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 </pre><pre class="rust"><code><span class="kw">use </span><span class="kw">crate</span>::std_alloc::Vec;
 <span class="kw">use </span>core::mem;
 <span class="kw">use </span>core::ops::Shl;
 <span class="kw">use </span>num_traits::{One, Zero};

 <span class="kw">use </span><span class="kw">crate</span>::big_digit::{<span class="self">self</span>, BigDigit, DoubleBigDigit, SignedDoubleBigDigit};
 <span class="kw">use </span><span class="kw">crate</span>::biguint::BigUint;

 <span class="kw">struct </span>MontyReducer {
     n0inv: BigDigit,
 }

 <span class="comment">// k0 = -m**-1 mod 2**BITS. Algorithm from: Dumas, J.G. &quot;On Newton–Raphson
 // Iteration for Multiplicative Inverses Modulo Prime Powers&quot;.
 </span><span class="kw">fn </span>inv_mod_alt(b: BigDigit) -&gt; BigDigit {
     <span class="macro">assert_ne!</span>(b &amp; <span class="number">1</span>, <span class="number">0</span>);

     <span class="kw">let </span><span class="kw-2">mut </span>k0 = <span class="number">2 </span>- b <span class="kw">as </span>SignedDoubleBigDigit;
     <span class="kw">let </span><span class="kw-2">mut </span>t = (b - <span class="number">1</span>) <span class="kw">as </span>SignedDoubleBigDigit;
     <span class="kw">let </span><span class="kw-2">mut </span>i = <span class="number">1</span>;
     <span class="kw">while </span>i &lt; big_digit::BITS {
         t = t.wrapping_mul(t);
         k0 = k0.wrapping_mul(t + <span class="number">1</span>);

         i &lt;&lt;= <span class="number">1</span>;
     }
     -k0 <span class="kw">as </span>BigDigit
 }

 <span class="kw">impl </span>MontyReducer {
     <span class="kw">fn </span>new(n: <span class="kw-2">&amp;</span>BigUint) -&gt; <span class="self">Self </span>{
         <span class="kw">let </span>n0inv = inv_mod_alt(n.data[<span class="number">0</span>]);
         MontyReducer { n0inv }
     }
 }

 <span class="doccomment">/// Computes z mod m = x * y * 2 ** (-n*_W) mod m
 /// assuming k = -1/m mod 2**_W
 /// See Gueron, &quot;Efficient Software Implementations of Modular Exponentiation&quot;.
 /// https://eprint.iacr.org/2011/239.pdf
 /// In the terminology of that paper, this is an &quot;Almost Montgomery Multiplication&quot;:
 /// x and y are required to satisfy 0 &lt;= z &lt; 2**(n*_W) and then the result
 /// z is guaranteed to satisfy 0 &lt;= z &lt; 2**(n*_W), but it may not be &lt; m.
 </span><span class="attribute">#[allow(clippy::many_single_char_names)]
 </span><span class="kw">fn </span>montgomery(x: <span class="kw-2">&amp;</span>BigUint, y: <span class="kw-2">&amp;</span>BigUint, m: <span class="kw-2">&amp;</span>BigUint, k: BigDigit, n: usize) -&gt; BigUint {
     <span class="comment">// This code assumes x, y, m are all the same length, n.
     // (required by addMulVVW and the for loop).
     // It also assumes that x, y are already reduced mod m,
     // or else the result will not be properly reduced.
     </span><span class="macro">assert!</span>(
         x.data.len() == n &amp;&amp; y.data.len() == n &amp;&amp; m.data.len() == n,
         <span class="string">&quot;{:?} {:?} {:?} {}&quot;</span>,
         x,
         y,
         m,
         n
     );

     <span class="kw">let </span><span class="kw-2">mut </span>z = BigUint::zero();
     z.data.resize(n * <span class="number">2</span>, <span class="number">0</span>);

     <span class="kw">let </span><span class="kw-2">mut </span>c: BigDigit = <span class="number">0</span>;
     <span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..n {
         <span class="kw">let </span>c2 = add_mul_vvw(<span class="kw-2">&amp;mut </span>z.data[i..n + i], <span class="kw-2">&amp;</span>x.data, y.data[i]);
         <span class="kw">let </span>t = z.data[i].wrapping_mul(k);
         <span class="kw">let </span>c3 = add_mul_vvw(<span class="kw-2">&amp;mut </span>z.data[i..n + i], <span class="kw-2">&amp;</span>m.data, t);
         <span class="kw">let </span>cx = c.wrapping_add(c2);
         <span class="kw">let </span>cy = cx.wrapping_add(c3);
         z.data[n + i] = cy;
         <span class="kw">if </span>cx &lt; c2 || cy &lt; c3 {
             c = <span class="number">1</span>;
         } <span class="kw">else </span>{
             c = <span class="number">0</span>;
         }
     }

     <span class="kw">if </span>c == <span class="number">0 </span>{
         z.data = z.data[n..].to_vec();
     } <span class="kw">else </span>{
         {
             <span class="kw">let </span>(<span class="kw-2">mut </span>first, second) = z.data.split_at_mut(n);
             sub_vv(<span class="kw-2">&amp;mut </span>first, <span class="kw-2">&amp;</span>second, <span class="kw-2">&amp;</span>m.data);
         }
         z.data = z.data[..n].to_vec();
     }

     z
 }

 <span class="attribute">#[inline(always)]
 </span><span class="kw">fn </span>add_mul_vvw(z: <span class="kw-2">&amp;mut </span>[BigDigit], x: <span class="kw-2">&amp;</span>[BigDigit], y: BigDigit) -&gt; BigDigit {
     <span class="kw">let </span><span class="kw-2">mut </span>c = <span class="number">0</span>;
     <span class="kw">for </span>(zi, xi) <span class="kw">in </span>z.iter_mut().zip(x.iter()) {
         <span class="kw">let </span>(z1, z0) = mul_add_www(<span class="kw-2">*</span>xi, y, <span class="kw-2">*</span>zi);
         <span class="kw">let </span>(c_, zi_) = add_ww(z0, c, <span class="number">0</span>);
         <span class="kw-2">*</span>zi = zi_;
         c = c_ + z1;
     }

     c
 }

 <span class="doccomment">/// The resulting carry c is either 0 or 1.
 </span><span class="attribute">#[inline(always)]
 </span><span class="kw">fn </span>sub_vv(z: <span class="kw-2">&amp;mut </span>[BigDigit], x: <span class="kw-2">&amp;</span>[BigDigit], y: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; BigDigit {
     <span class="kw">let </span><span class="kw-2">mut </span>c = <span class="number">0</span>;
     <span class="kw">for </span>(i, (xi, yi)) <span class="kw">in </span>x.iter().zip(y.iter()).enumerate().take(z.len()) {
         <span class="kw">let </span>zi = xi.wrapping_sub(<span class="kw-2">*</span>yi).wrapping_sub(c);
         z[i] = zi;
         <span class="comment">// see &quot;Hacker&#39;s Delight&quot;, section 2-12 (overflow detection)
         </span>c = ((yi &amp; !xi) | ((yi | !xi) &amp; zi)) &gt;&gt; (big_digit::BITS - <span class="number">1</span>)
     }

     c
 }

 <span class="doccomment">/// z1&lt;&lt;_W + z0 = x+y+c, with c == 0 or 1
 </span><span class="attribute">#[inline(always)]
 </span><span class="kw">fn </span>add_ww(x: BigDigit, y: BigDigit, c: BigDigit) -&gt; (BigDigit, BigDigit) {
     <span class="kw">let </span>yc = y.wrapping_add(c);
     <span class="kw">let </span>z0 = x.wrapping_add(yc);
     <span class="kw">let </span>z1 = <span class="kw">if </span>z0 &lt; x || yc &lt; y { <span class="number">1 </span>} <span class="kw">else </span>{ <span class="number">0 </span>};

     (z1, z0)
 }

 <span class="doccomment">/// z1 &lt;&lt; _W + z0 = x * y + c
 </span><span class="attribute">#[inline(always)]
 </span><span class="kw">fn </span>mul_add_www(x: BigDigit, y: BigDigit, c: BigDigit) -&gt; (BigDigit, BigDigit) {
     <span class="kw">let </span>z = x <span class="kw">as </span>DoubleBigDigit * y <span class="kw">as </span>DoubleBigDigit + c <span class="kw">as </span>DoubleBigDigit;
     ((z &gt;&gt; big_digit::BITS) <span class="kw">as </span>BigDigit, z <span class="kw">as </span>BigDigit)
 }

 <span class="doccomment">/// Calculates x ** y mod m using a fixed, 4-bit window.
 </span><span class="attribute">#[allow(clippy::many_single_char_names)]
 </span><span class="kw">pub</span>(<span class="kw">super</span>) <span class="kw">fn </span>monty_modpow(x: <span class="kw-2">&amp;</span>BigUint, y: <span class="kw-2">&amp;</span>BigUint, m: <span class="kw-2">&amp;</span>BigUint) -&gt; BigUint {
     <span class="macro">assert!</span>(m.data[<span class="number">0</span>] &amp; <span class="number">1 </span>== <span class="number">1</span>);
     <span class="kw">let </span>mr = MontyReducer::new(m);
     <span class="kw">let </span>num_words = m.data.len();

     <span class="kw">let </span><span class="kw-2">mut </span>x = x.clone();

     <span class="comment">// We want the lengths of x and m to be equal.
     // It is OK if x &gt;= m as long as len(x) == len(m).
     </span><span class="kw">if </span>x.data.len() &gt; num_words {
         x %= m;
         <span class="comment">// Note: now len(x) &lt;= numWords, not guaranteed ==.
     </span>}
     <span class="kw">if </span>x.data.len() &lt; num_words {
         x.data.resize(num_words, <span class="number">0</span>);
     }

     <span class="comment">// rr = 2**(2*_W*len(m)) mod m
     </span><span class="kw">let </span><span class="kw-2">mut </span>rr = BigUint::one();
     rr = (rr.shl(<span class="number">2 </span>* num_words <span class="kw">as </span>u64 * u64::from(big_digit::BITS))) % m;
     <span class="kw">if </span>rr.data.len() &lt; num_words {
         rr.data.resize(num_words, <span class="number">0</span>);
     }
     <span class="comment">// one = 1, with equal length to that of m
     </span><span class="kw">let </span><span class="kw-2">mut </span>one = BigUint::one();
     one.data.resize(num_words, <span class="number">0</span>);

     <span class="kw">let </span>n = <span class="number">4</span>;
     <span class="comment">// powers[i] contains x^i
     </span><span class="kw">let </span><span class="kw-2">mut </span>powers = Vec::with_capacity(<span class="number">1 </span>&lt;&lt; n);
     powers.push(montgomery(<span class="kw-2">&amp;</span>one, <span class="kw-2">&amp;</span>rr, m, mr.n0inv, num_words));
     powers.push(montgomery(<span class="kw-2">&amp;</span>x, <span class="kw-2">&amp;</span>rr, m, mr.n0inv, num_words));
     <span class="kw">for </span>i <span class="kw">in </span><span class="number">2</span>..<span class="number">1 </span>&lt;&lt; n {
         <span class="kw">let </span>r = montgomery(<span class="kw-2">&amp;</span>powers[i - <span class="number">1</span>], <span class="kw-2">&amp;</span>powers[<span class="number">1</span>], m, mr.n0inv, num_words);
         powers.push(r);
     }

     <span class="comment">// initialize z = 1 (Montgomery 1)
     </span><span class="kw">let </span><span class="kw-2">mut </span>z = powers[<span class="number">0</span>].clone();
     z.data.resize(num_words, <span class="number">0</span>);
     <span class="kw">let </span><span class="kw-2">mut </span>zz = BigUint::zero();
     zz.data.resize(num_words, <span class="number">0</span>);

     <span class="comment">// same windowed exponent, but with Montgomery multiplications
     </span><span class="kw">for </span>i <span class="kw">in </span>(<span class="number">0</span>..y.data.len()).rev() {
         <span class="kw">let </span><span class="kw-2">mut </span>yi = y.data[i];
         <span class="kw">let </span><span class="kw-2">mut </span>j = <span class="number">0</span>;
         <span class="kw">while </span>j &lt; big_digit::BITS {
             <span class="kw">if </span>i != y.data.len() - <span class="number">1 </span>|| j != <span class="number">0 </span>{
                 zz = montgomery(<span class="kw-2">&amp;</span>z, <span class="kw-2">&amp;</span>z, m, mr.n0inv, num_words);
                 z = montgomery(<span class="kw-2">&amp;</span>zz, <span class="kw-2">&amp;</span>zz, m, mr.n0inv, num_words);
                 zz = montgomery(<span class="kw-2">&amp;</span>z, <span class="kw-2">&amp;</span>z, m, mr.n0inv, num_words);
                 z = montgomery(<span class="kw-2">&amp;</span>zz, <span class="kw-2">&amp;</span>zz, m, mr.n0inv, num_words);
             }
             zz = montgomery(
                 <span class="kw-2">&amp;</span>z,
                 <span class="kw-2">&amp;</span>powers[(yi &gt;&gt; (big_digit::BITS - n)) <span class="kw">as </span>usize],
                 m,
                 mr.n0inv,
                 num_words,
             );
             mem::swap(<span class="kw-2">&amp;mut </span>z, <span class="kw-2">&amp;mut </span>zz);
             yi &lt;&lt;= n;
             j += n;
         }
     }

     <span class="comment">// convert to regular number
     </span>zz = montgomery(<span class="kw-2">&amp;</span>z, <span class="kw-2">&amp;</span>one, m, mr.n0inv, num_words);

     zz.normalize();
     <span class="comment">// One last reduction, just in case.
     // See golang.org/issue/13907.
     </span><span class="kw">if </span>zz &gt;= <span class="kw-2">*</span>m {
         <span class="comment">// Common case is m has high bit set; in that case,
         // since zz is the same length as m, there can be just
         // one multiple of m to remove. Just subtract.
         // We think that the subtract should be sufficient in general,
         // so do that unconditionally, but double-check,
         // in case our beliefs are wrong.
         // The div is not expected to be reached.
         </span>zz -= m;
         <span class="kw">if </span>zz &gt;= <span class="kw-2">*</span>m {
             zz %= m;
         }
     }

     zz.normalize();
     zz
 }
 </code></pre></div>
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