The Fourmilab Random Sequence Tester, ent, applies various tests to sequences of bytes stored in files and reports the results of those tests. The program is useful for evaluating pseudorandom number generators for encryption and statistical sampling applications, compression algorithms, and other applications where the information density of a file is of interest.
ent performs a variety of tests on the stream of bytes in its input file (or standard input if no input file is specified) and produces output as follows on the standard output stream:
Entropy = 7.980627 bits per character.
Optimum compression would reduce the size
of this 51768 character file by 0 percent.
Chi square distribution for 51768 samples is 1542.26, and randomly
would exceed this value less than 0.01 percent of the times.
Arithmetic mean value of data bytes is 125.93 (127.5 = random).
Monte Carlo value for Pi is 3.169834647 (error 0.90 percent).
Serial correlation coefficient is 0.004249 (totally uncorrelated = 0.0).
The values calculated are as follows:
The information density of the contents of the file, expressed as a number of bits per character. The results above, which resulted from processing an image file compressed with JPEG, indicate that the file is extremely dense in information—essentially random. Hence, compression of the file is unlikely to reduce its size. By contrast, the C source code of the program has entropy of about 4.9 bits per character, indicating that optimal compression of the file would reduce its size by 38%. [Hamming, pp. 104–108]
The chi-square test is the most commonly used test for the randomness of data, and is extremely sensitive to errors in pseudorandom sequence generators. The chi-square distribution is calculated for the stream of bytes in the file and expressed as an absolute number and a percentage which indicates how frequently a truly random sequence would exceed the value calculated. We interpret the percentage as the degree to which the sequence tested is suspected of being non-random. If the percentage is greater than 99% or less than 1%, the sequence is almost certainly not random. If the percentage is between 99% and 95% or between 1% and 5%, the sequence is suspect. Percentages between 90% and 95% and 5% and 10% indicate the sequence is “almost suspect”. Note that our JPEG file, while very dense in information, is far from random as revealed by the chi-square test.
Applying this test to the output of various pseudorandom sequence
generators is interesting. The low-order 8 bits returned by the
standard Unix rand() function, for example, yields:
Chi square distribution for 500000 samples is 0.01, and randomly
would exceed this value more than 99.99 percent of the times.
While an improved generator [Park & Miller] reports:
Chi square distribution for 500000 samples is 212.53, and randomly
would exceed this value 97.53 percent of the times.
Thus, the standard Unix generator (or at least the low-order bytes it returns) is unacceptably non-random, while the improved generator is much better but still sufficiently non-random to cause concern for demanding applications. Contrast both of these software generators with the chi-square result of a genuine random sequence created by timing radioactive decay events.
Chi square distribution for 500000 samples is 249.51, and randomly
would exceed this value 40.98 percent of the times.
See [Knuth, pp. 35–40] for more information on the chi-square test.
This is simply the result of summing the all the bytes (bits if the
-b option is specified) in the file and dividing by the file length.
If the data are close to random, this should be about 127.5 (0.5 for
-b option output). If the mean departs from this value, the values
are consistently high or low.
Each successive sequence of six bytes is used as 24 bit X and Y co-ordinates within a square. If the distance of the randomly-generated point is less than the radius of a circle inscribed within the square, the six-byte sequence is considered a “hit”. The percentage of hits can be used to calculate the value of π. For very large streams (this approximation converges very slowly), the value will approach the correct value of π if the sequence is close to random. A 500000 byte file created by radioactive decay yielded:
Monte Carlo value for Pi is 3.143580574 (error 0.06 percent).
This quantity measures the extent to which each byte in the file depends upon the previous byte. For random sequences, this value (which can be positive or negative) will, of course, be close to zero. A non-random byte stream such as a C program will yield a serial correlation coefficient on the order of 0.5. Wildly predictable data such as uncompressed bitmaps will exhibit serial correlation coefficients approaching 1. See [Knuth, pp. 64–65] for more details.
This software is licensed under the Creative Commons Attribution-ShareAlike license. Please see LICENSE.md in this repository for details.
[Hamming] Hamming, Richard W. Coding and Information Theory. Englewood Cliffs NJ: Prentice-Hall, 1980. ISBN 978-0-13-139139-0.
[Knuth] Knuth, Donald E. The Art of Computer Programming, Volume 2 / Seminumerical Algorithms. Reading MA: Addison-Wesley, 1969. ISBN 978-0-201-89684-8.
[Lempel & Ziv] Ziv J. and A. Lempel. “A Universal Algorithm for Sequential Data Compression”. IEEE Transactions on Information Theory 23, 3, pp. 337–343.
[Park & Miller] Park, Stephen K. and Keith W. Miller. “Random Number Generators: Good Ones Are Hard to Find”. Communications of the ACM, October 1988, p. 1192.
Introduction to Probability and Statistics at Fourmilab