NAME
ent - pseudorandom number sequence test
SYNOPSIS
ent [options] [file]
DESCRIPTION
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
〈mailto: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〉
20 July 2010 ent(1)