Generating Random Numbers

• Very simple algorithms exist that produce sequences of numbers that appear random.
• Since they are not truly random, they are often called pseudo-random numbers.
• Algorithm:
  • Select large positive integers A, B, and N
  • Select a starting value (seed) X(0) between 0 and N-1
  • Initialize n=0
  • Repeat as many times as desired
    • n=n+1
    • Y = A X(n-1) + B
    • X(n) is the remainder if Y is divided by N
• Example:
  • Let N=256, A=2,345, B=6,789, X(0)=123
  • n=1: $Y = 2,345 \cdot 123 + 6,789 = 295,224 = 1,153 \cdot 256 + 56$
    $\rightarrow X(1) = 56$
  • n=2: $Y = 2,345 \cdot 56 + 6,789 = 138,109 = 539 \cdot 256 + 125$
    $\rightarrow X(2) = 125$
  • n=3: $Y = 2,345 \cdot 125 + 6,789 = 299,914 = 1171 \cdot 256 + 138$
    $\rightarrow X(3) = 138$
• Hence, the above algorithm has produced the pseudo-random sequence of numbers 56, 125, 138.
• In general, all pseudo-random numbers will be between 0 and N-1.
• The random number generator will repeat itself after at most N numbers.
• The integer sequence above can now converted into a binary sequence
  00111000 01111101 10001010
• Thus, if A, B, N, X(0) are known to transmitter and receiver then they can generate identical pseudo-random sequences for encryption and decryption.