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Merge pull request #11 from NAThompson/develop

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Modular exponentiation, modular multiplicative inverse
This commit is contained in:
Andrey Semashev
2018-11-03 22:46:30 +03:00
committed by GitHub
parent 099cf261b0 cad4623876
commit aa68e17f2e
20 changed files with 865 additions and 232 deletions
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doc/gcd/math-gcd.qbk
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@ -15,7 +15,7 @@ programming problems.
namespace boost
{
namespace math
namespace integer
{
template < typename IntegerType >
@ -53,10 +53,10 @@ programming problems.
[section GCD Function Object]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]
template < typename IntegerType >
class boost::math::gcd_evaluator
class boost::integer::gcd_evaluator
{
public:
// Types
@ -65,12 +65,12 @@ programming problems.
typedef IntegerType second_argument_type;
// Function object interface
constexpr result_type operator ()(
constexpr result_type operator ()(
first_argument_type const &a,
second_argument_type const &b ) const;
};
The boost::math::gcd_evaluator class template defines a function object
The boost::integer::gcd_evaluator class template defines a function object
class to return the greatest common divisor of two integers.
The template is parameterized by a single type, called IntegerType here.
This type should be a numeric type that represents integers.
@ -82,17 +82,17 @@ the GCD function template. If a numeric type wants to customize evaluations
of its greatest common divisors, then the type should specialize on the
gcd_evaluator class template.
Note that these function objects are `constexpr` in C++14 and later only.
Note that these function objects are `constexpr` in C++14 and later only.
They are also declared `noexcept` when appropriate.
[endsect]
[section LCM Function Object]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]
template < typename IntegerType >
class boost::math::lcm_evaluator
class boost::integer::lcm_evaluator
{
public:
// Types
@ -101,12 +101,12 @@ They are also declared `noexcept` when appropriate.
typedef IntegerType second_argument_type;
// Function object interface
constexpr result_type operator ()(
constexpr result_type operator ()(
first_argument_type const &a,
second_argument_type const &b ) const;
};
The boost::math::lcm_evaluator class template defines a function object
The boost::integer::lcm_evaluator class template defines a function object
class to return the least common multiple of two integers. The template
is parameterized by a single type, called IntegerType here. This type
should be a numeric type that represents integers. The result of the
@ -126,13 +126,13 @@ They are also declared `noexcept` when appropriate.
[section:run_time Run-time GCD & LCM Determination]
[*Header: ] [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]
[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]
template < typename IntegerType >
constexpr IntegerType boost::math::gcd( IntegerType const &a, IntegerType const &b );
constexpr IntegerType boost::integer::gcd( IntegerType const &a, IntegerType const &b );
template < typename IntegerType >
constexpr IntegerType boost::math::lcm( IntegerType const &a, IntegerType const &b );
constexpr IntegerType boost::integer::lcm( IntegerType const &a, IntegerType const &b );
template < typename IntegerType, typename... Args >
constexpr IntegerType gcd( IntegerType const &a, IntegerType const &b, Args const&... );
@ -148,18 +148,18 @@ They are also declared `noexcept` when appropriate.
std::pair<typename std::iterator_traits<I>::value_type, I>
lcm_range(I first, I last);
The boost::math::gcd function template returns the greatest common
The boost::integer::gcd function template returns the greatest common
(nonnegative) divisor of the two integers passed to it.
`boost::math::gcd_range` is the iteration of the above gcd algorithm over a
range, returning the greatest common divisor of all the elements. The algorithm
terminates when the gcd reaches unity or the end of the range. Thus it also
returns the iterator after the last element inspected because this may not be
`boost::integer::gcd_range` is the iteration of the above gcd algorithm over a
range, returning the greatest common divisor of all the elements. The algorithm
terminates when the gcd reaches unity or the end of the range. Thus it also
returns the iterator after the last element inspected because this may not be
equal to the end of the range. The variadic version of `gcd` behaves similarly
but does not indicate which input value caused the gcd to reach unity.
The boost::math::lcm function template returns the least common
The boost::integer::lcm function template returns the least common
(nonnegative) multiple of the two integers passed to it.
As with gcd, there are range and variadic versions of the function for
As with gcd, there are range and variadic versions of the function for
more than 2 arguments.
Note that these functions are constexpr in C++14 and later only.
@ -171,17 +171,17 @@ They are also declared `noexcept` when appropriate.
[note These functions are deprecated in favor of constexpr `gcd` and `lcm` on C++14 capable compilers.]
[*Header: ] [@../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>]
[*Header: ] [@../../../../boost/integer/common_factor_ct.hpp <boost/integer/common_factor_ct.hpp>]
typedef ``['unspecified]`` static_gcd_type;
template < static_gcd_type Value1, static_gcd_type Value2 >
struct boost::math::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
struct boost::integer::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
{
};
template < static_gcd_type Value1, static_gcd_type Value2 >
struct boost::math::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
struct boost::integer::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
{
};
@ -190,7 +190,7 @@ for use in integral-constant-expressions by the compiler. Usually this
the same type as `boost::uintmax_t`, but may fall back to being `unsigned long`
for some older compilers.
The boost::math::static_gcd and boost::math::static_lcm class templates
The boost::integer::static_gcd and boost::integer::static_lcm class templates
take two value-based template parameters of the ['static_gcd_type] type
and inherit from the type `boost::mpl::integral_c`.
Inherited from the base class, they have a member /value/
@ -201,7 +201,7 @@ is beyond the range of `static_gcd_type`.
[h3 Example]
#include <boost/math/common_factor.hpp>
#include <boost/integer/common_factor.hpp>
#include <algorithm>
#include <iterator>
#include <iostream>
@ -212,28 +212,28 @@ is beyond the range of `static_gcd_type`.
using std::endl;
cout << "The GCD and LCM of 6 and 15 are "
<< boost::math::gcd(6, 15) << " and "
<< boost::math::lcm(6, 15) << ", respectively."
<< boost::integer::gcd(6, 15) << " and "
<< boost::integer::lcm(6, 15) << ", respectively."
<< endl;
cout << "The GCD and LCM of 8 and 9 are "
<< boost::math::static_gcd<8, 9>::value
<< boost::integer::static_gcd<8, 9>::value
<< " and "
<< boost::math::static_lcm<8, 9>::value
<< boost::integer::static_lcm<8, 9>::value
<< ", respectively." << endl;
int a[] = { 4, 5, 6 }, b[] = { 7, 8, 9 }, c[3];
std::transform( a, a + 3, b, c, boost::math::gcd_evaluator<int>() );
std::transform( a, a + 3, b, c, boost::integer::gcd_evaluator<int>() );
std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
}
[endsect]
[section:gcd_header Header <boost/math/common_factor.hpp>]
[section:gcd_header Header <boost/integer/common_factor.hpp>]
This header simply includes the headers
[@../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>]
and [@../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>].
[@../../../../boost/integer/common_factor_ct.hpp <boost/integer/common_factor_ct.hpp>]
and [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>].
Note this is a legacy header: it used to contain the actual implementation,
but the compile-time and run-time facilities
@ -243,7 +243,7 @@ were moved to separate headers (since they were independent of each other).
[section:demo Demonstration Program]
The program [@../../../../libs/math/test/common_factor_test.cpp common_factor_test.cpp] is a demonstration of the results from
The program [@../../../../libs/integer/test/common_factor_test.cpp common_factor_test.cpp] is a demonstration of the results from
instantiating various examples of the run-time GCD and LCM function
templates and the compile-time GCD and LCM class templates.
(The run-time GCD and LCM class templates are tested indirectly through
@ -256,13 +256,13 @@ the run-time function templates.)
The greatest common divisor and least common multiple functions are
greatly used in some numeric contexts, including some of the other
Boost libraries. Centralizing these functions to one header improves
code factoring and eases maintainence.
code factoring and eases maintenance.
[endsect]
[section:gcd_history History]
* 24th April 2017 Moved to Jeremy Murphy's improved algorithms, added constexpr and noexcept support,
* 24th April 2017 Moved to Jeremy Murphy's improved algorithms, added constexpr and noexcept support,
added compiler intrinsic support, added variadic and range based versions of the algorithms.
* 13 May 2013 Moved into main Boost.Math Quickbook documentation.
* 17 Dec 2005: Converted documentation to Quickbook Format.
@ -288,5 +288,3 @@ Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]

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<div class="toc"><dl class="toc">
<dt><span class="section"><a href="extended_euclidean.html#boost_integer.extended_euclidean.introduction">Introduction</a></span></dt>
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<div class="titlepage"><div><div><h3 class="title">
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</h3></div></div></div>
<p>
The extended Euclidean algorithm solves the integer relation <span class="emphasis"><em>mx
+ ny</em></span> = gcd(<span class="emphasis"><em>m</em></span>, <span class="emphasis"><em>n</em></span>) for
<span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>.
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</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">integer</span><span class="special">/</span><span class="identifier">extended_euclidean</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">integer</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">euclidean_result_t</span> <span class="special">{</span>
<span class="identifier">Z</span> <span class="identifier">gcd</span><span class="special">;</span>
<span class="identifier">Z</span> <span class="identifier">x</span><span class="special">;</span>
<span class="identifier">Z</span> <span class="identifier">y</span><span class="special">;</span>
<span class="special">};</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">&gt;</span>
<span class="identifier">euclidean_result_t</span><span class="special">&lt;</span><span class="identifier">Z</span><span class="special">&gt;</span> <span class="identifier">extended_euclidean</span><span class="special">(</span><span class="identifier">Z</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">Z</span> <span class="identifier">n</span><span class="special">);</span>
<span class="special">}}</span>
</pre>
</div>
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<a name="boost_integer.extended_euclidean.usage"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.usage" title="Usage">Usage</a>
</h3></div></div></div>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">15</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">res</span> <span class="special">=</span> <span class="identifier">extended_euclidean</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">n</span><span class="special">);</span>
<span class="keyword">int</span> <span class="identifier">gcd</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">gcd</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">x</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">y</span><span class="special">;</span>
<span class="comment">// mx + ny = gcd(m,n) should now hold</span>
</pre>
<p>
Unlike most of the library, the extended Euclidean algorithm requires C++11
features.
</p>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_integer.extended_euclidean.references"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.references" title="References">References</a>
</h3></div></div></div>
<p>
Wagstaff, Samuel S., <span class="emphasis"><em>The Joy of Factoring</em></span>, Vol. 68.
American Mathematical Soc., 2013.
</p>
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2001-2009 Beman
Dawes, Daryle Walker, Gennaro Prota, John Maddock<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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<div class="toc"><dl>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.introduction">Introduction</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.synopsis">Synopsis</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_function_object">GCD Function
@ -38,7 +38,7 @@
Determination</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.compile_time">Compile time GCD
and LCM determination</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_header">Header &lt;boost/math/common_factor.hpp&gt;</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_header">Header &lt;boost/integer/common_factor.hpp&gt;</a></span></dt>
<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.demo">Demonstration Program</a></span></dt>
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<dt><span class="section"><a href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_history">History</a></span></dt>
@ -61,7 +61,7 @@
</h3></div></div></div>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span>
<span class="special">{</span>
<span class="keyword">namespace</span> <span class="identifier">math</span>
<span class="keyword">namespace</span> <span class="identifier">integer</span>
<span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span> <span class="special">&gt;</span>
@ -102,10 +102,10 @@
Object</a>
</h3></div></div></div>
<p>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/math/common_factor_rt.hpp" target="_top">&lt;boost/math/common_factor_rt.hpp&gt;</a>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/integer/common_factor_rt.hpp" target="_top">&lt;boost/integer/common_factor_rt.hpp&gt;</a>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gcd_evaluator</span>
<span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">gcd_evaluator</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="comment">// Types</span>
@ -120,11 +120,11 @@
<span class="special">};</span>
</pre>
<p>
The boost::math::gcd_evaluator class template defines a function object class
to return the greatest common divisor of two integers. The template is parameterized
by a single type, called IntegerType here. This type should be a numeric
type that represents integers. The result of the function object is always
nonnegative, even if either of the operator arguments is negative.
The boost::integer::gcd_evaluator class template defines a function object
class to return the greatest common divisor of two integers. The template
is parameterized by a single type, called IntegerType here. This type should
be a numeric type that represents integers. The result of the function object
is always nonnegative, even if either of the operator arguments is negative.
</p>
<p>
This function object class template is used in the corresponding version
@ -144,10 +144,10 @@
Object</a>
</h3></div></div></div>
<p>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/math/common_factor_rt.hpp" target="_top">&lt;boost/math/common_factor_rt.hpp&gt;</a>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/integer/common_factor_rt.hpp" target="_top">&lt;boost/integer/common_factor_rt.hpp&gt;</a>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lcm_evaluator</span>
<span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">lcm_evaluator</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="comment">// Types</span>
@ -162,13 +162,13 @@
<span class="special">};</span>
</pre>
<p>
The boost::math::lcm_evaluator class template defines a function object class
to return the least common multiple of two integers. The template is parameterized
by a single type, called IntegerType here. This type should be a numeric
type that represents integers. The result of the function object is always
nonnegative, even if either of the operator arguments is negative. If the
least common multiple is beyond the range of the integer type, the results
are undefined.
The boost::integer::lcm_evaluator class template defines a function object
class to return the least common multiple of two integers. The template is
parameterized by a single type, called IntegerType here. This type should
be a numeric type that represents integers. The result of the function object
is always nonnegative, even if either of the operator arguments is negative.
If the least common multiple is beyond the range of the integer type, the
results are undefined.
</p>
<p>
This function object class template is used in the corresponding version
@ -188,13 +188,13 @@
Determination</a>
</h3></div></div></div>
<p>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/math/common_factor_rt.hpp" target="_top">&lt;boost/math/common_factor_rt.hpp&gt;</a>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/integer/common_factor_rt.hpp" target="_top">&lt;boost/integer/common_factor_rt.hpp&gt;</a>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span> <span class="special">&gt;</span>
<span class="keyword">constexpr</span> <span class="identifier">IntegerType</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gcd</span><span class="special">(</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">b</span> <span class="special">);</span>
<span class="keyword">constexpr</span> <span class="identifier">IntegerType</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">gcd</span><span class="special">(</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">b</span> <span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span> <span class="special">&gt;</span>
<span class="keyword">constexpr</span> <span class="identifier">IntegerType</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lcm</span><span class="special">(</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">b</span> <span class="special">);</span>
<span class="keyword">constexpr</span> <span class="identifier">IntegerType</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">lcm</span><span class="special">(</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">b</span> <span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span> <span class="keyword">typename</span> <span class="identifier">IntegerType</span><span class="special">,</span> <span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Args</span> <span class="special">&gt;</span>
<span class="keyword">constexpr</span> <span class="identifier">IntegerType</span> <span class="identifier">gcd</span><span class="special">(</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">IntegerType</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="identifier">b</span><span class="special">,</span> <span class="identifier">Args</span> <span class="keyword">const</span><span class="special">&amp;...</span> <span class="special">);</span>
@ -211,8 +211,8 @@
<span class="identifier">lcm_range</span><span class="special">(</span><span class="identifier">I</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">I</span> <span class="identifier">last</span><span class="special">);</span>
</pre>
<p>
The boost::math::gcd function template returns the greatest common (nonnegative)
divisor of the two integers passed to it. <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gcd_range</span></code>
The boost::integer::gcd function template returns the greatest common (nonnegative)
divisor of the two integers passed to it. <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">gcd_range</span></code>
is the iteration of the above gcd algorithm over a range, returning the greatest
common divisor of all the elements. The algorithm terminates when the gcd
reaches unity or the end of the range. Thus it also returns the iterator
@ -222,7 +222,7 @@
to reach unity.
</p>
<p>
The boost::math::lcm function template returns the least common (nonnegative)
The boost::integer::lcm function template returns the least common (nonnegative)
multiple of the two integers passed to it. As with gcd, there are range and
variadic versions of the function for more than 2 arguments.
</p>
@ -249,17 +249,17 @@
</p></td></tr>
</table></div>
<p>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/math/common_factor_ct.hpp" target="_top">&lt;boost/math/common_factor_ct.hpp&gt;</a>
<span class="bold"><strong>Header: </strong></span> <a href="../../../../../boost/integer/common_factor_ct.hpp" target="_top">&lt;boost/integer/common_factor_ct.hpp&gt;</a>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="emphasis"><em>unspecified</em></span> <span class="identifier">static_gcd_type</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</span> <span class="identifier">static_gcd_type</span> <span class="identifier">Value1</span><span class="special">,</span> <span class="identifier">static_gcd_type</span> <span class="identifier">Value2</span> <span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">static_gcd</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">integral_c</span><span class="special">&lt;</span><span class="identifier">static_gcd_type</span><span class="special">,</span> <span class="identifier">implementation_defined</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">static_gcd</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">integral_c</span><span class="special">&lt;</span><span class="identifier">static_gcd_type</span><span class="special">,</span> <span class="identifier">implementation_defined</span><span class="special">&gt;</span>
<span class="special">{</span>
<span class="special">};</span>
<span class="keyword">template</span> <span class="special">&lt;</span> <span class="identifier">static_gcd_type</span> <span class="identifier">Value1</span><span class="special">,</span> <span class="identifier">static_gcd_type</span> <span class="identifier">Value2</span> <span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">static_lcm</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">integral_c</span><span class="special">&lt;</span><span class="identifier">static_gcd_type</span><span class="special">,</span> <span class="identifier">implementation_defined</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">static_lcm</span> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">integral_c</span><span class="special">&lt;</span><span class="identifier">static_gcd_type</span><span class="special">,</span> <span class="identifier">implementation_defined</span><span class="special">&gt;</span>
<span class="special">{</span>
<span class="special">};</span>
</pre>
@ -271,8 +271,8 @@
<span class="keyword">long</span></code> for some older compilers.
</p>
<p>
The boost::math::static_gcd and boost::math::static_lcm class templates take
two value-based template parameters of the <span class="emphasis"><em>static_gcd_type</em></span>
The boost::integer::static_gcd and boost::integer::static_lcm class templates
take two value-based template parameters of the <span class="emphasis"><em>static_gcd_type</em></span>
type and inherit from the type <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">integral_c</span></code>. Inherited from the base class,
they have a member <span class="emphasis"><em>value</em></span> that is the greatest common
factor or least common multiple, respectively, of the template arguments.
@ -283,7 +283,7 @@
<a name="boost_integer.gcd_lcm.compile_time.h0"></a>
<span class="phrase"><a name="boost_integer.gcd_lcm.compile_time.example"></a></span><a class="link" href="gcd_lcm.html#boost_integer.gcd_lcm.compile_time.example">Example</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">common_factor</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">integer</span><span class="special">/</span><span class="identifier">common_factor</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">algorithm</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iterator</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
@ -294,29 +294,29 @@
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"The GCD and LCM of 6 and 15 are "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gcd</span><span class="special">(</span><span class="number">6</span><span class="special">,</span> <span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lcm</span><span class="special">(</span><span class="number">6</span><span class="special">,</span> <span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", respectively."</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">gcd</span><span class="special">(</span><span class="number">6</span><span class="special">,</span> <span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">lcm</span><span class="special">(</span><span class="number">6</span><span class="special">,</span> <span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">", respectively."</span>
<span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"The GCD and LCM of 8 and 9 are "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">static_gcd</span><span class="special">&lt;</span><span class="number">8</span><span class="special">,</span> <span class="number">9</span><span class="special">&gt;::</span><span class="identifier">value</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">static_gcd</span><span class="special">&lt;</span><span class="number">8</span><span class="special">,</span> <span class="number">9</span><span class="special">&gt;::</span><span class="identifier">value</span>
<span class="special">&lt;&lt;</span> <span class="string">" and "</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">static_lcm</span><span class="special">&lt;</span><span class="number">8</span><span class="special">,</span> <span class="number">9</span><span class="special">&gt;::</span><span class="identifier">value</span>
<span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">static_lcm</span><span class="special">&lt;</span><span class="number">8</span><span class="special">,</span> <span class="number">9</span><span class="special">&gt;::</span><span class="identifier">value</span>
<span class="special">&lt;&lt;</span> <span class="string">", respectively."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">int</span> <span class="identifier">a</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">4</span><span class="special">,</span> <span class="number">5</span><span class="special">,</span> <span class="number">6</span> <span class="special">},</span> <span class="identifier">b</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">7</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">9</span> <span class="special">},</span> <span class="identifier">c</span><span class="special">[</span><span class="number">3</span><span class="special">];</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">transform</span><span class="special">(</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">a</span> <span class="special">+</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gcd_evaluator</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;()</span> <span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">transform</span><span class="special">(</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">a</span> <span class="special">+</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">integer</span><span class="special">::</span><span class="identifier">gcd_evaluator</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;()</span> <span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">copy</span><span class="special">(</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">c</span> <span class="special">+</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream_iterator</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;(</span><span class="identifier">cout</span><span class="special">,</span> <span class="string">" "</span><span class="special">)</span> <span class="special">);</span>
<span class="special">}</span>
</pre>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_integer.gcd_lcm.gcd_header"></a><a class="link" href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_header" title="Header &lt;boost/math/common_factor.hpp&gt;">Header &lt;boost/math/common_factor.hpp&gt;</a>
<a name="boost_integer.gcd_lcm.gcd_header"></a><a class="link" href="gcd_lcm.html#boost_integer.gcd_lcm.gcd_header" title="Header &lt;boost/integer/common_factor.hpp&gt;">Header &lt;boost/integer/common_factor.hpp&gt;</a>
</h3></div></div></div>
<p>
This header simply includes the headers <a href="../../../../../boost/math/common_factor_ct.hpp" target="_top">&lt;boost/math/common_factor_ct.hpp&gt;</a>
and <a href="../../../../../boost/math/common_factor_rt.hpp" target="_top">&lt;boost/math/common_factor_rt.hpp&gt;</a>.
This header simply includes the headers <a href="../../../../../boost/integer/common_factor_ct.hpp" target="_top">&lt;boost/integer/common_factor_ct.hpp&gt;</a>
and <a href="../../../../../boost/integer/common_factor_rt.hpp" target="_top">&lt;boost/integer/common_factor_rt.hpp&gt;</a>.
</p>
<p>
Note this is a legacy header: it used to contain the actual implementation,
@ -329,7 +329,7 @@
<a name="boost_integer.gcd_lcm.demo"></a><a class="link" href="gcd_lcm.html#boost_integer.gcd_lcm.demo" title="Demonstration Program">Demonstration Program</a>
</h3></div></div></div>
<p>
The program <a href="../../../../../libs/math/test/common_factor_test.cpp" target="_top">common_factor_test.cpp</a>
The program <a href="../../../../../libs/integer/test/common_factor_test.cpp" target="_top">common_factor_test.cpp</a>
is a demonstration of the results from instantiating various examples of
the run-time GCD and LCM function templates and the compile-time GCD and
LCM class templates. (The run-time GCD and LCM class templates are tested
@ -344,7 +344,7 @@
The greatest common divisor and least common multiple functions are greatly
used in some numeric contexts, including some of the other Boost libraries.
Centralizing these functions to one header improves code factoring and eases
maintainence.
maintenance.
</p>
</div>
<div class="section">
@ -394,7 +394,7 @@
</tr></table>
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@ -26,7 +26,7 @@
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="boost_integer.integer"></a><a class="link" href="integer.html" title="Integer Type Selection">Integer Type Selection</a>
</h2></div></div></div>
<div class="toc"><dl>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="integer.html#boost_integer.integer.synopsis">Synopsis</a></span></dt>
<dt><span class="section"><a href="integer.html#boost_integer.integer.easiest">Easiest-to-Manipulate
Types</a></span></dt>
@ -353,10 +353,10 @@
<span class="special">{</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">int_t</span><span class="special">&lt;</span><span class="number">24</span><span class="special">&gt;::</span><span class="identifier">least</span> <span class="identifier">my_var</span><span class="special">;</span> <span class="comment">// my_var has at least 24-bits</span>
<span class="comment">//...</span>
<span class="comment">// This one is guarenteed not to be truncated:</span>
<span class="comment">// This one is guaranteed not to be truncated:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">int_max_value_t</span><span class="special">&lt;</span><span class="number">1000</span><span class="special">&gt;::</span><span class="identifier">least</span> <span class="identifier">my1000</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
<span class="comment">//...</span>
<span class="comment">// This one is guarenteed not to be truncated, and as fast</span>
<span class="comment">// This one is guaranteed not to be truncated, and as fast</span>
<span class="comment">// to manipulate as possible, its size may be greater than</span>
<span class="comment">// that of my1000:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">int_max_value_t</span><span class="special">&lt;</span><span class="number">1000</span><span class="special">&gt;::</span><span class="identifier">fast</span> <span class="identifier">my_fast1000</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>

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</h3></div></div></div>
<p>
The modular multiplicative inverse of a number <span class="emphasis"><em>a</em></span> is
that number <span class="emphasis"><em>x</em></span> which satisfies <span class="emphasis"><em>ax</em></span>
= 1 mod <span class="emphasis"><em>p</em></span>. A fast algorithm for computing modular multiplicative
inverses based on the extended Euclidean algorithm exists and is provided
by Boost.
</p>
</div>
<div class="section">
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<a name="boost_integer.mod_inverse.synopsis"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.synopsis" title="Synopsis">Synopsis</a>
</h3></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">integer</span><span class="special">/</span><span class="identifier">mod_inverse</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">integer</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">&gt;</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">optional</span><span class="special">&lt;</span><span class="identifier">Z</span><span class="special">&gt;</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="identifier">Z</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Z</span> <span class="identifier">m</span><span class="special">);</span>
<span class="special">}}</span>
</pre>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_integer.mod_inverse.usage"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.usage" title="Usage">Usage</a>
</h3></div></div></div>
<p>
Multiplicative modular inverses exist if and only if <span class="emphasis"><em>a</em></span>
and <span class="emphasis"><em>m</em></span> are coprime. So for example
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">x</span><span class="special">)</span>
<span class="special">{</span>
<span class="keyword">int</span> <span class="identifier">should_be_three</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">value</span><span class="special">();</span>
<span class="special">}</span>
<span class="keyword">auto</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">4</span><span class="special">);</span>
<span class="keyword">if</span> <span class="special">(!</span><span class="identifier">y</span><span class="special">)</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"There is no inverse of 2 mod 4\n"</span><span class="special">;</span>
<span class="special">}</span>
</pre>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_integer.mod_inverse.references"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.references" title="References">References</a>
</h3></div></div></div>
<p>
Wagstaff, Samuel S., <span class="emphasis"><em>The Joy of Factoring</em></span>, Vol. 68.
American Mathematical Soc., 2013.
</p>
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2001-2009 Beman
Dawes, Daryle Walker, Gennaro Prota, John Maddock<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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<div class="toc">
<p><b>Table of Contents</b></p>
<dl>
<dl class="toc">
<dt><span class="section"><a href="index.html#boost_integer.overview">Overview</a></span></dt>
<dt><span class="section"><a href="boost_integer/traits.html">Integer Traits</a></span></dt>
<dt><span class="section"><a href="boost_integer/integer.html">Integer Type Selection</a></span></dt>
<dt><span class="section"><a href="boost_integer/gcd_lcm.html">Greatest Common Divisor and Least
Common Multiple</a></span></dt>
<dt><span class="section"><a href="boost_integer/extended_euclidean.html">Extended Euclidean Algorithm</a></span></dt>
<dt><span class="section"><a href="boost_integer/mod_inverse.html">Modular Multiplicative Inverse</a></span></dt>
<dt><span class="section"><a href="boost_integer/mask.html">Integer Masks</a></span></dt>
<dt><span class="section"><a href="boost_integer/log2.html">Compile Time log2 Calculation</a></span></dt>
<dt><span class="section"><a href="boost_integer/minmax.html">Compile time min/max calculation</a></span></dt>
@ -237,12 +239,50 @@
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="boost_integer/extended_euclidean.html" title="Extended Euclidean Algorithm">Extended Euclidean
algorithm</a>.
</p>
</td>
<td>
<p>
<code class="literal"><a href="../../../../boost/integer/extended_euclidean.hpp" target="_top">&lt;boost/integer/extended_euclidean.hpp&gt;</a></code>
</p>
</td>
<td>
<p>
Solves <span class="emphasis"><em>mx + ny = gcd(x,y)</em></span> for <span class="emphasis"><em>x</em></span>
and <span class="emphasis"><em>y</em></span>.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="boost_integer/mod_inverse.html" title="Modular Multiplicative Inverse">Modular multiplicative
inverse</a>.
</p>
</td>
<td>
<p>
<code class="literal"><a href="../../../../boost/integer/mod_inverse.hpp" target="_top">&lt;boost/integer/mod_inverse.hpp&gt;</a></code>
</p>
</td>
<td>
<p>
Given <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>m</em></span>, solves
<span class="emphasis"><em>ax</em></span> = 1 mod <span class="emphasis"><em>m</em></span> for <span class="emphasis"><em>x</em></span>.
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"><p><small>Last revised: April 24, 2017 at 17:49:59 GMT</small></p></td>
<td align="left"><p><small>Last revised: November 02, 2018 at 19:38:28 GMT</small></p></td>
<td align="right"><div class="copyright-footer"></div></td>
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270
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@ -2,7 +2,6 @@
[quickbook 1.6]
[compatibility-mode 1.5]
[copyright 2001-2009 Beman Dawes, Daryle Walker, Gennaro Prota, John Maddock]
[purpose Integer Type Selection]
[license
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
@ -17,13 +16,13 @@
[section:overview Overview]
Boost.Integer provides integer type support, particularly helpful in generic programming.
It provides the means to select an integer type based upon its properties, like the number of bits or
It provides the means to select an integer type based upon its properties, like the number of bits or
the maximum supported value, as well as compile-time bit mask selection. There is a derivative of
std::numeric_limits that provides integral constant expressions for `min` and `max`.
Finally, it provides two compile-time algorithms: determining the highest power of two in a
std::numeric_limits that provides integral constant expressions for `min` and `max`.
Finally, it provides two compile-time algorithms: determining the highest power of two in a
compile-time value; and computing min and max of constant expressions.
[table
[table
[[Component][Header][Purpose]]
[
[Forward Declarations.]
@ -38,8 +37,8 @@ compile-time value; and computing min and max of constant expressions.
[
[[link boost_integer.integer Integer Type Selection].]
[[^[@../../../../boost/integer.hpp <boost/integer.hpp>]]]
[Templates for integer type selection based on properties such as maximum value or number of bits:
Use to select the type of an integer when some property such as maximum value or number of bits is known.
[Templates for integer type selection based on properties such as maximum value or number of bits:
Use to select the type of an integer when some property such as maximum value or number of bits is known.
Useful for generic programming. ]
]
[
@ -50,21 +49,33 @@ compile-time value; and computing min and max of constant expressions.
[
[[link boost_integer.mask Integer Masks].]
[[^[@../../../../boost/integer/integer_mask.hpp <boost/integer/integer_mask.hpp>]]]
[Templates for the selection of integer masks, single or lowest group, based on the number of bits:
[Templates for the selection of integer masks, single or lowest group, based on the number of bits:
Use to select a particular mask when the bit position(s) are based on a compile-time variable. Useful for generic programming. ]
]
[
[[link boost_integer.log2 Compile time log2 Calculation].]
[[^[@../../../../boost/integer/static_log2.hpp <boost/integer/static_log2.hpp>]]]
[Template for finding the highest power of two in a number:
[Template for finding the highest power of two in a number:
Use to find the bit-size/range based on a maximum value. Useful for generic programming. ]
]
[
[[link boost_integer.minmax Compile time min/max calculation].]
[[^[@../../../../boost/integer/static_min_max.hpp <boost/integer/static_min_max.hpp>]]]
[Templates for finding the extrema of two numbers:
[Templates for finding the extrema of two numbers:
Use to find a bound based on a minimum or maximum value. Useful for generic programming. ]
]
[
[[link boost_integer.extended_euclidean Extended Euclidean algorithm].]
[[^[@../../../../boost/integer/extended_euclidean.hpp <boost/integer/extended_euclidean.hpp>]]]
[Solves /mx + ny = gcd(x,y)/ for /x/ and /y/.]
]
[
[[link boost_integer.mod_inverse Modular multiplicative inverse].]
[[^[@../../../../boost/integer/mod_inverse.hpp <boost/integer/mod_inverse.hpp>]]]
[Given /a/ and /m/, solves /ax/ = 1 mod /m/ for /x/.]
]
]
[endsect]
@ -75,7 +86,7 @@ compile-time value; and computing min and max of constant expressions.
The C++ Standard Library <limits> header supplies a class template `numeric_limits<>` with specializations for each fundamental type.
For integer types, the interesting members of `std::numeric_limits<>` are:
For integer types, the interesting members of `std::numeric_limits<>` are:
static const bool is_specialized; // Will be true for integer types.
static T min() throw(); // Smallest representable value.
@ -85,13 +96,13 @@ For integer types, the interesting members of `std::numeric_limits<>` are:
static const bool is_signed; // True if the type is signed.
static const bool is_integer; // Will be true for all integer types.
For many uses, these are sufficient.
But min() and max() are problematical because they are not constant expressions (std::5.19),
yet some usages require constant expressions.
For many uses, these are sufficient.
But min() and max() are problematical because they are not constant expressions (std::5.19),
yet some usages require constant expressions.
The template class [^integer_traits] addresses this problem.
The template class [^integer_traits] addresses this problem.
[endsect]
[endsect]
[section Synopsis]
@ -110,33 +121,33 @@ The template class [^integer_traits] addresses this problem.
};
}
[endsect]
[endsect]
[section Description]
Template class [^integer_traits] is derived from [^std::numeric_limits]. The primary specialization adds the single
[^bool] member [^is_integral] with the compile-time constant value [^false].
However, for all integral types [^T] (std::3.9.1/7 [basic.fundamental]), there are specializations
provided with the following compile-time constants defined:
Template class [^integer_traits] is derived from [^std::numeric_limits]. The primary specialization adds the single
[^bool] member [^is_integral] with the compile-time constant value [^false].
However, for all integral types [^T] (std::3.9.1/7 [basic.fundamental]), there are specializations
provided with the following compile-time constants defined:
[table
[table
[[member][type][value]]
[[[^is_integral]][bool][[^true]]]
[[[^const_min]][[^T]][equivalent to [^std::numeric_limits<T>::min()]]]
[[[^const_max]][[^T]][equivalent to [^std::numeric_limits<T>::max()]]]
]
Note: The /is_integral/ flag is provided, because a user-defined integer class should specialize
[^std::numeric_limits<>::is_integer = true], while compile-time constants
[^const_min] and [^const_max] are not provided for that user-defined class, unless boost::integer_traits is also specialized.
Note: The /is_integral/ flag is provided, because a user-defined integer class should specialize
[^std::numeric_limits<>::is_integer = true], while compile-time constants
[^const_min] and [^const_max] are not provided for that user-defined class, unless boost::integer_traits is also specialized.
[endsect]
[endsect]
[section Test Program]
The program [^[@../../test/integer_traits_test.cpp integer_traits_test.cpp]] exercises the [^integer_traits] class.
The program [^[@../../test/integer_traits_test.cpp integer_traits_test.cpp]] exercises the [^integer_traits] class.
[endsect]
[endsect]
[section Acknowledgements]
@ -147,8 +158,8 @@ Beman Dawes, Ed Brey, Steve Cleary, and Nathan Myers discussed the integer trait
[section:integer Integer Type Selection]
The [@../../../../boost/integer.hpp <boost/integer.hpp>] type selection templates allow
integer types to be selected based on desired characteristics such as number of bits or maximum value.
The [@../../../../boost/integer.hpp <boost/integer.hpp>] type selection templates allow
integer types to be selected based on desired characteristics such as number of bits or maximum value.
This facility is particularly useful for solving generic programming problems.
[section:synopsis Synopsis]
@ -164,7 +175,7 @@ This facility is particularly useful for solving generic programming problems.
// signed
template<int Bits>
struct int_t
struct int_t
{
/* Member exact may or may not be defined depending upon Bits */
typedef ``['implementation-defined-type]`` exact;
@ -174,7 +185,7 @@ This facility is particularly useful for solving generic programming problems.
// unsigned
template<int Bits>
struct uint_t
struct uint_t
{
/* Member exact may or may not be defined depending upon Bits */
typedef ``['implementation-defined-type]`` exact;
@ -184,14 +195,14 @@ This facility is particularly useful for solving generic programming problems.
// signed
template<long long MaxValue>
struct int_max_value_t
struct int_max_value_t
{
typedef ``['implementation-defined-type]`` least;
typedef int_fast_t<least>::fast fast;
};
template<long long MinValue>
struct int_min_value_t
struct int_min_value_t
{
typedef ``['implementation-defined-type]`` least;
typedef int_fast_t<least>::fast fast;
@ -199,7 +210,7 @@ This facility is particularly useful for solving generic programming problems.
// unsigned
template<unsigned long long Value>
struct uint_value_t
struct uint_value_t
{
typedef ``['implementation-defined-type]`` least;
typedef int_fast_t<least>::fast fast;
@ -210,25 +221,25 @@ This facility is particularly useful for solving generic programming problems.
[section:easiest Easiest-to-Manipulate Types]
The [^int_fast_t] class template maps its input type to the next-largest type that the processor
can manipulate the easiest, or to itself if the input type is already an easy-to-manipulate type.
For instance, processing a bunch of [^char] objects may go faster if they were converted to [^int] objects before processing.
The input type, passed as the only template parameter, must be a built-in integral type, except [^bool].
Unsigned integral types can be used, as well as signed integral types.
The [^int_fast_t] class template maps its input type to the next-largest type that the processor
can manipulate the easiest, or to itself if the input type is already an easy-to-manipulate type.
For instance, processing a bunch of [^char] objects may go faster if they were converted to [^int] objects before processing.
The input type, passed as the only template parameter, must be a built-in integral type, except [^bool].
Unsigned integral types can be used, as well as signed integral types.
The output type is given as the nested type [^fast].
[*Implementation Notes:]
By default, the output type is identical to the input type. Eventually, this code's implementation should
be customized for each platform to give accurate mappings between the built-in types and the easiest-to-manipulate
[*Implementation Notes:]
By default, the output type is identical to the input type. Eventually, this code's implementation should
be customized for each platform to give accurate mappings between the built-in types and the easiest-to-manipulate
built-in types. Also, there is no guarantee that the output type actually is easier to manipulate than the input type.
[endsect]
[section:sized Sized Types]
The [^int_t], [^uint_t], [^int_max_value_t], [^int_min_value_t], and [^uint_value_t] class templates find
the most appropiate built-in integral type for the given template parameter. This type is given by the
nested type [^least]. The easiest-to-manipulate version of that type is given by the nested type [^fast].
The [^int_t], [^uint_t], [^int_max_value_t], [^int_min_value_t], and [^uint_value_t] class templates find
the most appropiate built-in integral type for the given template parameter. This type is given by the
nested type [^least]. The easiest-to-manipulate version of that type is given by the nested type [^fast].
The following table describes each template's criteria.
[table Criteria for the Sized Type Class Templates
@ -237,60 +248,60 @@ The following table describes each template's criteria.
]
[
[[^boost::int_t<N>::least]]
[The smallest, built-in, signed integral type with at least /N/ bits, including the sign bit.
The parameter should be a positive number. A compile-time error results if the parameter is
[The smallest, built-in, signed integral type with at least /N/ bits, including the sign bit.
The parameter should be a positive number. A compile-time error results if the parameter is
larger than the number of bits in the largest integer type.]
]
[
[[^boost::int_t<N>::fast]]
[The easiest-to-manipulate, built-in, signed integral type with at least /N/ bits, including the sign bit.
The parameter should be a positive number. A compile-time error results if the parameter is
[The easiest-to-manipulate, built-in, signed integral type with at least /N/ bits, including the sign bit.
The parameter should be a positive number. A compile-time error results if the parameter is
larger than the number of bits in the largest integer type.]
]
[
[[^boost::int_t<N>::exact]]
[A built-in, signed integral type with exactly /N/ bits, including the sign bit.
[A built-in, signed integral type with exactly /N/ bits, including the sign bit.
The parameter should be a positive number. Note that the member /exact/ is defined
[*only] if there exists a type with exactly /N/ bits.]
]
[
[[^boost::uint_t<N>::least]]
[The smallest, built-in, unsigned integral type with at least /N/ bits.
The parameter should be a positive number. A compile-time error results if the
[The smallest, built-in, unsigned integral type with at least /N/ bits.
The parameter should be a positive number. A compile-time error results if the
parameter is larger than the number of bits in the largest integer type.]
]
[
[[^boost::uint_t<N>::fast]]
[The easiest-to-manipulate, built-in, unsigned integral type with at least /N/ bits.
The parameter should be a positive number. A compile-time error results if the
[The easiest-to-manipulate, built-in, unsigned integral type with at least /N/ bits.
The parameter should be a positive number. A compile-time error results if the
parameter is larger than the number of bits in the largest integer type.]
]
[
[[^boost::uint_t<N>::exact]]
[A built-in, unsigned integral type with exactly /N/ bits.
The parameter should be a positive number. A compile-time error results if the
parameter is larger than the number of bits in the largest integer type.
[A built-in, unsigned integral type with exactly /N/ bits.
The parameter should be a positive number. A compile-time error results if the
parameter is larger than the number of bits in the largest integer type.
Note that the member /exact/ is defined
[*only] if there exists a type with exactly N bits.]
]
[
[[^boost::int_max_value_t<V>::last]]
[The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
[The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
The parameter should be a positive number.]
]
[
[[^boost::int_max_value_t<V>::fast]]
[The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
[The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
The parameter should be a positive number.]
]
[
[[^boost::int_min_value_t<V>::least]]
[The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
[The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
The parameter should be a negative number.]
]
[
[[^boost::int_min_value_t<V>::fast]]
[The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
[The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
The parameter should be a negative number.]
]
[
@ -317,10 +328,10 @@ The following table describes each template's criteria.
{
boost::int_t<24>::least my_var; // my_var has at least 24-bits
//...
// This one is guarenteed not to be truncated:
// This one is guaranteed not to be truncated:
boost::int_max_value_t<1000>::least my1000 = 1000;
//...
// This one is guarenteed not to be truncated, and as fast
// This one is guaranteed not to be truncated, and as fast
// to manipulate as possible, its size may be greater than
// that of my1000:
boost::int_max_value_t<1000>::fast my_fast1000 = 1000;
@ -330,7 +341,7 @@ The following table describes each template's criteria.
[section Demonstration Program]
The program [@../../test/integer_test.cpp integer_test.cpp] is a simplistic demonstration of the results from instantiating
The program [@../../test/integer_test.cpp integer_test.cpp] is a simplistic demonstration of the results from instantiating
various examples of the sized type class templates.
[endsect]
@ -347,31 +358,32 @@ The rationale for the design of the templates in this header includes:
[section Alternative]
If the number of bits required is known beforehand, it may be more appropriate to use the types supplied
If the number of bits required is known beforehand, it may be more appropriate to use the types supplied
in [@../../../../boost/cstdint.hpp <boost/cstdint.hpp>].
[endsect]
[section Credits]
The author of most of the Boost integer type choosing templates is
[@http://www.boost.org/people/beman_dawes.html Beman Dawes].
He gives thanks to Valentin Bonnard and [@http://www.boost.org/people/kevlin_henney.htm Kevlin Henney]
for sharing their designs for similar templates.
The author of most of the Boost integer type choosing templates is
[@http://www.boost.org/people/beman_dawes.html Beman Dawes].
He gives thanks to Valentin Bonnard and [@http://www.boost.org/people/kevlin_henney.htm Kevlin Henney]
for sharing their designs for similar templates.
[@http://www.boost.org/people/daryle_walker.html Daryle Walker] designed the value-based sized templates.
[endsect]
[endsect]
[include gcd/math-gcd.qbk]
[include modular_arithmetic/extended_euclidean.qbk]
[include modular_arithmetic/mod_inverse.qbk]
[section:mask Integer Masks]
[section Overview]
The class templates in [@../../../../boost/integer/integer_mask.hpp <boost/integer/integer_mask.hpp>]
provide bit masks for a certain bit position or a contiguous-bit pack of a certain size.
The class templates in [@../../../../boost/integer/integer_mask.hpp <boost/integer/integer_mask.hpp>]
provide bit masks for a certain bit position or a contiguous-bit pack of a certain size.
The types of the masking constants come from the [link boost_integer.integer integer type selection templates] header.
[endsect]
@ -411,14 +423,14 @@ The types of the masking constants come from the [link boost_integer.integer int
} // namespace boost
[endsect]
[endsect]
[section Single Bit-Mask Class Template]
The [^boost::high_bit_mask_t] class template provides constants for bit masks representing the bit at a
certain position. The masks are equivalent to the value 2[super Bit], where [^Bit] is the template parameter.
The bit position must be a nonnegative number from zero to ['Max], where Max is one less than the
number of bits supported by the largest unsigned built-in integral type. The following table describes
The [^boost::high_bit_mask_t] class template provides constants for bit masks representing the bit at a
certain position. The masks are equivalent to the value 2[super Bit], where [^Bit] is the template parameter.
The bit position must be a nonnegative number from zero to ['Max], where Max is one less than the
number of bits supported by the largest unsigned built-in integral type. The following table describes
the members of an instantiation of [^high_bit_mask_t].
[table Members of the `boost::high_bit_mask_t` Class Template
@ -430,14 +442,14 @@ the members of an instantiation of [^high_bit_mask_t].
[[[^bit_position]][The value of the template parameter, in case its needed from a renamed instantiation of the class template.]]
]
[endsect]
[endsect]
[section Group Bit-Mask Class Template]
The [^boost::low_bits_mask_t] class template provides constants for bit masks
equivalent to the value (2[super Bits] - 1), where [^Bits] is the template parameter.
The parameter [^Bits] must be a non-negative integer from
zero to ['Max], where Max is the number of bits supported by the largest, unsigned, built-in integral type.
The [^boost::low_bits_mask_t] class template provides constants for bit masks
equivalent to the value (2[super Bits] - 1), where [^Bits] is the template parameter.
The parameter [^Bits] must be a non-negative integer from
zero to ['Max], where Max is the number of bits supported by the largest, unsigned, built-in integral type.
The following table describes the members of [^low_bits_mask_t].
[table Members of the [^boost::low_bits_mask_t] Class Template
@ -453,7 +465,7 @@ The following table describes the members of [^low_bits_mask_t].
[section Implementation Notes]
When [^Bits] is the exact size of a built-in unsigned type, the implementation has to change to
When [^Bits] is the exact size of a built-in unsigned type, the implementation has to change to
prevent undefined behavior. Therefore, there are specializations of [^low_bits_mask_t] at those bit counts.
[endsect]
@ -479,23 +491,23 @@ prevent undefined behavior. Therefore, there are specializations of [^low_bits_m
//...
}
[endsect]
[endsect]
[section Demonstration Program]
The program [@../../test/integer_mask_test.cpp integer_mask_test.cpp] is a simplistic demonstration of the
The program [@../../test/integer_mask_test.cpp integer_mask_test.cpp] is a simplistic demonstration of the
results from instantiating various examples of the bit mask class templates.
[endsect]
[endsect]
[section Rationale]
The class templates in this header are an extension of the [link boost_integer.integer integer type selection class templates].
The new class templates provide the same sized types, but also convenient masks to use when extracting the
highest or all the significant bits when the containing built-in type contains more bits.
The class templates in this header are an extension of the [link boost_integer.integer integer type selection class templates].
The new class templates provide the same sized types, but also convenient masks to use when extracting the
highest or all the significant bits when the containing built-in type contains more bits.
This prevents contamination of values by the higher, unused bits.
[endsect]
[endsect]
[section Credits]
@ -506,7 +518,7 @@ The author of the Boost bit mask class templates is [@http://www.boost.org/peopl
[section:log2 Compile Time log2 Calculation]
The class template in [@../../../../boost/integer/static_log2.hpp <boost/integer/static_log2.hpp>]
The class template in [@../../../../boost/integer/static_log2.hpp <boost/integer/static_log2.hpp>]
determines the position of the highest bit in a given value. This facility is useful for solving generic programming problems.
[section Synopsis]
@ -533,47 +545,47 @@ determines the position of the highest bit in a given value. This facility is us
} // namespace boost
[endsect]
[endsect]
[section Usage]
The [^boost::static_log2] class template takes one template parameter, a value of type
[^static_log2_argument_type]. The template only defines one member, [^value], which gives the
The [^boost::static_log2] class template takes one template parameter, a value of type
[^static_log2_argument_type]. The template only defines one member, [^value], which gives the
truncated, base-two logarithm of the template argument.
Since the logarithm of zero, for any base, is undefined, there is a specialization of [^static_log2]
for a template argument of zero. This specialization has no members, so an attempt to use the base-two
Since the logarithm of zero, for any base, is undefined, there is a specialization of [^static_log2]
for a template argument of zero. This specialization has no members, so an attempt to use the base-two
logarithm of zero results in a compile-time error.
Note:
Note:
* [^static_log2_argument_type] is an ['unsigned integer type] (C++ standard, 3.9.1p3).
* [^static_log2_result_type] is an ['integer type] (C++ standard, 3.9.1p7).
[endsect]
[endsect]
[section Demonstration Program]
The program [@../../test/static_log2_test.cpp static_log2_test.cpp] is a simplistic
The program [@../../test/static_log2_test.cpp static_log2_test.cpp] is a simplistic
demonstration of the results from instantiating various examples of the binary logarithm class template.
[endsect]
[section Rationale]
The base-two (binary) logarithm, abbreviated lb, function is occasionally used to give order-estimates
of computer algorithms. The truncated logarithm can be considered the highest power-of-two in a value,
which corresponds to the value's highest set bit (for binary integers). Sometimes the highest-bit position
The base-two (binary) logarithm, abbreviated lb, function is occasionally used to give order-estimates
of computer algorithms. The truncated logarithm can be considered the highest power-of-two in a value,
which corresponds to the value's highest set bit (for binary integers). Sometimes the highest-bit position
could be used in generic programming, which requires the position to be available statically (['i.e.] at compile-time).
[endsect]
[endsect]
[section Credits]
The original version of the Boost binary logarithm class template was
written by [@http://www.boost.org/people/daryle_walker.html Daryle Walker] and then
enhanced by Giovanni Bajo with support for compilers without partial template specialization.
The current version was suggested, together with a reference implementation, by Vesa Karvonen.
The original version of the Boost binary logarithm class template was
written by [@http://www.boost.org/people/daryle_walker.html Daryle Walker] and then
enhanced by Giovanni Bajo with support for compilers without partial template specialization.
The current version was suggested, together with a reference implementation, by Vesa Karvonen.
Gennaro Prota wrote the actual source file.
[endsect]
@ -581,15 +593,15 @@ Gennaro Prota wrote the actual source file.
[section:minmax Compile time min/max calculation]
The class templates in [@../../../../boost/integer/static_min_max.hpp <boost/integer/static_min_max.hpp>]
provide a compile-time evaluation of the minimum or maximum of two integers. These facilities are useful
The class templates in [@../../../../boost/integer/static_min_max.hpp <boost/integer/static_min_max.hpp>]
provide a compile-time evaluation of the minimum or maximum of two integers. These facilities are useful
for generic programming problems.
[section Synopsis]
namespace boost
{
typedef ``['implementation-defined]`` static_min_max_signed_type;
typedef ``['implementation-defined]`` static_min_max_unsigned_type;
@ -607,15 +619,15 @@ for generic programming problems.
}
[endsect]
[endsect]
[section Usage]
The four class templates provide the combinations for finding the minimum or maximum of two [^signed] or
[^unsigned] ([^long]) parameters, /Value1/ and /Value2/, at compile-time. Each template has a single static data member,
The four class templates provide the combinations for finding the minimum or maximum of two [^signed] or
[^unsigned] ([^long]) parameters, /Value1/ and /Value2/, at compile-time. Each template has a single static data member,
[^value], which is set to the respective minimum or maximum of the template's parameters.
[endsect]
[endsect]
[section Example]
@ -653,14 +665,14 @@ The four class templates provide the combinations for finding the minimum or max
[section Demonstration Program]
The program [@../../test/static_min_max_test.cpp static_min_max_test.cpp] is a simplistic demonstration of
The program [@../../test/static_min_max_test.cpp static_min_max_test.cpp] is a simplistic demonstration of
various comparisons using the compile-time extrema class templates.
[endsect]
[endsect]
[section Rationale]
Sometimes the minimum or maximum of several values needs to be found for later compile-time processing,
Sometimes the minimum or maximum of several values needs to be found for later compile-time processing,
['e.g.] for a bound for another class template.
[endsect]
@ -682,23 +694,23 @@ The author of the Boost compile-time extrema class templates is [@http://www.boo
[h4 1.42.0]
* Reverted Trunk to release branch state (i.e. a "known good state").
* Fixed issues: [@https://svn.boost.org/trac/boost/ticket/653 653],
[@https://svn.boost.org/trac/boost/ticket/3084 3084],
[@https://svn.boost.org/trac/boost/ticket/3177 3177],
[@https://svn.boost.org/trac/boost/ticket/3180 3180],
[@https://svn.boost.org/trac/boost/ticket/3548 3568],
[@https://svn.boost.org/trac/boost/ticket/3657 3657],
* Fixed issues: [@https://svn.boost.org/trac/boost/ticket/653 653],
[@https://svn.boost.org/trac/boost/ticket/3084 3084],
[@https://svn.boost.org/trac/boost/ticket/3177 3177],
[@https://svn.boost.org/trac/boost/ticket/3180 3180],
[@https://svn.boost.org/trac/boost/ticket/3548 3568],
[@https://svn.boost.org/trac/boost/ticket/3657 3657],
[@https://svn.boost.org/trac/boost/ticket/2134 2134].
* Added long long support to [^boost::static_log2], [^boost::static_signed_min], [^boost::static_signed_max],
* Added long long support to [^boost::static_log2], [^boost::static_signed_min], [^boost::static_signed_max],
[^boost::static_unsigned_min][^boost::static_unsigned_max], when available.
* The argument type and the result type of [^boost::static_signed_min] etc are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
* The argument type and the result type of [^boost::static_signed_min] etc are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
provided typedefs in new code (and update old code as soon as possible).
[h4 1.32.0]
* The argument type and the result type of [^boost::static_log2] are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
* The argument type and the result type of [^boost::static_log2] are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
provided typedefs in new code (and update old code as soon as possible).
[endsect]

55
doc/modular_arithmetic/extended_euclidean.qbk Normal file
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@ -0,0 +1,55 @@
[section:extended_euclidean Extended Euclidean Algorithm]
[section Introduction]
The extended Euclidean algorithm solves the integer relation /mx + ny/ = gcd(/m/, /n/) for /x/ and /y/.
[endsect]
[section Synopsis]
#include <boost/integer/extended_euclidean.hpp>
namespace boost { namespace integer {
template<class Z>
struct euclidean_result_t {
Z gcd;
Z x;
Z y;
};
template<class Z>
euclidean_result_t<Z> extended_euclidean(Z m, Z n);
}}
[endsect]
[section Usage]
int m = 12;
int n = 15;
auto res = extended_euclidean(m, n);
int gcd = res.gcd;
int x = res.x;
int y = res.y;
// mx + ny = gcd(m,n) should now hold
Unlike most of the library, the extended Euclidean algorithm requires C++11 features.
[endsect]
[section References]
Wagstaff, Samuel S., ['The Joy of Factoring], Vol. 68. American Mathematical Soc., 2013.
[endsect]
[endsect]
[/
Copyright 2018 Nick Thompson.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]

51
doc/modular_arithmetic/mod_inverse.qbk Normal file
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@ -0,0 +1,51 @@
[section:mod_inverse Modular Multiplicative Inverse]
[section Introduction]
The modular multiplicative inverse of a number /a/ is that number /x/ which satisfies /ax/ = 1 mod /p/.
A fast algorithm for computing modular multiplicative inverses based on the extended Euclidean algorithm exists and is provided by Boost.
[endsect]
[section Synopsis]
#include <boost/integer/mod_inverse.hpp>
namespace boost { namespace integer {
template<class Z>
boost::optional<Z> mod_inverse(Z a, Z m);
}}
[endsect]
[section Usage]
Multiplicative modular inverses exist if and only if /a/ and /m/ are coprime.
So for example
auto x = mod_inverse(2, 5);
if (x)
{
int should_be_three = x.value();
}
auto y = mod_inverse(2, 4);
if (!y)
{
std::cout << "There is no inverse of 2 mod 4\n";
}
[endsect]
[section References]
Wagstaff, Samuel S., ['The Joy of Factoring], Vol. 68. American Mathematical Soc., 2013.
[endsect]
[endsect]
[/
Copyright 2018 Nick Thompson.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]

72
include/boost/integer/extended_euclidean.hpp Normal file
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@ -0,0 +1,72 @@
/*
* (C) Copyright Nick Thompson 2018.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#ifndef BOOST_INTEGER_EXTENDED_EUCLIDEAN_HPP
#define BOOST_INTEGER_EXTENDED_EUCLIDEAN_HPP
#include <limits>
#include <stdexcept>
#include <boost/throw_exception.hpp>
namespace boost { namespace integer {
// From "The Joy of Factoring", Algorithm 2.7, with a small optimization to remove tmps from Wikipedia.
// Solves mx + ny = gcd(m,n). Returns tuple with (gcd(m,n), x, y).
template<class Z>
struct euclidean_result_t {
Z gcd;
Z x;
Z y;
};
template<class Z>
euclidean_result_t<typename std::enable_if<std::numeric_limits< Z >::is_signed, Z>::type>
extended_euclidean(Z m, Z n)
{
if (m < 1 || n < 1)
{
BOOST_THROW_EXCEPTION(std::domain_error("Arguments must be strictly positive."));
}
bool swapped = false;
if (m < n)
{
swapped = true;
using std::swap;
swap(m, n);
}
Z u0 = m;
Z u1 = 1;
Z u2 = 0;
Z v0 = n;
Z v1 = 0;
Z v2 = 1;
Z w0;
Z w1;
Z w2;
while(v0 > 0)
{
Z q = u0/v0;
w0 = u0 - q*v0;
w1 = u1 - q*v1;
w2 = u2 - q*v2;
u0 = v0;
u1 = v1;
u2 = v2;
v0 = w0;
v1 = w1;
v2 = w2;
}
if (swapped)
{
return {u0, u2, u1};
}
return {u0, u1, u2};
}
}}
#endif

54
include/boost/integer/mod_inverse.hpp Normal file
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@ -0,0 +1,54 @@
/*
* (C) Copyright Nick Thompson 2018.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#ifndef BOOST_INTEGER_MOD_INVERSE_HPP
#define BOOST_INTEGER_MOD_INVERSE_HPP
#include <stdexcept>
#include <boost/throw_exception.hpp>
#include <boost/optional.hpp>
#include <boost/integer/extended_euclidean.hpp>
namespace boost { namespace integer {
// From "The Joy of Factoring", Algorithm 2.7.
// Here's some others names I've found for this function:
// PowerMod[a, -1, m] (Mathematica)
// mpz_invert (gmplib)
// modinv (some dude on stackoverflow)
// Would mod_inverse be sometimes mistaken as the modular *additive* inverse?
// In any case, I think this is the best name we can get for this function without agonizing.
template<class Z>
boost::optional<Z> mod_inverse(Z a, Z modulus)
{
if (modulus < 2)
{
BOOST_THROW_EXCEPTION(std::domain_error("Modulus must be > 1."));
}
// make sure a < modulus:
a = a % modulus;
if (a == 0)
{
// a doesn't have a modular multiplicative inverse:
return boost::none;
}
euclidean_result_t<Z> u = extended_euclidean(a, modulus);
if (u.gcd > 1)
{
return boost::none;
}
// x might not be in the range 0 < x < m, let's fix that:
while (u.x <= 0)
{
u.x += modulus;
}
// While indeed this is an inexpensive and comforting check,
// the multiplication overflows and hence makes the check itself buggy.
//BOOST_ASSERT(u.x*a % modulus == 1);
return u.x;
}
}}
#endif

3
test/Jamfile.v2
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@ -3,6 +3,7 @@
#~ (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
import testing ;
import ../../config/checks/config : requires ;
project : requirements <warnings>all <toolset>gcc:<cxxflags>-Wextra ;
@ -16,6 +17,8 @@ test-suite integer
[ run integer_mask_test.cpp ]
[ run static_log2_test.cpp ]
[ run static_min_max_test.cpp ]
[ run extended_euclidean_test.cpp : : : [ requires cxx11_unified_initialization_syntax sfinae_expr ] ]
[ run mod_inverse_test.cpp : : : [ requires cxx11_unified_initialization_syntax sfinae_expr ] ]
[ compile integer_traits_include_test.cpp ]
[ compile integer_include_test.cpp ]
[ compile integer_mask_include_test.cpp ]

61
test/extended_euclidean_test.cpp Normal file
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/*
* (C) Copyright Nick Thompson 2018.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#include <boost/config.hpp>
#if (defined(BOOST_MSVC) && (BOOST_MSVC < 1500)) || \
(defined(__clang_major__) && (__clang_major__ == 3) && (__clang_minor__ < 2)) || \
(defined(BOOST_GCC) && defined(BOOST_GCC_CXX11) && BOOST_GCC < 40800)
#define DISABLE_MP_TESTS
#endif
#ifndef DISABLE_MP_TESTS
#include <boost/core/lightweight_test.hpp>
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/integer/common_factor.hpp>
#include <boost/integer/extended_euclidean.hpp>
using boost::multiprecision::int128_t;
using boost::multiprecision::int256_t;
using boost::integer::extended_euclidean;
using boost::integer::gcd;
template<class Z>
void test_extended_euclidean()
{
// Stress test:
//Z max_arg = std::numeric_limits<Z>::max();
Z max_arg = 500;
for (Z m = max_arg; m > 0; --m)
{
for (Z n = max_arg; n > 0; --n)
{
boost::integer::euclidean_result_t<Z> u = extended_euclidean(m, n);
int256_t gcdmn = gcd(m, n);
int256_t x = u.x;
int256_t y = u.y;
BOOST_TEST_EQ(u.gcd, gcdmn);
BOOST_TEST_EQ(m*x + n*y, gcdmn);
}
}
}
int main()
{
test_extended_euclidean<int16_t>();
test_extended_euclidean<int32_t>();
test_extended_euclidean<int64_t>();
test_extended_euclidean<int128_t>();
return boost::report_errors();;
}
#else
int main()
{
return 0;
}
#endif

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test/mod_inverse_test.cpp Normal file
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/*
* (C) Copyright Nick Thompson 2018.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#include <boost/config.hpp>
#if (defined(BOOST_MSVC) && (BOOST_MSVC < 1500)) || \
(defined(__clang_major__) && (__clang_major__ == 3) && (__clang_minor__ < 2)) || \
(defined(BOOST_GCC) && defined(BOOST_GCC_CXX11) && BOOST_GCC < 40800)
#define DISABLE_MP_TESTS
#endif
#ifndef DISABLE_MP_TESTS
#include <boost/core/lightweight_test.hpp>
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/integer/common_factor.hpp>
#include <boost/integer/mod_inverse.hpp>
using boost::multiprecision::int128_t;
using boost::multiprecision::int256_t;
using boost::integer::mod_inverse;
using boost::integer::gcd;
template<class Z>
void test_mod_inverse()
{
//Z max_arg = std::numeric_limits<Z>::max();
Z max_arg = 500;
for (Z modulus = 2; modulus < max_arg; ++modulus)
{
if (modulus % 1000 == 0)
{
std::cout << "Testing all inverses modulo " << modulus << std::endl;
}
for (Z a = 1; a < modulus; ++a)
{
Z gcdam = gcd(a, modulus);
boost::optional<Z> inv_a = mod_inverse(a, modulus);
// Should fail if gcd(a, mod) != 1:
if (gcdam > 1)
{
BOOST_TEST(!inv_a);
}
else
{
BOOST_TEST(inv_a.value() > 0);
// Cast to a bigger type so the multiplication won't overflow.
int256_t a_inv = inv_a.value();
int256_t big_a = a;
int256_t m = modulus;
int256_t outta_be_one = (a_inv*big_a) % m;
BOOST_TEST_EQ(outta_be_one, 1);
}
}
}
}
int main()
{
test_mod_inverse<int16_t>();
test_mod_inverse<int32_t>();
test_mod_inverse<int64_t>();
test_mod_inverse<int128_t>();
return boost::report_errors();
}
#else
int main()
{
return 0;
}
#endif