This page describes a program, `ent`

, 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.

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 (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:

**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 about 4.9 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. An interactive chi-square calculator is available at this site.

**Arithmetic Mean**- 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. **Monte Carlo Value for Pi**- 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 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.

**-b**- The input is treated as a stream of bits rather
than of 8-bit bytes. Statistics reported reflect
the properties of the bitstream.
**-c**- Print a table of the number of occurrences of
each possible byte (or bit, if the
**-b**option is also specified) value, and the fraction of the overall file made up by that value. Printable characters in the ISO 8859-1 Latin1 character set are shown along with their decimal byte values. In non-terse output mode, values with zero occurrences are not printed. **-f**- Fold upper case letters to lower case before
computing statistics. Folding is done based on the
ISO 8859-1 Latin1 character set, with accented
letters correctly processed.
**-t**- Terse mode: output is written in Comma
Separated Value (CSV) format, suitable for
loading into a spreadsheet and easily read
by any programming language. See
Terse Mode Output Format
below for additional details.
**-u**- Print how-to-call information.

0,File-bytes,Entropy,Chi-square,Mean,Monte-Carlo-Pi,Serial-Correlation 1,where the italicised values in the type 1 record are the numerical values for the quantities named in the type 0 column title record. If thefile_length,entropy,chi_square,mean,Pi_value,correlation

2,Value,Occurrences,Fraction 3,If thev,count,fraction. . .

`ent.exe`

(compiled using Microsoft Visual
C++ 1.52, creating
a 16-bit 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.
- Introduction
to Probability and Statistics
- [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.
ISBN 0-201-89684-2.
- [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.

This software is in the public domain. Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, without any conditions or restrictions. This software is provided "as is" without express or implied warranty.

by John Walker

October 20th, 1998