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<h1><img src="../../boost.png" alt="boost.png (6897 bytes)"
align="middle" width="277" height="86">Header <cite>&lt;<a
href="../../boost/crc.hpp">boost/crc.hpp</a>&gt;</cite></h1>

<p>The header <cite>&lt;<a
href="../../boost/crc.hpp">boost/crc.hpp</a>&gt;</cite> supplies two
class templates in namespace <code>boost</code>.  These templates define
objects that can compute the <dfn>CRC</dfn>, or cyclic redundancy code
(or check), of a given stream of data.  The header also supplies
function templates to compute a CRC in one step.</p>

<h2><a name="contents">Contents</a></h2>

<ol>
	<li><a href="#contents">Contents</a></li>
	<li><a href="#header">Header Synopsis</a></li>
	<li><a href="#rationale">Rationale</a></li>
	<li><a href="#background">Background</a>
		<ul>
			<li><a href="#parameters">CRC Parameters</a></li>
		</ul></li>
	<li><a href="#crc_basic">Theoretical CRC Computer</a></li>
	<li><a href="#crc_optimal">Optimized CRC Computer</a></li>
	<li><a href="#usage">Computer Usage</a></li>
	<li><a href="#crc_func">CRC Function</a></li>
	<li><a href="#a_crc_func">Augmented-CRC Function</a></li>
	<li><a href="#crc_ex">Pre-Defined CRC Samples</a></li>
	<li><a href="#references">References</a></li>
	<li><a href="#credits">Credits</a>
		<ul>
			<li><a href="#contributors">Contributors</a></li>
			<li><a href="#acknowledgements">Acknowledgements</a></li>
			<li><a href="#history">History</a></li>
		</ul></li>
</ol>

<h2><a name="header">Header Synopsis</a></h2>

<blockquote><pre>#include &lt;boost/integer.hpp&gt;  <i>// for boost::uint_t</i>
#include &lt;cstddef&gt;            <i>// for std::size_t</i>

namespace boost
{

template &lt; std::size_t Bits &gt;
    class crc_basic;

template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly = 0u,
           <em>impl_def</em> InitRem = 0u,
           <em>impl_def</em> FinalXor = 0u, bool ReflectIn = false,
           bool ReflectRem = false &gt;
    class crc_optimal;

template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly,
           <em>impl_def</em> InitRem, <em>impl_def</em> FinalXor,
           bool ReflectIn, bool ReflectRem &gt;
    typename uint_t&lt;Bits&gt;::fast  crc( void const *buffer,
     std::size_t byte_count );

template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly &gt;
    typename uint_t&lt;Bits&gt;::fast  augmented_crc( void const *buffer,
     std::size_t byte_count,
     typename uint_t&lt;Bits&gt;::fast initial_remainder = 0u );

typedef crc_optimal&lt;16, 0x8005, 0, 0, true, true&gt;         crc_16_type;
typedef crc_optimal&lt;16, 0x1021, 0xFFFF, 0, false, false&gt;  crc_ccitt_type;
typedef crc_optimal&lt;16, 0x8408, 0, 0, true, true&gt;         crc_xmodem_type;

typedef crc_optimal&lt;32, 0x04C11DB7, 0xFFFFFFFF, 0xFFFFFFFF, true, true&gt;
  crc_32_type;

}
</pre></blockquote>

<p>The implementation-defined type <var>impl_def</var> stands for the
quickest-to-manipulate built-in unsigned integral type that can
represent at least <var>Bits</var> bits.</p>

<h2><a name="rationale">Rationale</a></h2>

<p>A common error detection technique, especially with electronic
communications, is an appended checksum.  The transmitter sends its data
bits, followed by the bits of the checksum.  The checksum is based on
operations done on the data bit stream.  The receiver applies the same
operations on the bits it gets, and then gets the checksum.  If the
computed checksum doesn't match the received checksum, then an error
ocurred in the transmission.  There is the slight chance that the error
is only in the checksum, and an actually-correct data stream is
rejected.  There is also the chance of an error occurring that does not
change the checksum, making that error invisible.  CRC is a common
checksum type, used for error detection for hardware interfaces and
encoding formats.</p>

<h2><a name="background">Background</a></h2>

<p>CRCs work by computing the remainder of a modulo-2 polynominal
division.  The message is treated as the (binary) coefficents of a long
polynominal for the dividend, with the earlier bits of the message fed
first as the polynominal's highest coefficents.  A particular CRC
algorithm has another polynominal associated with it to be used as the
divisor.  The quotient is ignored.  The remainder of the division
considered the checksum.  However, the division uses modulo-2 rules (no
carries) for the coefficents.</p>

<p>See <cite><a href="http://www.ross.net/crc/crcpaper.html">A
Painless Guide to CRC Error Detection Algorithms</a></cite> for complete
information.  A clearer guide is at the <a
href="http://www.netrino.com/Embedded-Systems/How-To/CRC-Calculation-C-Code">CRC
Implementation Code in C</a> web page.</p>

<h3><a name="parameters">CRC Parameters</a></h3>

<dl>
	<dt>Truncated polynominal
	<dd>The divisor polynominal has a degree one bit larger than the
		checksum (remainder) size.  That highest bit is always one, so
		it is ignored when describing a particular CRC type.  Excluding
		this bit makes the divisor fit in the same data type as the
		checksum.

	<dt>Initial remainder
	<dd>The interim CRC remainder changes as each bit is processed. 
		Usually, the interim remainder starts at zero, but some CRCs use
		a different initial value to avoid &quot;blind spots.&quot;  A
		blind spot is when a common sequence of message bits does not
		change certain interim remainder values.

	<dt>Final XOR value
	<dd>A CRC remainder can be combined with a defined value, <i>via</i>
		a bitwise exclusive-or operation, before being returned to the
		user.  The value is usually zero, meaning the interim remainder
		is returned unchanged.  The other common value is an all-ones
		value, meaning that the bitwise complement of the interim
		remainder is returned.

	<dt>Reflected input
	<dd>A message's bits are usually fed a byte at a time, with the
		highest bits of the byte treated as the higher bits of the
		dividend polynominal.  Some CRCs reflect the bits (about the
		byte's center, so the first and last bits are switched,
		<i>etc.</i>) before feeding.

	<dt>Reflected (remainder) output
	<dd>Some CRCs return the reflection of the interim remainder (taking
		place <em>before</em> the final XOR value stage).
</dl>

<h2><a name="crc_basic">Theoretical CRC Computer</a></h2>

<blockquote><pre>template &lt; std::size_t Bits &gt;
class boost::crc_basic
{
public:
    // Type
    typedef <em>implementation_defined</em>  value_type;

    // Constant reflecting template parameter
    static  std::size_t const  bit_count = Bits;

    // Constructor
    explicit  crc_basic( value_type truncated_polynominal,
               value_type initial_remainder = 0, value_type final_xor_value = 0,
               bool reflect_input = false, bool reflect_remainder = false );

    // Internal Operations
    value_type  get_truncated_polynominal() const;
    value_type  get_initial_remainder() const;
    value_type  get_final_xor_value() const;
    bool        get_reflect_input() const;
    bool        get_reflect_remainder() const;

    value_type  get_interim_remainder() const;
    void        reset( value_type new_rem );
    void        reset();

    // External Operations
    void  process_bit( bool bit );
    void  process_bits( unsigned char bits, std::size_t bit_count );
    void  process_byte( unsigned char byte );
    void  process_block( void const *bytes_begin, void const *bytes_end );
    void  process_bytes( void const *buffer, std::size_t byte_count );

    value_type  checksum() const;

};
</pre></blockquote>

<p>The <code>value_type</code> is the smallest built-in type that can
hold the specified (by <code>Bits</code>) number of bits.  This should
be <code>boost::uint_t&lt;Bits&gt;::least</code>, see the <a
href="../integer/doc/html/boost_integer/integer.html">documentation for integer type
selection</a> for details.</p>

<p>This implementation is slow since it computes its CRC the same way as
in theory, bit by bit.  No optimizations are performed.  It wastes space
since most of the CRC parameters are specified at run-time as
constructor parameters.</p>

<h2><a name="crc_optimal">Optimized CRC Computer</a></h2>

<blockquote><pre>template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly,
           <em>impl_def</em> InitRem, <em>impl_def</em> FinalXor,
           bool ReflectIn, bool ReflectRem &gt;
class boost::crc_optimal
{
public:
    // Type
    typedef <em>implementation_defined</em>  value_type;

    // Constants reflecting template parameters
    static  std::size_t const  bit_count = Bits;
    static  value_type const   truncated_polynominal = TruncPoly;
    static  value_type const   initial_remainder = InitRem;
    static  value_type const   final_xor_value = FinalXor;
    static  bool const         reflect_input = ReflectIn;
    static  bool const         reflect_remainder = ReflectRem;

    // Constructor
    explicit  crc_optimal( value_type init_rem = InitRem );

    // Internal Operations
    value_type  get_truncated_polynominal() const;
    value_type  get_initial_remainder() const;
    value_type  get_final_xor_value() const;
    bool        get_reflect_input() const;
    bool        get_reflect_remainder() const;

    value_type  get_interim_remainder() const;
    void        reset( value_type new_rem = InitRem );

    // External Operations
    void  process_byte( unsigned char byte );
    void  process_block( void const *bytes_begin, void const *bytes_end );
    void  process_bytes( void const *buffer, std::size_t byte_count );

    value_type  checksum() const;

    // Operators
    void        operator ()( unsigned char byte );
    value_type  operator ()() const;

};
</pre></blockquote>

<p>The <code>value_type</code> is the quickest-to-manipulate built-in
type that can hold at least the specified (by <code>Bits</code>) number
of bits.  This should be <code>boost::uint_t&lt;Bits&gt;::fast</code>. 
See the <a href="../integer/doc/html/boost_integer/integer.html">integer type selection
documentation</a> for details.  The <code>TruncPoly</code>,
<code>InitRem</code>, and <code>FinalXor</code> template parameters also
are of this type.</p>

<p>This implementation is fast since it uses as many optimizations as
practical.  All of the CRC parameters are specified at compile-time as
template parameters.  No individual bits are considered; only whole
bytes are passed.  A table of interim CRC values versus byte values is
pre-computed when the first object using a particular bit size,
truncated polynominal, and input reflection state is processed.</p>

<h2><a name="usage">Computer Usage</a></h2>

<p>The two class templates have different policies on where the CRC's
parameters go.  Both class templates use the number of bits in the CRC
as the first template parameter.  The theoretical computer class
template has the bit count as its only template parameter, all the other
CRC parameters are entered through the constructor.  The optimized
computer class template obtains all its CRC parameters as template
parameters, and instantiated objects are usually
default-constructed.</p>

<p>The CRC parameters can be inspected at run-time with the following
member functions: <code>get_truncated_polynominal</code>,
<code>get_initial_remainder</code>, <code>get_final_xor_value</code>,
<code>get_reflect_input</code>, and <code>get_reflect_remainder</code>. 
The fast computer also provides compile-time constants for its CRC
parameters.</p>

<p>The <code>get_interim_remainder</code> member function returns the
internal state of the CRC remainder.  It represents the unreflected
remainder of the last division.  Saving an interim remainder allows the
freezing of CRC processing, as long as the other CRC parameters and the
current position of the bit stream are saved.  Restarting a frozen
stream involves constructing a new computer with the most of the old
computer's parameters.  The only change is to use the frozen remainder
as the new computer's initial remainder.  Then the interrupted bit
stream can be fed as if nothing happened.  The fast CRC computer has a
special constructor that takes one argument, an interim remainder, for
this purpose (overriding the initial remainder CRC parameter).</p>

<p>The <code>reset</code> member functions reset the internal state of
the CRC remainder to the given value.  If no value is given, then the
internal remainder is set to the initial remainder value when the object
was created.  The remainder must be unreflected.  When a CRC calculation
is finished, calling <code>reset</code> lets the object be reused for a
new session.</p>

<p>After any construction, both CRC computers work the same way. 
Feeding new data to a computer is in a seperate operation(s) from
extracting the current CRC value from the computer.  The following table
lists the feeding and extracting operations.</p>

<table cellpadding="5" border="1">
	<caption>Regular CRC Operations</caption>
	<tr>
		<th>Operation</th>
		<th>Description</th>
	</tr>
	<tr>
		<td><code>void process_bit( bool bit );</code></td>
		<td>Feeds the single <var>bit</var> to the computer, updating
			the interim CRC.  It is only defined for the slow CRC
			computer.</td>
	</tr>
	<tr>
		<td><code>void process_bits( unsigned char bits, std::size_t
			bit_count );</code></td>
		<td>Acts as applying <code>process_bit</code> to the lowest
			<var>bit_count</var> bits given in <var>bits</var>, most
			significant relevant bit first.  The results are undefined
			if <var>bit_count</var> exceeds the number of bits per byte.
			It is only defined for the slow CRC computer.</td>
	</tr>
	<tr>
		<td><code>void process_byte( unsigned char byte );</code></td>
		<td>Acts as applying <code>process_bit</code> to the all the
			bits in <var>byte</var>.  If reflection is not desired, the
			bits are fed from the most to least significant.  The bits
			are fed in the opposite order if reflection is desired.</td>
	</tr>
	<tr>
		<td><code>void process_block( void const *bytes_begin, void
			const *bytes_end );</code></td>
		<td>Acts as applying <code>process_byte</code> to each byte in
			the given memory block.  This memory block starts at
			<var>bytes_begin</var> and finishes before
			<var>bytes_end</var>.  The bytes are processed in that
			order.</td>
	</tr>
	<tr>
		<td><code>void  process_bytes( void const *buffer, std::size_t
			byte_count );</code></td>
		<td>Acts as applying <code>process_byte</code> to each byte in
			the given memory block.  This memory block starts at
			<var>buffer</var> and lasts for <var>byte_count</var> bytes.
			 The bytes are processed in ascending order.</td>
	</tr>
	<tr>
		<td><code>value_type checksum() const;</code></td>
		<td>Returns the CRC checksum of the data passed in so far,
			possibly after applying the remainder-reflection and
			exclusive-or operations.</td>
	</tr>
	<tr>
		<td><code>void  operator ()( unsigned char byte );</code></td>
		<td>Calls <code>process_byte</code>.  This member function lets
			its object act as a (stateful) function object.  It is only
			defined for the fast CRC computer.</td>
	</tr>
	<tr>
		<td><code>value_type  operator ()() const;</code></td>
		<td>Calls <code>checksum</code>.  This member function lets
			its object act as a generator function object.  It is only
			defined for the fast CRC computer.</td>
	</tr>
</table>

<p>You can use them like this:</p>

<blockquote><pre>#include &lt;boost/crc.hpp&gt;      <i>// for boost::crc_basic, boost::crc_optimal</i>
#include &lt;boost/cstdint.hpp&gt;  <i>// for boost::uint16_t</i>

#include &lt;algorithm&gt;  <i>// for std::for_each</i>
#include &lt;cassert&gt;    <i>// for assert</i>
#include &lt;cstddef&gt;    <i>// for std::size_t</i>
#include &lt;iostream&gt;   <i>// for std::cout</i>
#include &lt;ostream&gt;    <i>// for std::endl</i>


// Main function
int
main ()
{
    // This is "123456789" in ASCII
    unsigned char const  data[] = { 0x31, 0x32, 0x33, 0x34, 0x35, 0x36, 0x37,
     0x38, 0x39 };
    std::size_t const    data_len = sizeof( data ) / sizeof( data[0] );

    // The expected CRC for the given data
    boost::uint16_t const  expected = 0x29B1;

    // Simulate CRC-CCITT
    boost::crc_basic&lt;16&gt;  crc_ccitt1( 0x1021, 0xFFFF, 0, false, false );
    crc_ccitt1.process_bytes( data, data_len );
    assert( crc_ccitt1.checksum() == expected );

    // Repeat with the optimal version (assuming a 16-bit type exists)
    boost::crc_optimal&lt;16, 0x1021, 0xFFFF, 0, false, false&gt;  crc_ccitt2;
    crc_ccitt2 = std::for_each( data, data + data_len, crc_ccitt2 );
    assert( crc_ccitt2() == expected );

    std::cout &lt;&lt; "All tests passed." &lt;&lt; std::endl;
    return 0;
}
</pre></blockquote>

<h2><a name="crc_func">CRC Function</a></h2>

<blockquote><pre>template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly,
 <em>impl_def</em> InitRem, <em>impl_def</em> FinalXor,
 bool ReflectIn, bool ReflectRem &gt;
typename boost::uint_t&lt;Bits&gt;::fast
boost::crc( void const *buffer, std::size_t byte_count );
</pre></blockquote>

<p>The <code>boost::crc</code> function template computes the CRC of a
given data block.  The data block starts at the address given by
<var>buffer</var> and lasts for <var>byte_count</var> bytes.  The CRC
parameters are passed through template arguments, identical to the
optimized CRC computer (<a href="#crc_optimal">see above</a>).  In fact,
such a computer is used to implement this function.</p>

<h2><a name="a_crc_func">Augmented-CRC Function</a></h2>

<blockquote><pre>template &lt; std::size_t Bits, <em>impl_def</em> TruncPoly &gt;
typename boost::uint_t&lt;Bits&gt;::fast
boost::augmented_crc( void const *buffer, std::size_t byte_count,
 typename boost::uint_t&lt;Bits&gt;::fast initial_remainder = 0u );
</pre></blockquote>

<p>All the other CRC-computing function or class templates work assuming
that the division steps start immediately on the first message bits. 
The <code>boost::augmented_crc</code> function template has a
different division order.  Instead of combining (<i>via</i> bitwise
exclusive-or) the current message bit with the highest bit of a separate
remainder, these templates shift a new message bit into the low bit of a
remainder register as the highest bit is shifted out.  The new method
means that the bits in the inital remainder value are processed before
any of the actual message bits are processed.  To compensate, the real
CRC can only be extracted after feeding enough zero bits (the same count
as the register size) after the message bits.</p>

<p>The template parameters of the function template are
the CRC's bit size (<code>Bits</code>) and the truncated polynominal
(<code>TruncPoly</code>).  The function parameters are the starting address of
the data block to be worked on (<var>buffer</var>), the number of bytes in that
data block (<var>byte_count</var>), and the incoming value of the remainder
(<var>initial_remainder</var>).  That last parameter defaults to zero if it is
ommitted.</p>

<p>This function template is useful if the bytes of the CRC directly
follow the message's bytes.  First, set the bytes of where the CRC will
go to zero.  Then use <code>augmented_crc</code> over the augmented
message, <i>i.e.</i> the message bytes and the appended CRC bytes.  Then
assign the result to the CRC.  To later check a received message, either
use <code>augmented_crc</code> (with the same parameters as
transmission, of course) on the received <em>unaugmented</em> message
and check if the result equals the CRC, or use
<code>augmented_crc</code> on the received <em>augmented</em> message
and check if the result equals zero.  Note that the CRC has to be stored
with the more-significant bytes first (big-endian).</p>

<p>Interruptions in the CRC data can be handled by feeding the result of
<code>augmented_crc</code> of the previous data block as the
<var>initial_remainder</var> when calling <code>augmented_crc</code> on
the next data block.  Remember that the actual CRC can only be
determined after feeding the augmented bytes.  Since this method uses
modulo-2 polynominal division at its most raw, neither final XOR values
nor reflection can be used.</p>

<p>Note that for the same CRC system, the initial remainder for
augmented message method will be different than for the unaugmented
message method.  The main exception is zero; if the augmented-CRC
algorithm uses a zero initial remainder, the equivalent unaugmented-CRC
algorithm will also use a zero initial remainder.  Given an initial
remainder for a augmented-CRC algorithm, the result from processing just
zero-valued CRC bytes without any message bytes is the equivalent inital
remainder for the unaugmented-CRC algorithm.  An example follows:</p>

<blockquote><pre>#include &lt;boost/crc.hpp&gt;      <i>// for boost::crc_basic, boost::augmented_crc</i>
#include &lt;boost/cstdint.hpp&gt;  <i>// for boost::uint16_t</i>

#include &lt;cassert&gt;    <i>// for assert</i>
#include &lt;iostream&gt;   <i>// for std::cout</i>
#include &lt;ostream&gt;    <i>// for std::endl</i>


// Main function
int
main ()
{
    using boost::uint16_t;
    using boost::augmented_crc;

    uint16_t        data[6] = { 2, 4, 31, 67, 98, 0 };
    uint16_t const  init_rem = 0x123;

    uint16_t  crc1 = augmented_crc&lt;16, 0x8005&gt;( data, sizeof(data), init_rem );

    uint16_t const  zero = 0;
    uint16_t const  new_init_rem = augmented_crc&lt;16, 0x8005&gt;( &amp;zero, sizeof(zero) );

    boost::crc_basic&lt;16&gt;  crc2( 0x8005, new_init_rem );
    crc2.process_block( data, &amp;data[5] );  // don't include CRC
    assert( crc2.checksum() == crc1 );

    std::cout &lt;&lt; "All tests passed." &lt;&lt; std::endl;
    return 0;
}
</pre></blockquote>

<h2><a name="crc_ex">Pre-Defined CRC Samples</a></h2>

<p>Four sample CRC types are given, representing several common CRC
algorithms.  For example, computations from <code>boost::crc_32_type</code>
can be used for implementing the PKZip standard.  Note that, in general, this
library is concerned with CRC implementation, and not with determining
&quot;good&quot; sets of CRC parameters.</p>

<table cellpadding="5" border="1">
	<caption>Common CRCs</caption>
	<tr>
		<th>Algorithm</th>
		<th>Example Protocols</th>
	</tr>
	<tr>
		<td><code>crc_16_type</code></td>
		<td>BISYNCH, ARC</td>
	</tr>
	<tr>
		<td><code>crc_ccitt_type</code></td>
		<td>designated by CCITT (Comit&eacute; Consultatif International
		    T&eacute;l&eacute;graphique et T&eacute;l&eacute;phonique)</td>
	</tr>
	<tr>
		<td><code>crc_xmodem_type</code></td>
		<td>XMODEM</td>
	</tr>
	<tr>
		<td><code>crc_32_type</code></td>
		<td>PKZip, AUTODIN II, Ethernet, FDDI</td>
	</tr>
</table>

<hr>

<h2><a name="references">References</a></h2>

<ul>
	<li>The CRC header itself: <cite><a href="../../boost/crc.hpp">crc.hpp</a></cite>
	<li>Some test code: <cite><a href="test/crc_test.cpp">crc_test.cpp</a></cite>
	<li>Some example code: <cite><a href="crc_example.cpp">crc_example.cpp</a></cite>
</ul>

<h2><a name="credits">Credits</a></h2>

<h3><a name="contributors">Contributors</a></h3>

<dl>
	<dt>Michael Barr (<a
		href="mailto:mbarr@netrino.com">mbarr@netrino.com</a>)
	<dd>Wrote <a
		href="http://www.netrino.com/Embedded-Systems/How-To/CRC-Calculation-C-Code">CRC 
		Implementation Code in C</a>, a less-confusing guide to implementing CRC
		algorithms.  (Originally published as &quot;Slow and Steady
		Never Lost the Race&quot; in the January 2000 issue of <cite><a
		href="http://www.embedded.com/">Embedded Systems
		Programming</a></cite>, pages 37&ndash;46.  The web version used to be
		known as <cite><a href="http://www.netrino.com/Connecting/2000-01/">Easier
		Said Than Done</a></cite>.)

	<dt>Daryle Walker
	<dd>Started the library and contributed the theoretical and optimal
		CRC computation class templates and the CRC computing function
		template.  Contributed <cite><a
		href="test/crc_test.cpp">crc_test.cpp</a></cite> and <cite><a
		href="crc_example.cpp">crc_example.cpp</a></cite>.

	<dt>Ross N. Williams
	<dd>Wrote <cite><a href="http://www.ross.net/crc/crcpaper.html">A
		Painless Guide to CRC Error Detection Algorithms</a></cite>, a
		definitive source of CRC information.
</dl>

<h3><a name="acknowledgements">Acknowledgements</a></h3>

<p>For giving advice on compiler/C++ compliance, implementation,
interface, algorithms, and bug reports:</p>

<ul>
	<li>Darin Adler</li>
	<li>Beman Dawes</li>
	<li>Doug Gregor</li>
	<li>John Maddock</li>
	<li>Joe Mariadassou</li>
	<li>Jens Maurer</li>
	<li>Vladimir Prus</li>
	<li>Joel Young</li>
</ul>

<h3><a name="history">History</a></h3>

<dl>
	<dt>18 Dec 2011, Daryle Walker
	<dd>Folded the two versions of <code>boost::augmented_crc</code> together.

	<dt>15 Jun 2003, Daryle Walker
	<dd>Added example program.

	<dt>14 May 2001, Daryle Walker
	<dd>Initial version.
</dl>

<hr>

<p>Revised: 18 December 2011</p>

<p>Copyright 2001, 2003, 2011 Daryle Walker. Use, modification, and distribution
are subject to the Boost Software License, Version 1.0.  (See accompanying
file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or a copy at
&lt;<a href="http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>&gt;.)</p>

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