CRC RevEng: arbitrary-precision CRC calculator and algorithm finder
Copyright (C) 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017
Gregory Cook

This file is part of CRC RevEng.

CRC RevEng is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

CRC RevEng is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with CRC RevEng.  If not, see <https://www.gnu.org/licenses/>.

THIRD-PARTY SOFTWARE

CRC RevEng incorporates source code from ASPEX under the terms of
version 3 of the GNU General Public License (GPLv3).  ASPEX is:

Copyright (C) 1998, 2003 by David A. Hinds -- All Rights Reserved

Homepage: <http://aspex.sourceforge.net/>

ASPEX is licensed under version 2 and all later versions of the GNU
General Public License, therefore CRC RevEng remains licensed under
version 3 and all later versions.

Depending on one's interpretation of the License, the author created a
modified copy of ASPEX as a whole by extracting files getopt.c and
getopt.h verbatim on 19 December 2010.  These are found in the contrib/
directory.  A compatible getopt module is available as standard in many
environments and will work just as well, but this copy is included for
those ANSI C environments that lack one.

A copy of the RISC OS Shared C Library (CLib), and a patch to make
certain versions of RISC OS compatible with this version of CLib, are
supplied with the RISC OS binary under licence from Castle Technology
Limited for distribution to end users for the purpose of upgrading, if
required.  These are found in the bin/riscos/ directory.  The ARM Tube
OS binary, located in the bin/armtubeos/ directory, is statically linked
with (that is, it incorporates) the RISC OS Shared C Library.
CLib qualifies as a "System Library" and GPLv3 permits licensees to
"combine GPLed software with GPL-incompatible System Libraries, [...]
and distribute them both together."  See Sections 1, 5 and 6 and A Quick
Guide to GPLv3, <https://www.gnu.org/licenses/quick-guide-gplv3.html>
The patch is an independent program and is not bound by GPLv3 by mere
inclusion.  See Section 5.

CRC REVENG

CRC RevEng is an arbitrary-precision, machine word length-independent,
byte order-independent CRC calculator and algorithm finder in ANSI C.
It is a port and expansion of the author's crcbfs.pl script from 2007,
and runs up to 200 times faster on equivalent problems.  It is also a
reference implementation of the author's "Catalogue of parametrised CRC
algorithms", with the 91 currently listed models available as presets.

To the author's knowledge CRC RevEng is the first published compiled
application to address the general case of CRC algorithm reversal and
reverse engineering, its predecessor crcbfs.pl being the first published
application of any type to do so.  Greg Ewing of Canterbury University
in New Zealand solved a CRC algorithm manually on similar principles in
2010, but partly by feeding chosen plaintexts into an implementation at
hand.

CRC RevEng is hosted by SourceForge, from where the latest version can
be downloaded and documentation browsed:

	<http://reveng.sourceforge.net/>

COMPILING

Compiling CRC RevEng is straightforward: in the i386 GNU/Linux, MinGW
and Raspbian environments, simply cd to the directory containing the
source files, and enter

	make

A special makefile can also be used in MinGW to make a Windows
executable with version information and an icon:

	make -f Mk.Win32

In RISC OS, with the Acorn Desktop Development Environment (DDE)
installed, click Menu on the file RISCOSify, and set its type to
Obey.  Double-click Select first on RISCOSify, then on Mk/RISCOS.

In non-ASCII compatible environments, the array aliases[] in file
preset.c may need to be reordered to suit the local string collation
order.  The correct order of models[] and aliases[] can be verified at
compile time by the error-free completion of this command:

	make clean pretst

Otherwise, enter commands similar to the following to compile CRC RevEng
on any ANSI C compliant compiler:

	gcc -O3 -Wall -ansi -c bmpbit.c cli.c model.c poly.c preset.c \
		reveng.c
	cd contrib
	gcc -O3 -Wall -ansi -c getopt.c
	cd ..
	gcc -o reveng bmpbit.o cli.o model.o poly.o preset.o reveng.o \
		contrib/getopt.o

The platform-independent method does not compile the preset models.  To
compile them, you will need to edit the configuration options in
config.h to suit your architecture.  Having done so, define the macro
PRESETS in config.h and recompile as above.

Some enterprise users may wish to disable the -F switch to minimise CPU
usage.  To do this, define the macro NOFORCE in config.h or on the
command line.

SYNOPSIS

Usage:	reveng	-cdDesvhu? [-bBfFGlLMrStVXyz]
		[-a BITS] [-A OBITS] [-i INIT] [-k KPOLY] [-m MODEL]
		[-p POLY] [-P RPOLY] [-q QPOLY] [-w WIDTH] [-x XOROUT]
		[STRING...]
Options:
	-a BITS		bits per character (1 to n)
	-A OBITS	bits per output character (1 to n)
	-i INIT		initial register value
	-k KPOLY	generator in Koopman notation (implies WIDTH)
	-m MODEL	preset CRC algorithm
	-p POLY		generator or search range start polynomial
	-P RPOLY	reversed generator polynomial (implies WIDTH)
	-q QPOLY	search range end polynomial
	-w WIDTH	register size, in bits
	-x XOROUT	final register XOR value
Modifier switches:
	-b big-endian CRC		-B big-endian CRC output
	-f read files named in STRINGs	-F skip preset model check pass
	-G skip brute force search pass	-l little-endian CRC
	-L little-endian CRC output	-M non-augmenting algorithm
	-r right-justified output	-S print spaces between characters
	-t left-justified output	-V reverse algorithm only
	-X print uppercase hexadecimal	-y low bytes first in files
	-z raw binary STRINGs
Mode switches:
	-c calculate CRCs		-d dump algorithm parameters
	-D list preset algorithms	-e echo (and reformat) input
	-s search for algorithm		-v calculate reversed CRCs
	-h | -u | -? show this help

SPECIFYING A MODEL

You can use one of the preset models or specify your own.

	reveng -m crc-32

selects the CRC-32 algorithm used in PKZIP and elsewhere.  You can dump
any preset model as a Williams model record using -d:

	reveng -m crc-32 -d
	width=32  poly=0x04c11db7  init=0xffffffff  refin=true
	refout=true  xorout=0xffffffff  check=0xcbf43926  name="CRC-32"

You can specify the parameters of the model instead:

	reveng -w 32 -p 04c11db7 -i ffffffff -l -x ffffffff

This is equivalent to

	reveng -m crc-32

except the model will have no name when dumped.  (The -l switch sets
both RefIn = True and RefOut = True.  To set RefOut separately, use
switches -L and -B.)

The options and switches for specifying a model are:

	-b
		Big-endian input and output (RefIn = False, RefOut =
		False).  Implies -B and -r.  This is the default.
	-B
		Big-endian output (RefOut = False).  Implies -r.
	-i INIT
		A hexadecimal digit string specifying the initial value
		to set in the shift register before computing the CRC.
		The width must be specified with -k, -m, -P or -w
		before -i.
	-k KPOLY
		The generator polynomial (see -p), written in the
		hexadecimal, reversed-reciprocal notation found in
		Philip Koopman's papers.  The highest-order term is
		included and the lowest-order term is omitted, shifting
		the other terms to the right.  Thus 0x18005 is specified
		as C002.  This automatically provides the Width value.
	-l
		Little-endian input and output (RefIn = True, RefOut =
		True).  Implies -L and -t.
	-L
		Little-endian output (RefOut = True).  Implies -t.
	-M
		Specifies that the algorithm does not multiply the
		message polynomial by x^n before division.  The
		resulting algorithm does not comply with the Williams
		model, and cannot be dumped with -d.
	-m MODEL
		Set the Width, Poly, Init, RefIn, RefOut and XorOut
		values to the preset named MODEL.  MODEL is matched
		case-insensitively with the Name and Alias fields in the
		author's "Catalogue of parametrised CRC algorithms", as
		of the date of this release.  The preset models can be
		listed with -D.
		-M, -r, -S, -t, -X, -y and -z must not precede -m MODEL.
	-p POLY
		The Poly value, that is, the generator polynomial that
		sets the feedback tap positions of the shift register.
		POLY is written in the hexadecimal, direct notation
		found in MSB-first code.  The highest-order term is
		omitted, thus 0x18005 is specified as 8005.
		-p also sets the start of the range for polynomial range
		searching, see -q.
		The width must be specified with -k, -m, -P or -w
		before -p.
	-P RPOLY
		The generator polynomial (see -p), written in the
		hexadecimal, reversed notation found in LSB-first code.
		The highest-order term is omitted before reversal, thus
		0x18005 is specified as A001.  This automatically
		provides the Width value.
	-V
		Reverse the current model.  If RefOut = True, the Init
		value is reversed otherwise the XorOut value is
		reversed.  Then the Init and XorOut values are swapped,
		the generator polynomial is reciprocated and the RefIn
		and RefOut values are negated.
		This switch is distinct from -v, see below.
	-w WIDTH
		The width, that is, the number of bits in the shift
		register.
	-x XOROUT
		A hexadecimal digit string specifying the value to be
		added to the final shift register value before output.
		The width must be specified with -k, -m, -P or -w
		before -x.

Other model-related options:

	-d
		Dump a Williams model record of the specified model on
		standard output.  Though formatted differently, this
		single-line record has identical semantics to the
		multi-line records presented in "A Painless Guide to CRC
		Error Detection Algorithms".  Disabled for the
		non-compliant models created by -M.
	-D
		Dump Williams model records of all preset models on
		standard output.

INPUT AND OUTPUT

Messages for CRC RevEng to process can be specified as files, as raw
binary strings, or as numerical (typically hexadecimal) string arguments
on the command line.  Output from CRC RevEng is either as Williams model
records (having their own fixed format) or as numerical string arguments
printed one per line on standard output.

When passing numerical arguments on the command line, each argument is
conceptually divided into characters, each character consisting of one
or more hexadecimal digits.  For each character, enough hex digits to
specify it are read then a number of bits (specified by the -a option)
are taken from the least significant end, reversed (if RefIn = True) and
appended to the binary representation of the argument; any excess bits
are discarded.  The -a option (q.v.) permits a number of useful
representations of a given underlying binary sequence.

When passing messages as files or as raw binary strings, the same
division into characters applies; bytes of the message are read until
enough bits have been collected, then a specified number of bits are
taken from the specified least significant end of the collection (see
-y), reflected (if RefIn = True) and added to the binary representation.

When printing output, the binary representation is again divided into
characters, each of which is reversed (if RefOut = True) and printed
with the minimum sufficient number of hex digits.

INPUT/OUTPUT OPTIONS

There are a few more options for controlling the presentation of input
and output:

	-a BITS
		Specifies the number of bits per character in input and
		output.  Implies -A BITS.  Raw input messages are
		selected by -f or -z, otherwise messages are numerical
		arguments.  Particular values of BITS cause the input to
		be interpreted as a sequence of:
		-a 16	16-bit words, raw or as quartets of hex digits.
			When RefIn or RefOut are True, this is
			equivalent to swapping the bytes of each pair
			before input to an 8-bit calculator.
		-a  8	8-bit characters, raw or as pairs of hex digits.
			This is the default.
		-a  7	7-bit characters, one per 8-bit byte (raw) or as
			pairs of hex digits.
		-a  4	Single hex digits (numerical arguments only).
		-a  3	Single octal digits 0-7.
		-a  1	Single binary digits 0-1.
		Note that raw messages (files or arguments) containing
		digits will only be read as expected if there are no
		characters except digits (not even a final newline or
		other whitespace), and if the digits are maskable in the
		system character encoding, as digits 0-9 are in ASCII,
		EBCDIC and Unicode.  This is because the lowest bits of
		each byte are extracted as described above.
	-A OBITS
		Specifies the number of bits per character in output.
	-f
		Arguments are file names; input binary messages from the
		file data.
	-r
		Right-justified output.  If a binary output message does
		not consist of a whole number of characters, this switch
		arranges for padding zeroes to be added to the start of
		the message.  The padding will appear in the MSB of the
		first character (RefOut = False) or the LSB of the first
		character (RefOut = True).  -r is the default when
		RefOut = False.
	-S
		Print spaces between the characters of the output
		string(s).
	-t
		Left-justified output.  If a binary output message does
		not consist of a whole number of characters, this switch
		arranges for padding zeroes to be added to the end of
		the message.  The padding will appear in the LSB of the
		last character (RefOut = False) or the MSB of the last
		character (RefOut = True).  -t is the default when
		RefOut = True.
	-X
		Print uppercase hexadecimal characters.
	-y
		When reading binary messages from files or arguments,
		and the -a option is more than 8, this option specifies
		that the first byte of each character contains the LSB.
	-z
		Arguments are raw binary strings; input binary messages
		from the arguments.

CALCULATING AND REVERSING CRCs

When a model has been specified, use -c or -v to calculate CRCs for
input messages.

	-c
		Calculate a CRC for each argument and print it on
		standard output, one per line.
	-v
		Reverse the current model (as for -V) AND the order of
		characters in each argument, calculate a CRC for each
		reversed argument, reverse the order of characters in
		each CRC and print it on standard output, one per line.

If -V and -v are given together, their respective model reversals cancel
out.  CRC RevEng then calculates an ordinary CRC for each argument,
processing the characters from right to left and likewise emitting the
CRC characters in reverse order.
Correspondingly, to obtain the same effect as -v using a model reversed
by -V, the user must present the characters of his or her message, and
process those of the returned CRC, in reverse order.

Take care when the CRC width (-w) is not a multiple of the character
width (-a).  If the result of a calculation is not what you expect, try
selecting left-justification (with -t) or right-justification (with -r).

The -c mode is, of course, useful for creating a checksum to append to a
message so that the combination will pass a particular CRC check.  The
-v mode, on the other hand, is useful for editing a message so as to
force its checksum to a desired or at least predetermined value.  In
order to do this there must be some part of the message's data that can
be modified freely without observable effect; many network protocols and
file formats, including executables, images and word processor
documents, have (or can be altered to have) reserved fields or comment
sections that cannot be easily viewed, and whose contents are entirely
ignored.

Among the simplest ways to control a CRC calculation is to find one such
unused space that is both contiguous and large enough to hold a
checksum.  For example, suppose we have an existing message with an X.25
CRC:

  0: 44 6F 67 73 2F 2A 12 34  2A 2F 72 6F 63 6B 4E 47 | Dogs/*.4*/rockNG

Here, 4E 47 is the X.25 checksum, and we wish to alter the message
without either changing the checksum or failing the CRC.  We notice that
the 7th and 8th bytes can be replaced at will, and these can contain a
calculated value to force the CRC.  Firstly we change the text as we
wish:

  0: 43 61 74 73 2F 2A 12 34  2A 2F 72 75 6C 65 4E 47 | Cats/*.4*/ruleNG

Calculate the CRC of the part on the left of the unused space with
XorOut = 0:

	reveng -m x-25 -x 0 -c 436174732f2a
	9dc5

Then reverse-calculate the CRC of the part on the right, including the
old CRC, with Init = 0:

	reveng -m x-25 -i 0 -v 2a2f72756c654e47
	1505

Now exclusive-OR the two returned CRCs together, and insert the result
in the unused space.  CRC RevEng can be used to do the exclusive-OR if a
hex calculator is not to hand:

	reveng -w 16 -p 0001 -c 9dc51505
	88c0

Our edited message now looks like this:

  0: 43 61 74 73 2F 2A 88 C0  2A 2F 72 75 6C 65 4E 47 | Cats/*.A*/ruleNG

To confirm that it still passes the X.25 CRC:

	reveng -m x-25 -c 436174732f2a88c02a2f72756c65
	4e47

For more flexible editing options, see Mark Adler's source file spoof.c
(2014). "spoof.c takes an abbreviated description of the CRC, the
exclusive-or of the current CRC of the message and the desired CRC, the
length of the message, and a list of bit locations in a message, and
tells you which of those bits should be inverted in the message to get
the desired CRC. Note that it does not need the message itself, due to
the linearity property of CRCs."

				* * *

In Stigge et al. (section 4.1) a polynomial q(x) is calculated as the
multiplicative inverse of x^N such that (x^N) q(x) = 1 (mod pCRC(x)).
The authors promote the extended Euclidean algorithm as a means of
calculating q(x), however any CRC calculator can also produce it.  The
reciprocal of the CRC-32 polynomial is 0xdb710641, as output by:

	reveng -w 32 -p 04c11db7 -V -d

The authors' constant CRCINV, a reflected representation of q(x), is the
CRC of the reversal of the desired remainder, 0x00000001:

	reveng -w 32 -p db710641 -c 80000000
	5b358fd3

Equivalently, the -v function returns q(x) in direct order from the
unreflected parameters:

	reveng -w 32 -p 04c11db7 -v 00000001
	cbf1acda

SEARCHING FOR CRC MODELS

The most important feature of CRC RevEng is the ability to recover the
parameters of a CRC algorithm from a handful of codewords created by
that algorithm.  In general at least four data points are needed, either
as known parameters or as message-CRC pairs.  Extra data points help to
eliminate false results and to confirm models that are found.

Known parameters are specified using -w, -p, -i and -x (see SPECIFYING A
MODEL above).  The width, -w, is a required parameter for all searches
and counts as one of the data points.  The size of characters (words) in
the protocol must also be known and set with -a if this is not 8 bits.

The search function is selected with -s, and message-CRC pairs are given
as arguments, each message and CRC combined into one argument.  There
must not be any non-participating characters between each message and
its CRC, or the search will not work.  Typically, end-of-message markers
do not participate in the CRC.

As non-standard algorithms are comparatively rare, the program first
tries all the preset models of the given width, reporting and exiting if
one is found.  Otherwise it commences a full search.  As it proceeds it
prints a progress message from time to time, allowing the search to
be restarted from that point:

	reveng: searching: width=32  poly=0x50000001  refin=false
	refout=false

If -b or -l are specified, CRC RevEng only searches for algorithms with
that bit ordering.  Otherwise, it tries RefIn = False, RefOut = False
then RefIn = True, RefOut = True.  Crossed-endian algorithms are also
uncommon and the program will not search for them.

To find the Poly value when Init is not known, at least two arguments
must have the same length.

To find both the Init and XorOut values, at least two arguments must
have different lengths; otherwise there is only enough information to
determine one value, given the other.  If all arguments have the same
length then, by default,  CRC RevEng fixes XorOut at zero and calculates
Init accordingly.  (In hardware it is easier to set a non-zero Init than
to apply a non-zero XorOut.)  To set XorOut to another value, specify
-x XOROUT; to fix Init and calculate XorOut instead, use -i INIT.

When bit strings are input with -a 1, there is no information on
endianness.  In such cases -s returns the big- and little-endian forms
of each algorithm found.  The Check values of these forms will differ,
as they are always calculated on the 8-bit UTF-8 string "123456789".

POLYNOMIAL RANGE SEARCHING

To restart a stopped search, or to divide a search between several
processors, CRC RevEng can be instructed to search within a specified
range of generator polynomial values.

The full search space comprises all 'odd' polynomials of the specified
WIDTH, that is, polynomials of the form x^n + ... + 1.  Treating the
concatenated coefficients as a binary integer, the range can be up to
(but excluding) a specified polynomial, from a specified polynomial
upwards, or from one polynomial up to (but excluding) another.

Polynomial range searching is enabled using [-p POLY] -q QPOLY, where
POLY and QPOLY are hex strings.  -p POLY, if given, must precede
-q QPOLY.  To start searching at a polynomial, use -p POLY -q 0.  To
stop searching at a polynomial (exclusive), use -q QPOLY.  To search
between two polynomial values, use -p POLY -q QPOLY.

Range limiting does not apply to the initial check against the preset
models, or to Init or XorOut values, which are computed using a fast,
efficient algorithm.

For example, to split a 32-bit search into four processes:

	reveng -w 32 -q 40000000 -s c98964f6b9 a5fa49f2fd 13370aee7df0
	reveng -w 32 -p 40000000 -q 80000000 \
		-s c98964f6b9 a5fa49f2fd 13370aee7df0
	reveng -w 32 -p 80000000 -q c0000000 \
		-s c98964f6b9 a5fa49f2fd 13370aee7df0
	reveng -w 32 -p c0000000 -q 0 \
		-s c98964f6b9 a5fa49f2fd 13370aee7df0

To continue an interrupted search:
NB: If an either-endian search is stopped while RefIn/RefOut = False
then it takes two further command lines to complete the search: one
big-endian range search, and one little-endian full search.

	reveng -w 32 -p 50000001 -q 0 -b \
		-s c98964f6b9 a5fa49f2fd 13370aee7df0
	reveng -w 32 -l -s c98964f6b9 a5fa49f2fd 13370aee7df0

The full list of search options is as follows:

	-F
		Skip the preset model check pass.  (Not recommended.)
	-G
		Skip the brute force search pass.
	-p POLY
		When followed by -q QPOLY, sets the start of the range
		(inclusive) for polynomial range searching.  POLY is
		written in hexadecimal direct notation.  The LSB is
		forced to 1 as only 'odd' polynomials, with a +1 term,
		are tested.
	-q QPOLY
		Enables polynomial range searching and sets the end of
		the range (exclusive).  A previous -p POLY is no longer
		treated as a known generator polynomial and is taken as
		the start of the range; if there is no previous -p POLY,
		the start of the range defaults to the lowest odd
		polynomial.
		QPOLY is written in hexadecimal direct notation.  If
		QPOLY is zero, the range extends up to (and including)
		the highest odd polynomial.  Unlike -p POLY, the LSB is
		significant.
	-s
		Search for and display Williams model records of CRC
		models matching the arguments and given parameters.

OTHER FEATURES

CRC RevEng provides a few additional options for convenience:

	-e
		Echo arguments to standard output.  Useful to check that
		files are being read correctly and, together with -a,
		-A, -b, -B, -l, -L, -r, -S, -t, -X and -y, to reformat
		argument strings.
		The Init value is exclusive-ORed with the beginning of
		each argument, so that when the argument is not a whole
		number of bytes long, an equivalent string can be
		produced for input to a bytewise calculator (which has
		Init set to 0).
	-h
	-u
	-?
		Print a summary of options and switches to standard
		error, and exit.

SEARCH EXAMPLES

	reveng -w 16 -l -F -s 31816b 32c16a 31326a0a
	reveng -w 32 -p 04c11db7 -l -s c98964f6b9 a5fa49f2fd 13370aee7df0
	reveng -w 32 -l -s c98964f6b9 a5fa49f2fd 13370aee7df0

A comprehensive list is being compiled.

CAVEATS

In addition to the disclaimers listed at the top of this file and in the
GNU General Public License (see file COPYING), remember that CRC RevEng
is merely a search tool, and not authoritative.  Searching is only
statistical and any particular result may be a fluke, especially from
a small number of samples.  Also any output is only as accurate as the
input.

Model reversal (-V, -v) makes little sense on crossed-endian models.

CRC RevEng lacks a facility to generate code to implement specified
algorithms.  Pycrc by Thomas Pircher (2014) is one suitable code
generator; Mark Adler's crcany source package (2014) also produces
source code when compiled and run, and references the author's CRC
catalogue.

REFERENCES

Adler, Mark (15 January 2017). "zlib Home Site" (section "CRC (Cyclic
Redundancy Check) Bonus Information").  Contains links to spoof.c.gz and
crcany.tar.gz.
  <http://zlib.net/>
  <https://github.com/madler?tab=repositories>

Bies, Lammert; et al.  "Computer Interfacing Forum" (section "Error
detection and correction").
  <https://www.lammertbies.nl/forum/viewforum.php?f=11>

Cook, Greg (6 February 2017).  "Catalogue of parametrised CRC
algorithms".
  <http://reveng.sourceforge.net/crc-catalogue/>

Ewing, Gregory C. (March 2010). "Reverse-Engineering a CRC Algorithm".
Christchurch: University of Canterbury.
  <https://www.cosc.canterbury.ac.nz/greg.ewing/essays/
    CRC-Reverse-Engineering.html>

Koopman, Philip (July 2002).  "32-Bit Cyclic Redundancy Codes for
Internet Applications".  The International Conference on Dependable
Systems and Networks: 459-468.  doi:10.1109/DSN.2002.1028931.
  <https://users.ece.cmu.edu/~koopman/networks/dsn02/dsn02_koopman.pdf>

Koopman, Philip (23 January 2017). "Best CRC Polynomials". Pittsburgh:
Carnegie Mellon University.
  <https://users.ece.cmu.edu/~koopman/crc/>

Koopman, Philip; Chakravarty, Tridib (June 2004).  "Cyclic Redundancy
Code (CRC) Polynomial Selection For Embedded Networks".  The
International Conference on Dependable Systems and Networks: 145-154.
doi:10.1109/DSN.2004.1311885.
  <https://users.ece.cmu.edu/~koopman/roses/dsn04/
    koopman04_crc_poly_embedded.pdf>

Pircher, Thomas (12 December 2016). pycrc. Python based parametrised CRC
calculator and C code generator.
  <https://pycrc.org/>

Stigge, Martin; Ploetz, Henryk; Mueller, Wolf; Redlich, Jens-Peter
(May 2006).  "Reversing CRC -- Theory and Practice".  Berlin: Humboldt
University Berlin.
  <https://sar.informatik.hu-berlin.de/research/publications/
    SAR-PR-2006-05/SAR-PR-2006-05_.pdf>

Williams, Ross N. (24 September 1996). "A Painless Guide to CRC Error
Detection Algorithms V3.00".
  <http://www.ross.net/crc/download/crc_v3.txt>
  <http://www.wolfgang-ehrhardt.de/crc_v3.html>

INSPIRATION

CRC RevEng came about from the coincidence of four events:

* Finding the following problems on Lammert Bies' forum, which were
  beyond the precision or speed of crcbfs.pl:
  o "baexps_pr1", 3 August 2010, reverse CRC-64 calculation
    <https://www.lammertbies.nl/forum/viewtopic.php?t=1589>
  o "movilsystem", 20 August 2010, unknown CRC-16 in 2055-byte codewords
    <https://www.lammertbies.nl/forum/viewtopic.php?t=1595>
* Meeting Llamasoft founder Jeff Minter at the R3PLAY Expo 2010 in
  Blackpool, and subsequently reading his posts on Twitter that November
  on the progress of code for his game, "Minotaur Rescue".
    <https://twitter.com/llamasoft_ox/status/3040146031116288>
* Heavy snowfall in London from 30 November 2010, and:
* An Internet outage on 2 December 2010, diverting attention to
  rainy-day projects.

THANKS

The author would like to thank Dr. Mark Adler, Lammert Bies, Wolfgang
Ehrhardt, Greg Ewing, Prof. Philip Koopman, Thomas Pircher, Dr. Ross
Williams, and all contributors to CRC RevEng and the CRC Catalogue.

AUTHOR

Greg Cook
<debounce@yahoo.co.uk>
<http://regregex.bbcmicro.net/>

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