Middle-square method

To generate a sequence of n-digit pseudorandom numbers, an n-digit starting value is created and squared, producing a 2n-digit number. If the result has fewer than 2n digits, leading zeroes are added to compensate. The middle n digits of the result would be the next number in the sequence, and returned as the result. This process is then repeated to generate more numbers.

The value of n must be even in order for the method to work-- if the value of n is odd then there will not necessarily be a uniquely defined 'middle n-digits' to select from. Consider the following: If a 3-digit number is squared it can yield a 6 digit number (eg: 540² = 291600). If there were to be a middle three digit that would leave 6 − 3 = 3 digits to be distributed to the left and right of the middle. It is impossible to evenly distribute these digits equally on both sides of the middle number and therefore there are no 'middle digits.' It is acceptable to pad the seeds with zeros to the left in order to create an even valued n-digit (eg: 540 → 0540).

For a generator of n-digit numbers, the period can be no longer than 8ⁿ. If the middle n digits are all zeroes, the generator then outputs zeroes forever. If the first half of a number in the sequence is zeroes, the subsequent numbers will be decreasing to zero. While these runs of zero are easy to detect, they occur too frequently for this method to be of practical use. The middle-squared method can also get stuck on a number other than zero. For n = 4, this occurs with the values 0100, 2500, 3792, and 7600. Other seed values form very short repeating cycles, e.g., 0540 → 2916 → 5030 → 3009. These phenomena are even more obvious when n = 2, as none of the 100 possible seeds generates more than 14 iterations without reverting to 0, 10, 50, 60, or a 24 ↔ 57 loop.

seed_number = int(input("Please enter a four digit number:\n[####] "))
number = seed_number
already_seen = set()
counter = 0

while number not in already_seen:
    counter += 1
    already_seen.add(number)
    number = int(str(number * number).zfill(8)[2:6])  # zfill adds padding of zeroes
    print(f"#{counter}: {number}")

print(f"We began with {seed_number}, and"
      f" have repeated ourselves after {counter} steps"
      f" with {number}.")

The defects associated with the original middle-square generator can be rectified by running the middle square with a Weyl sequence.[4][5] The Weyl sequence prevents convergence to zero. The Weyl sequence also prevents the repeating cycle problem described above. A C implementation is shown below.

#include <stdint.h>

uint64_t x = 0, w = 0, s = 0xb5ad4eceda1ce2a9;

inline static uint32_t msws() {
    x *= x; 
    x += (w += s); 
    return x = (x>>32) | (x<<32);
}

The Weyl sequence (w += s) is added to the square of x. The middle is extracted by shifting right 32 bits. This generator passes BigCrush.[6][7] and PractRand.[8] This may be the fastest PRNG that passes all the statistical tests. A free software package is available here.[5]

A counter-based version of this generator called "squares" is now available. See arXiv article "Squares: A Fast Counter-Based RNG".[9][10]

• Linear congruential generator
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