Ubuntu Manpage: ent - pseudorandom number sequence test

Provided by: ent_1.2debian-1build1_amd64

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.