ENT

This directory contains a program which applies various tests to sequences of bytes stored in files and reports the results of those tests. The program is useful for those 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.

The program is provided as random.tar.gz, a GZIPped TAR archive containing an MS-DOS executable program (compiled using Microsoft Quick C for Windows, creating an MS-DOS file which does not require Windows to execute), and in source code form along with a Makefile to build the program under Unix.

The program 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 0.01 percent of the times.
  
    Arithmetic mean value of data bytes is 125.93 (128 = 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:

ENTROPY

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 4.931743 bits per character, indicating that optimal compression of the file would reduce its size by 38%. [Hamming, pp. 104-108]

CHI-SQUARE TEST

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 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 95.00 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 32768 samples is 237.05, and randomly would exceed this value 75.00 percent of the times.

See [Knuth, pp. 35-40] for more information on the chi-square test.

ARITHMETIC MEAN

This is simply the result of summing the all the bytes in the file and dividing by the file length. If the data are close to random, this should be about 128. If the mean departs from this value, the values are consistently high or low.

MONTE CARLO VALUE FOR PI

Each successive sequence of four bytes is used as 16 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 four-byte sequence is considered a "hit". The percentage of hits can be used to calculate the value of Pi. For very large streams (this approximation converges very slowly), the value will approach the correct value of Pi if the sequence is close to random. A 32768 byte file created by radioactive decay yielded:

Monte Carlo value for PI is 3.139648438 (error 0.06 percent).

SERIAL CORRELATION COEFFICIENT

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.

Distribution

References

[Hamming]

Hamming, Richard W., Coding and Information Theory, Englewood Cliffs NJ: Prentice-Hall, 1980.

[Knuth]

Knuth, Donald E., The Art of Computer Programming, Volume 2/Seminumerical Algorithms, Reading MA: Addison-Wesley, 1969.

[Park & Miller]

Park, Stephen K. and Miller, Keith W., "Random Number Generators: Good Ones Are Hard to Find". Communications of the ACM, New York, October 1988, p. 1192.