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</pre><pre class="rust ">
<span class="kw">use</span> <span class="ident">integer</span>::<span class="ident">Integer</span>;
<span class="kw">use</span> <span class="ident">traits</span>::<span class="ident">Zero</span>;

<span class="kw">use</span> <span class="ident">biguint</span>::<span class="ident">BigUint</span>;

<span class="kw">struct</span> <span class="ident">MontyReducer</span><span class="op">&lt;</span><span class="lifetime">&#39;a</span><span class="op">&gt;</span> {
    <span class="ident">n</span>: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a</span> <span class="ident">BigUint</span>,
    <span class="ident">n0inv</span>: <span class="ident">u32</span>
}

<span class="comment">// Calculate the modular inverse of `num`, using Extended GCD.</span>
<span class="comment">//</span>
<span class="comment">// Reference:</span>
<span class="comment">// Brent &amp; Zimmermann, Modern Computer Arithmetic, v0.5.9, Algorithm 1.20</span>
<span class="kw">fn</span> <span class="ident">inv_mod_u32</span>(<span class="ident">num</span>: <span class="ident">u32</span>) <span class="op">-&gt;</span> <span class="ident">u32</span> {
    <span class="comment">// num needs to be relatively prime to 2**32 -- i.e. it must be odd.</span>
    <span class="macro">assert</span><span class="macro">!</span>(<span class="ident">num</span> <span class="op">%</span> <span class="number">2</span> <span class="op">!=</span> <span class="number">0</span>);

    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">a</span>: <span class="ident">i64</span> <span class="op">=</span> <span class="ident">num</span> <span class="kw">as</span> <span class="ident">i64</span>;
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">b</span>: <span class="ident">i64</span> <span class="op">=</span> (<span class="ident">u32</span>::<span class="ident">max_value</span>() <span class="kw">as</span> <span class="ident">i64</span>) <span class="op">+</span> <span class="number">1</span>;

    <span class="comment">// ExtendedGcd</span>
    <span class="comment">// Input: positive integers a and b</span>
    <span class="comment">// Output: integers (g, u, v) such that g = gcd(a, b) = ua + vb</span>
    <span class="comment">// As we don&#39;t need v for modular inverse, we don&#39;t calculate it.</span>

    <span class="comment">// 1: (u, w) &lt;- (1, 0)</span>
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">u</span> <span class="op">=</span> <span class="number">1</span>;
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">w</span> <span class="op">=</span> <span class="number">0</span>;
    <span class="comment">// 3: while b != 0</span>
    <span class="kw">while</span> <span class="ident">b</span> <span class="op">!=</span> <span class="number">0</span> {
        <span class="comment">// 4: (q, r) &lt;- DivRem(a, b)</span>
        <span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">a</span> <span class="op">/</span> <span class="ident">b</span>;
        <span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">a</span> <span class="op">%</span> <span class="ident">b</span>;
        <span class="comment">// 5: (a, b) &lt;- (b, r)</span>
        <span class="ident">a</span> <span class="op">=</span> <span class="ident">b</span>; <span class="ident">b</span> <span class="op">=</span> <span class="ident">r</span>;
        <span class="comment">// 6: (u, w) &lt;- (w, u - qw)</span>
        <span class="kw">let</span> <span class="ident">m</span> <span class="op">=</span> <span class="ident">u</span> <span class="op">-</span> <span class="ident">w</span><span class="kw-2">*</span><span class="ident">q</span>;
        <span class="ident">u</span> <span class="op">=</span> <span class="ident">w</span>; <span class="ident">w</span> <span class="op">=</span> <span class="ident">m</span>;
    }

    <span class="macro">assert</span><span class="macro">!</span>(<span class="ident">a</span> <span class="op">==</span> <span class="number">1</span>);
    <span class="comment">// Downcasting acts like a mod 2^32 too.</span>
    <span class="ident">u</span> <span class="kw">as</span> <span class="ident">u32</span>
}

<span class="kw">impl</span><span class="op">&lt;</span><span class="lifetime">&#39;a</span><span class="op">&gt;</span> <span class="ident">MontyReducer</span><span class="op">&lt;</span><span class="lifetime">&#39;a</span><span class="op">&gt;</span> {
    <span class="kw">fn</span> <span class="ident">new</span>(<span class="ident">n</span>: <span class="kw-2">&amp;</span><span class="lifetime">&#39;a</span> <span class="ident">BigUint</span>) <span class="op">-&gt;</span> <span class="self">Self</span> {
        <span class="kw">let</span> <span class="ident">n0inv</span> <span class="op">=</span> <span class="ident">inv_mod_u32</span>(<span class="ident">n</span>.<span class="ident">data</span>[<span class="number">0</span>]);
        <span class="ident">MontyReducer</span> { <span class="ident">n</span>: <span class="ident">n</span>, <span class="ident">n0inv</span>: <span class="ident">n0inv</span> }
    }
}

<span class="comment">// Montgomery Reduction</span>
<span class="comment">//</span>
<span class="comment">// Reference:</span>
<span class="comment">// Brent &amp; Zimmermann, Modern Computer Arithmetic, v0.5.9, Algorithm 2.6</span>
<span class="kw">fn</span> <span class="ident">monty_redc</span>(<span class="ident">a</span>: <span class="ident">BigUint</span>, <span class="ident">mr</span>: <span class="kw-2">&amp;</span><span class="ident">MontyReducer</span>) <span class="op">-&gt;</span> <span class="ident">BigUint</span> {
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">c</span> <span class="op">=</span> <span class="ident">a</span>.<span class="ident">data</span>;
    <span class="kw">let</span> <span class="ident">n</span> <span class="op">=</span> <span class="kw-2">&amp;</span><span class="ident">mr</span>.<span class="ident">n</span>.<span class="ident">data</span>;
    <span class="kw">let</span> <span class="ident">n_size</span> <span class="op">=</span> <span class="ident">n</span>.<span class="ident">len</span>();

    <span class="comment">// Allocate sufficient work space</span>
    <span class="ident">c</span>.<span class="ident">resize</span>(<span class="number">2</span> <span class="op">*</span> <span class="ident">n_size</span> <span class="op">+</span> <span class="number">2</span>, <span class="number">0</span>);

    <span class="comment">// β is the size of a word, in this case 32 bits. So &quot;a mod β&quot; is</span>
    <span class="comment">// equivalent to masking a to 32 bits.</span>
    <span class="comment">// mu &lt;- -N^(-1) mod β</span>
    <span class="kw">let</span> <span class="ident">mu</span> <span class="op">=</span> <span class="number">0u32</span>.<span class="ident">wrapping_sub</span>(<span class="ident">mr</span>.<span class="ident">n0inv</span>);

    <span class="comment">// 1: for i = 0 to (n-1)</span>
    <span class="kw">for</span> <span class="ident">i</span> <span class="kw">in</span> <span class="number">0</span>..<span class="ident">n_size</span> {
        <span class="comment">// 2: q_i &lt;- mu*c_i mod β</span>
        <span class="kw">let</span> <span class="ident">q_i</span> <span class="op">=</span> <span class="ident">c</span>[<span class="ident">i</span>].<span class="ident">wrapping_mul</span>(<span class="ident">mu</span>);

        <span class="comment">// 3: C &lt;- C + q_i * N * β^i</span>
        <span class="kw">super</span>::<span class="ident">algorithms</span>::<span class="ident">mac_digit</span>(<span class="kw-2">&amp;</span><span class="kw-2">mut</span> <span class="ident">c</span>[<span class="ident">i</span>..], <span class="ident">n</span>, <span class="ident">q_i</span>);
    }

    <span class="comment">// 4: R &lt;- C * β^(-n)</span>
    <span class="comment">// This is an n-word bitshift, equivalent to skipping n words.</span>
    <span class="kw">let</span> <span class="ident">ret</span> <span class="op">=</span> <span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">c</span>[<span class="ident">n_size</span>..].<span class="ident">to_vec</span>());

    <span class="comment">// 5: if R &gt;= β^n then return R-N else return R.</span>
    <span class="kw">if</span> <span class="kw-2">&amp;</span><span class="ident">ret</span> <span class="op">&lt;</span> <span class="ident">mr</span>.<span class="ident">n</span> {
        <span class="ident">ret</span>
    } <span class="kw">else</span> {
        <span class="ident">ret</span> <span class="op">-</span> <span class="ident">mr</span>.<span class="ident">n</span>
    }
}

<span class="comment">// Montgomery Multiplication</span>
<span class="kw">fn</span> <span class="ident">monty_mult</span>(<span class="ident">a</span>: <span class="ident">BigUint</span>, <span class="ident">b</span>: <span class="kw-2">&amp;</span><span class="ident">BigUint</span>, <span class="ident">mr</span>: <span class="kw-2">&amp;</span><span class="ident">MontyReducer</span>) <span class="op">-&gt;</span> <span class="ident">BigUint</span> {
    <span class="ident">monty_redc</span>(<span class="ident">a</span> <span class="op">*</span> <span class="ident">b</span>, <span class="ident">mr</span>)
}

<span class="comment">// Montgomery Squaring</span>
<span class="kw">fn</span> <span class="ident">monty_sqr</span>(<span class="ident">a</span>: <span class="ident">BigUint</span>, <span class="ident">mr</span>: <span class="kw-2">&amp;</span><span class="ident">MontyReducer</span>) <span class="op">-&gt;</span> <span class="ident">BigUint</span> {
    <span class="comment">// TODO: Replace with an optimised squaring function</span>
    <span class="ident">monty_redc</span>(<span class="kw-2">&amp;</span><span class="ident">a</span> <span class="op">*</span> <span class="kw-2">&amp;</span><span class="ident">a</span>, <span class="ident">mr</span>)
}

<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">monty_modpow</span>(<span class="ident">a</span>: <span class="kw-2">&amp;</span><span class="ident">BigUint</span>, <span class="ident">exp</span>: <span class="kw-2">&amp;</span><span class="ident">BigUint</span>, <span class="ident">modulus</span>: <span class="kw-2">&amp;</span><span class="ident">BigUint</span>) <span class="op">-&gt;</span> <span class="ident">BigUint</span>{
    <span class="kw">let</span> <span class="ident">mr</span> <span class="op">=</span> <span class="ident">MontyReducer</span>::<span class="ident">new</span>(<span class="ident">modulus</span>);

    <span class="comment">// Calculate the Montgomery parameter</span>
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">v</span> <span class="op">=</span> <span class="macro">vec</span><span class="macro">!</span>[<span class="number">0</span>; <span class="ident">modulus</span>.<span class="ident">data</span>.<span class="ident">len</span>()];
    <span class="ident">v</span>.<span class="ident">push</span>(<span class="number">1</span>);
    <span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">v</span>);

    <span class="comment">// Map the base to the Montgomery domain</span>
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">apri</span> <span class="op">=</span> <span class="ident">a</span> <span class="op">*</span> <span class="kw-2">&amp;</span><span class="ident">r</span> <span class="op">%</span> <span class="ident">modulus</span>;

    <span class="comment">// Binary exponentiation</span>
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">ans</span> <span class="op">=</span> <span class="kw-2">&amp;</span><span class="ident">r</span> <span class="op">%</span> <span class="ident">modulus</span>;
    <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">e</span> <span class="op">=</span> <span class="ident">exp</span>.<span class="ident">clone</span>();
    <span class="kw">while</span> <span class="op">!</span><span class="ident">e</span>.<span class="ident">is_zero</span>() {
        <span class="kw">if</span> <span class="ident">e</span>.<span class="ident">is_odd</span>() {
            <span class="ident">ans</span> <span class="op">=</span> <span class="ident">monty_mult</span>(<span class="ident">ans</span>, <span class="kw-2">&amp;</span><span class="ident">apri</span>, <span class="kw-2">&amp;</span><span class="ident">mr</span>);
        }
        <span class="ident">apri</span> <span class="op">=</span> <span class="ident">monty_sqr</span>(<span class="ident">apri</span>, <span class="kw-2">&amp;</span><span class="ident">mr</span>);
        <span class="ident">e</span> <span class="op">=</span> <span class="ident">e</span> <span class="op">&gt;&gt;</span> <span class="number">1</span>;
    }

    <span class="comment">// Map the result back to the residues domain</span>
    <span class="ident">monty_redc</span>(<span class="ident">ans</span>, <span class="kw-2">&amp;</span><span class="ident">mr</span>)
}
</pre>
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