Middle-square method: Difference between revisions

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{{Short description|Pseudorandom number generator}}

[[Image:Middle-square method.svg|right|250px|thumb|One iteration of the middle-square method, showing a six 6-digit seed, which is then squared, and the resulting value has its middle six 6 digits as the output value (and also as the next seed for the sequence).]]

[[Image:Middle_square_method_2_digits.svg|right|250px|thumb|[[Directed graph]] of all 100 2-digit pseudorandom numbers obtained using the middle-square method with ''n'' = 2.]]

In [[mathematics]] and [[computer science]], the '''middle-square method''' is a method of generating [[pseudorandom number]]s. In practice it is not a goodhighly flawed method for many practical purposes, since its [[Pseudorandom number generator#Periodicity|period]] is usually very short and it has some cripplingsevere weaknesses,; suchrepeated asenough times, the outputmiddle-square sequencemethod alwayswill convergingeither tobegin zero.repeatedly Thegenerating methodthe originatedsame withnumber [[Johnor voncycle Neumann]], and was notably described atto a conferenceprevious in 1949.<ref name="vonneumann">The 1949 papers were not reprinted until 1951. John von Neumann, “Various techniques usednumber in connectionthe with random digits,” in A.S. Householder, G.E. Forsythe,sequence and H.H.loop Germond, eds., ''Monte Carlo Method, National Bureau of Standards Applied Mathematics Series'', vol. 12 (Washington, D.C.: U.S. Government Printing Office, 1951): pp. 36-38indefinitely.</ref>

== History ==

To generate a sequence of ten-digit pseudorandom numbers, a 4-digit starting value is created and squared, producing an 8-digit number (if the result is less than 8 digits, [[leading zero]]es are added to compensate). The middle 4 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.

===In mathematics===

The method was invented by [[John von Neumann]], and was described by him at a conference in 1949.<ref name="vonneumann">The 1949 papers were not reprinted until 1951. John von Neumann, “Various techniques used in connection with random digits”, in A.&nbsp;S. Householder, G.&nbsp;E. Forsythe, and H.&nbsp;H. Germond, eds., ''Monte Carlo Method, National Bureau of Standards Applied Mathematics Series'', vol.&nbsp;12 (Washington, D.C.: U.S. Government Printing Office, 1951): pp.&nbsp;36–38.</ref>

In the 1949 talk, Von Neumann famously quipped that, "Any oneAnyone who considers arithmetical methods of producing random digits is, of course, in a state of [[sin]]." What he meant, he elaborated, was that there were no true "random numbers,", just means to produce them, and "a strict arithmetic procedure"," like the onemiddle-square described abovemethod, "is not such a method.". Nevertheless, he found these kinds of methods much quicker (hundreds of times faster) than reading "truly" random numbers off [[punch cards]], which had practical importance for his [[ENIAC]] work. He found the "destruction" of middle-square sequences to be a factor in their favor, because it could be easily detected: "one always fears the appearance of undetected short cycles.".<ref name="vonneumann"/> [[Nicholas Metropolis]] reported sequences of 750,000 digits before "destruction" by means of using 38-bit numbers with the "middle-square" method.<ref>Donald E. Knuth, ''The art of computer programming, Vol. &nbsp;2, Seminumerical algorithms,'', 2nd edn. (Reading, Mass.: Addison-Wesley, 1981), ch. &nbsp;3, section &nbsp;3.1.</ref>

For a generator of ''n''-digit numbers, the period can be no longer than 8^''n''. If the middle 4 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, 60, or a 24-57 loop.

The book ''The Broken Dice'' by [[Ivar Ekeland]] gives an extended account of how the method was invented by a Franciscan friar known only as Brother Edvin sometime between 1240 and 1250.<ref name="Ekeland1996">{{cite book |author=Ivar Ekeland |title=The Broken Dice, and Other Mathematical Tales of Chance |date=15 June 1996 |publisher=University of Chicago Press |isbn=978-0-226-19992-4}}</ref> Supposedly, the manuscript is now lost, but [[Jorge Luis Borges]] sent Ekeland a copy that he made at the [[Vatican Library]].

In the 1949 talk, Von Neumann famously quipped that, "Any one who considers arithmetical methods of producing random digits is, of course, in a state of [[sin]]." What he meant, he elaborated, was that there were no true "random numbers," just means to produce them, and "a strict arithmetic procedure," like the one described above, "is not such a method." Nevertheless he found these kinds of methods much quicker (hundreds of times faster) than reading "truly" random numbers off [[punch cards]], which had practical importance for his [[ENIAC]] work. He found the "destruction" of middle-square sequences to be a factor in their favor, because it could be easily detected: "one always fears the appearance of undetected short cycles."<ref name="vonneumann"/> [[Nicholas Metropolis]] reported sequences of 750,000 digits before "destruction" by means of using 38-bit numbers with the "middle-square" method.<ref>Donald E. Knuth, ''The art of computer programming, Vol. 2, Seminumerical algorithms,'' 2nd edn. (Reading, Mass.: Addison-Wesley, 1981), ch. 3, section 3.1.</ref>

Modifying the middle-square algorithm with a [[Weyl sequence]] improves period and randomness.<ref>{{cite book | title = Random Numbers and Computers | last = Kneusel | first = Ron | publisher = Springer | year = 2018 | edition = 1 | pages = 13–14 }}</ref><ref>{{cite arXiv | last=Widynski | first=Bernard | eprint=1704.00358 | title=Middle-Square Weyl Sequence RNG | date=April 2017| class=cs.CR }}</ref>

==Example Source Code==

<source lang="c">

#include<math.h>

== The method ==

int main(void){

To generate a sequence of ten''n''-digit pseudorandom numbers, aan 4''n''-digit starting value is created and squared, producing ana 82''n''-digit number. (ifIf the result ishas lessfewer than 82''n'' digits, [[leading zero]]es are added to compensate). The middle 4''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{{snd}} 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 (e.g. 540^''2'' = 291600). If there were to be middle 3&nbsp;digits, that would leave 6 − 3 = 3&nbsp;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 number (e.g. 540&nbsp;→&nbsp;0540).

for( i = 0; i < samplesToProduce; i++){

//select the middle six digits for the next seed

newseed = (x / 1000) % 100000;

For a generator of ''n''-digit numbers, the period can be no longer than 8^''n''. If the middle 4''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''&nbsp;=&nbsp;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''&nbsp;=&nbsp;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.

//The sequence will almost always go to zero, so an improvement

//to the algorithm would be reseeding with the time as seen below.

/*

if(newseed < threshHold){

newseed = time(NULL);

}

*/

=== Example implementation ===

x = pow(newseed,2);

Here, the algorithm is rendered in [[Python 3|Python 3.12]].

storageArray[i] = x;

}

<syntaxhighlight lang="python">

}

seed_number = int(input("Please enter a four-digit number:\n[####] "))

</source>

number = seed_number

already_seen = set()

counter = 0

while number not in already_seen:

==References==

counter += 1

<references/>

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}.")

</syntaxhighlight>

==See also==

* [[Linear congruential generator]]

* [[Blum Blum Shub]]

* [[Hash function#Mid-squares | middle-square hash function ]]

==References==

<references/>

{{DEFAULTSORT:Middle-Square Method}}

[[Category:Pseudorandom number generators]]

[[Category:Articles with example C code]]

[[Category:Articles with example Python (programming language) code]]

[[Category:John von Neumann]]

{{Compu-sci-stub}}

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