ent(1)
======
Wesley J. Landaker <wjl@icecavern.net>

NAME
----
ent - pseudorandom number sequence test

SYNOPSIS
--------
_ent_ [options] [file]

DESCRIPTION
-----------
image:entitle.gif[ENT Logo]

_ent_ performs a variety of tests on the stream of bytes in _file_ (or
standard input if no _file_ is specified) and produces output on standard 
output; for example:
..........................................................................
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(1) 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[1]:
...................................................................
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[2] 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
coordinates 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.

OPTIONS
-------

-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 (Latin-1) 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 (Latin-1)
  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.

FILES
-----
If no _file_ is specified, _ent_ obtains its input from standard input.
Output is always written to standard output.

TERSE MODE
----------
Terse mode is selected by specifying the _-t_ option on the command
line. Terse mode output is written in Comma Separated Value (CSV)
format, which can be directly loaded into most spreadsheet programs
and is easily read by any programming language. Each record in the CSV
file begins with a record type field, which identifies the content of
the following fields. If the _-c_ option is not specified, the terse
mode output will consist of two records, as follows:
........................................................................
0,File-bytes,Entropy,Chi-square,Mean,Monte-Carlo-Pi,Serial-Correlation
1,file_length,entropy,chi_square,mean,Pi_value,correlation
........................................................................
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
the_ -b_ option is specified, the second field of the type 0 record will
be "File-bits", and the file_length field in type 1 record will be
given in bits instead of bytes. If the _-c_ option is specified,
additional records are appended to the terse mode output which contain
the character counts:
....................................................
2,Value,Occurrences,Fraction
3,v,count,fraction
...
....................................................

If the _-b_ option is specified, only two type 3 records will appear for
the two bit values v=0 and v=1. Otherwise, 256 type 3 records are
included, one for each possible byte value. The second field of a type
3 record indicates how many bytes (or bits) of value v appear in the
input, and fraction gives the decimal fraction of the file which has
value v (which is equal to the count value of this record divided by
the file_length field in the type 1 record).

BUGS
----
Note that the "optimal compression" shown for the file is computed
from the byte- or bit-stream entropy and thus reflects compressibility
based on a reading frame of the chosen width (8-bit bytes or
individual bits if the _-b_ option is specified). Algorithms which use a
larger reading frame, such as the Lempel-Ziv [Lempel & Ziv] algorithm,
may achieve greater compression if the file contains repeated
sequences of multiple bytes.

COPYING
-------
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.

Original text and program by John Walker[3]
October 20th, 1998

Modifications by Wesley J. Landaker <wjl@icecavern.net>, released
under the same terms as above.

SEE ALSO
--------
Introduction to Probability and Statistics[4]

[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.

[1]::
   http://www.fourmilab.ch/hotbits/

[2]::
   http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html

[3]::
   http://www.fourmilab.ch/

[4]::
  http://www.fourmilab.ch/rpkp/experiments/statistics.html

