Cryptography: True Randomness and Pseudorandomness

Sunday 24 May 2026

A random number (binary string) is a number drawn (selected) from a set of numbers in an unpredictable, independent manner. So each member of the set has an equal chance of being selected. This means the outcome (selected number) has nothing to do with the past or future selection. In computing, this is true randomness. Pseudorandomness is a number selection from the set of numbers that looks random-like. The application areas include cryptography, computer simulation, gaming, and statistics, to name a few. 

Example: Random string selection from a set of 3-bit binary strings 

Set = {0, 1}^3 = {000, 001, 010, 011, 100, 101, 110, 111}

Probability [ each 3-bit binary string ] = 1/8 

Probability distribution = {1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8}

Example: Pseudorandom string selection from a set of 3-bit binary strings according to a given selection process Pr [ string ].

Pr [ string ] = ¼, for strings 001, 011, 101, 111

                   = 0,  for strings 000, 010, 100, 110

Probability distribution = {0, 1/4, 0, 1/4, 0, 1/4, 0, 1/4}

Collatz Conjecture or 3n+1 Problem

Saturday 09 May 2026

Well, it is not a solution as the great mathematicians tried! After some rough work and thinking about probabilities in my mind. Calculation of probabilities for even or odd outside the loop and inside the loop reveals a difference of half. If input is even, there are two possibilities inside the loop, whereas for odd input, there is only even output. Considering internal probabilities equally likely, that is, 1/3.