[ci skip] Modular exponentiation, modular multiplicative inverse, extended Euclidean algorithm, discrete logarithm.

include/boost/integer/extended_euclidean.hpp

@@ -0,0 +1,67 @@
+/*
+ *  (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 <tuple>
+#include <limits>
+
+namespace boost { namespace integer {
+
+// From "The Joy of Factoring", Algorithm 2.7.
+// Should the tuple be a named tuple? Is that possible?
+// Solves mx + ny = gcd(m,n). Returns tuple with (gcd(m,n), x, y).
+template<class Z>
+std::tuple<Z, Z, Z> extended_euclidean(Z m, Z n)
+{
+    using std::numeric_limits;
+    static_assert(numeric_limits<Z>::is_integer,
+                  "The extended Euclidean algorithm works on integral types.\n");
+
+    static_assert(numeric_limits<Z>::is_signed,
+                  "The extended Euclidean algorithm only works on signed integer types.\n");
+
+    if (m < 1 || n < 1)
+    {
+        throw std::domain_error("Arguments must be strictly positive.\n");
+    }
+    bool swapped = false;
+    if (m < n)
+    {
+        swapped = true;
+        std::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)
+    {
+        std::swap(u1, u2);
+    }
+    return std::make_tuple(u0, u1, u2);
+}
+
+}}
+#endif