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Middle Square Method

The Middle Square Method is one of the earliest pseudorandom number generators, invented by John von Neumann in 1946. It generates numbers by repeatedly squaring a value and extracting the middle digits. This method has serious statistical flaws including short cycles and can degenerate to zero, making it unsuitable for any practical use beyond historical study.

Properties

Property	Value
Category	Random Number Generators
Sub-category	Pseudorandom Number Generator
Security status	🎓 Educational Only
Complexity	Beginner
Inventor	John von Neumann
Year	1949
Origin	🇺🇸 United States
Source	algorithms/random/middle-square.js

Parameters

Parameter	Supported values
Seed sizes	1 byte (8 bits) to 16 bytes (128 bits)

Capabilities

Flag	Value
IsDeterministic	Yes
IsCryptographicallySecure	No

Security

Status: 🎓 Educational Only

No vulnerabilities are recorded for this implementation.

Documentation

• Wikipedia: Middle-square method
• von Neumann, J.: Various techniques used in connection with random digits (1951)
• History of Random Number Generation

References

• Applied and Computational Complex Analysis, Volume 1
• The Art of Computer Programming, Vol. 2: Seminumerical Algorithms

Test vectors

4 vectors ship with this algorithm and run in the test suite. Byte values are hexadecimal.

Vector 1 — Middle Square: seed=5772156649 (von Neumann’s example, 10-digit number from 1951 paper)

Field	Value
seed	00000001580c1ee9
outputSize	40
input	null
expected	ce6093a7418fc1a9f04b87d30da6de35 c9f9f581000000000000000000000000 0000000000000000

Vector 2 — Middle Square: seed=1 (minimal seed, demonstrates rapid degeneration to zero)

Field	Value
seed	01
outputSize	40
input	null
expected	00000000000000000000000000000000 00000001000000000000000000000000 0000000000000000

Vector 3 — Middle Square: seed=12345 (short cycle demonstration)

Field	Value
seed	00003039
outputSize	40
input	null
expected	0000000000528a47d24d034ba27d5fd6 fb9e5281000000000000000000000000 0000000000000000

Vector 4 — Middle Square: seed=675248 (historical 6-digit example from literature)

Field	Value
seed	000a4db0
outputSize	40
input	null
expected	0000006a50c8d6d1ecb41ac6a5b7c1f7 94a5b306dbc8c93e65fb09ab90a5fee0 daf7da3cbbbc1925

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