing cipher2
• first key encrypts cipher2 producing the
  ciphertext

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Pseudorandom Numbers

• Output of X9.17 is a 64-bit pseudorandom number
  and a 64-bit seed value
• Another popular PRNG is
• Blum-Blum-Shub (BBS) method, named after the
  authors who developed this in 1986
• BBS is a secure PRNG method
• BBS involves choosing two primes p and q that
  both have a reminder of 3 when divided by 4
• E.g., 7 and 11 have this property

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BBS method

• Take large primes p and q that have a reminder of
  3 upon division by 4
• Let n p q
• Choose a seed s relatively prime with n
• Algorithm
• X0 s2 mod n
• For i 1 to ?
• Xi (Xi-1)2 mod n
• Bi Xi mod 2

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BBS method

• Bi denotes the least significant bit in each
  iteration
• The algorithm generates the bits continuously
• User decides how many of the bits to choose to
  form the number, e.g., 64-bits chosen to form a
  key

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Uses for Random Numbers

• Where is a random number used?
• Key generation in RSA
• Key generation by a CA
• Authentication schemes (use of nonce is common
  with the random numbers)