Ubuntu Manpage: ent - pseudorandom number sequence test

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 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.