Random Numbers and Pseudorandom Numbers: Generation, Properties, and Uses, Slides of Network security

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1

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

• 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.

• INT(0.520647 * 100) = 52
• Add 1
• So the random number is 1 + 52 = 53

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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
• Fortuna’s generator is a block cipher in

CTR mode. Recall that CTR generates a

stream of data.

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Pseudorandom Numbers

• s (^) i+1 = (ns (^) i + c) mod m
• 0 ≤ s (^) i < m
• Choice of n, c and m are critical
• n = c = s 0 = 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 s 0 = 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

• 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 cipher
• second key decrypts cipher1 producing cipher
• 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

• B (^) i 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)