| <!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="Source of the Rust file `/root/.cargo/registry/src/github.com-1ecc6299db9ec823/num-bigint-0.2.6/src/monty.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>monty.rs - source</title><link rel="preload" as="font" type="font/woff2" crossorigin href="../../SourceSerif4-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../FiraSans-Regular.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../FiraSans-Medium.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../SourceCodePro-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../SourceSerif4-Bold.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../SourceCodePro-Semibold.ttf.woff2"><link rel="stylesheet" href="../../normalize.css"><link rel="stylesheet" href="../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" href="../../ayu.css" disabled><link rel="stylesheet" href="../../dark.css" disabled><link rel="stylesheet" href="../../light.css" id="themeStyle"><script id="default-settings" ></script><script src="../../storage.js"></script><script defer src="../../source-script.js"></script><script defer src="../../source-files.js"></script><script defer src="../../main.js"></script><noscript><link rel="stylesheet" href="../../noscript.css"></noscript><link rel="alternate icon" type="image/png" href="../../favicon-16x16.png"><link rel="alternate icon" type="image/png" href="../../favicon-32x32.png"><link rel="icon" type="image/svg+xml" href="../../favicon.svg"></head><body class="rustdoc source"><!--[if lte IE 11]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><a class="sidebar-logo" href="../../num_bigint/index.html"><div class="logo-container"><img class="rust-logo" src="../../rust-logo.svg" alt="logo"></div></a></nav><main><div class="width-limiter"><nav class="sub"><a class="sub-logo-container" href="../../num_bigint/index.html"><img class="rust-logo" src="../../rust-logo.svg" alt="logo"></a><form class="search-form"><div class="search-container"><span></span><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"><div id="help-button" title="help" tabindex="-1"><a href="../../help.html">?</a></div><div id="settings-menu" tabindex="-1"><a href="../../settings.html" title="settings"><img width="22" height="22" alt="Change settings" src="../../wheel.svg"></a></div></div></form></nav><section id="main-content" class="content"><div class="example-wrap"><pre class="src-line-numbers"><span id="1">1</span> |
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| </pre><pre class="rust"><code><span class="kw">use </span>integer::Integer; |
| <span class="kw">use </span>traits::Zero; |
| |
| <span class="kw">use </span>biguint::BigUint; |
| |
| <span class="kw">struct </span>MontyReducer<<span class="lifetime">'a</span>> { |
| n: <span class="kw-2">&</span><span class="lifetime">'a </span>BigUint, |
| n0inv: u32, |
| } |
| |
| <span class="comment">// Calculate the modular inverse of `num`, using Extended GCD. |
| // |
| // Reference: |
| // Brent & Zimmermann, Modern Computer Arithmetic, v0.5.9, Algorithm 1.20 |
| </span><span class="kw">fn </span>inv_mod_u32(num: u32) -> u32 { |
| <span class="comment">// num needs to be relatively prime to 2**32 -- i.e. it must be odd. |
| </span><span class="macro">assert!</span>(num % <span class="number">2 </span>!= <span class="number">0</span>); |
| |
| <span class="kw">let </span><span class="kw-2">mut </span>a: i64 = i64::from(num); |
| <span class="kw">let </span><span class="kw-2">mut </span>b: i64 = i64::from(u32::max_value()) + <span class="number">1</span>; |
| |
| <span class="comment">// ExtendedGcd |
| // Input: positive integers a and b |
| // Output: integers (g, u, v) such that g = gcd(a, b) = ua + vb |
| // As we don't need v for modular inverse, we don't calculate it. |
| |
| // 1: (u, w) <- (1, 0) |
| </span><span class="kw">let </span><span class="kw-2">mut </span>u = <span class="number">1</span>; |
| <span class="kw">let </span><span class="kw-2">mut </span>w = <span class="number">0</span>; |
| <span class="comment">// 3: while b != 0 |
| </span><span class="kw">while </span>b != <span class="number">0 </span>{ |
| <span class="comment">// 4: (q, r) <- DivRem(a, b) |
| </span><span class="kw">let </span>q = a / b; |
| <span class="kw">let </span>r = a % b; |
| <span class="comment">// 5: (a, b) <- (b, r) |
| </span>a = b; |
| b = r; |
| <span class="comment">// 6: (u, w) <- (w, u - qw) |
| </span><span class="kw">let </span>m = u - w * q; |
| u = w; |
| w = m; |
| } |
| |
| <span class="macro">assert!</span>(a == <span class="number">1</span>); |
| <span class="comment">// Downcasting acts like a mod 2^32 too. |
| </span>u <span class="kw">as </span>u32 |
| } |
| |
| <span class="kw">impl</span><<span class="lifetime">'a</span>> MontyReducer<<span class="lifetime">'a</span>> { |
| <span class="kw">fn </span>new(n: <span class="kw-2">&</span><span class="lifetime">'a </span>BigUint) -> <span class="self">Self </span>{ |
| <span class="kw">let </span>n0inv = inv_mod_u32(n.data[<span class="number">0</span>]); |
| MontyReducer { n: n, n0inv: n0inv } |
| } |
| } |
| |
| <span class="comment">// Montgomery Reduction |
| // |
| // Reference: |
| // Brent & Zimmermann, Modern Computer Arithmetic, v0.5.9, Algorithm 2.6 |
| </span><span class="kw">fn </span>monty_redc(a: BigUint, mr: <span class="kw-2">&</span>MontyReducer) -> BigUint { |
| <span class="kw">let </span><span class="kw-2">mut </span>c = a.data; |
| <span class="kw">let </span>n = <span class="kw-2">&</span>mr.n.data; |
| <span class="kw">let </span>n_size = n.len(); |
| |
| <span class="comment">// Allocate sufficient work space |
| </span>c.resize(<span class="number">2 </span>* n_size + <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 "a mod β" is |
| // equivalent to masking a to 32 bits. |
| // mu <- -N^(-1) mod β |
| </span><span class="kw">let </span>mu = <span class="number">0u32</span>.wrapping_sub(mr.n0inv); |
| |
| <span class="comment">// 1: for i = 0 to (n-1) |
| </span><span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..n_size { |
| <span class="comment">// 2: q_i <- mu*c_i mod β |
| </span><span class="kw">let </span>q_i = c[i].wrapping_mul(mu); |
| |
| <span class="comment">// 3: C <- C + q_i * N * β^i |
| </span><span class="kw">super</span>::algorithms::mac_digit(<span class="kw-2">&mut </span>c[i..], n, q_i); |
| } |
| |
| <span class="comment">// 4: R <- C * β^(-n) |
| // This is an n-word bitshift, equivalent to skipping n words. |
| </span><span class="kw">let </span>ret = BigUint::new(c[n_size..].to_vec()); |
| |
| <span class="comment">// 5: if R >= β^n then return R-N else return R. |
| </span><span class="kw">if </span>ret < <span class="kw-2">*</span>mr.n { |
| ret |
| } <span class="kw">else </span>{ |
| ret - mr.n |
| } |
| } |
| |
| <span class="comment">// Montgomery Multiplication |
| </span><span class="kw">fn </span>monty_mult(a: BigUint, b: <span class="kw-2">&</span>BigUint, mr: <span class="kw-2">&</span>MontyReducer) -> BigUint { |
| monty_redc(a * b, mr) |
| } |
| |
| <span class="comment">// Montgomery Squaring |
| </span><span class="kw">fn </span>monty_sqr(a: BigUint, mr: <span class="kw-2">&</span>MontyReducer) -> BigUint { |
| <span class="comment">// TODO: Replace with an optimised squaring function |
| </span>monty_redc(<span class="kw-2">&</span>a * <span class="kw-2">&</span>a, mr) |
| } |
| |
| <span class="kw">pub fn </span>monty_modpow(a: <span class="kw-2">&</span>BigUint, exp: <span class="kw-2">&</span>BigUint, modulus: <span class="kw-2">&</span>BigUint) -> BigUint { |
| <span class="kw">let </span>mr = MontyReducer::new(modulus); |
| |
| <span class="comment">// Calculate the Montgomery parameter |
| </span><span class="kw">let </span><span class="kw-2">mut </span>v = <span class="macro">vec!</span>[<span class="number">0</span>; modulus.data.len()]; |
| v.push(<span class="number">1</span>); |
| <span class="kw">let </span>r = BigUint::new(v); |
| |
| <span class="comment">// Map the base to the Montgomery domain |
| </span><span class="kw">let </span><span class="kw-2">mut </span>apri = a * <span class="kw-2">&</span>r % modulus; |
| |
| <span class="comment">// Binary exponentiation |
| </span><span class="kw">let </span><span class="kw-2">mut </span>ans = <span class="kw-2">&</span>r % modulus; |
| <span class="kw">let </span><span class="kw-2">mut </span>e = exp.clone(); |
| <span class="kw">while </span>!e.is_zero() { |
| <span class="kw">if </span>e.is_odd() { |
| ans = monty_mult(ans, <span class="kw-2">&</span>apri, <span class="kw-2">&</span>mr); |
| } |
| apri = monty_sqr(apri, <span class="kw-2">&</span>mr); |
| e >>= <span class="number">1</span>; |
| } |
| |
| <span class="comment">// Map the result back to the residues domain |
| </span>monty_redc(ans, <span class="kw-2">&</span>mr) |
| } |
| </code></pre></div> |
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