| <!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-0.3.3/src/biguint/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><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. "On Newton–Raphson |
| // Iteration for Multiplicative Inverses Modulo Prime Powers". |
| </span><span class="kw">fn </span>inv_mod_alt(b: BigDigit) -> BigDigit { |
| <span class="macro">assert_ne!</span>(b & <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 < big_digit::BITS { |
| t = t.wrapping_mul(t); |
| k0 = k0.wrapping_mul(t + <span class="number">1</span>); |
| |
| i <<= <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">&</span>BigUint) -> <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, "Efficient Software Implementations of Modular Exponentiation". |
| /// https://eprint.iacr.org/2011/239.pdf |
| /// In the terminology of that paper, this is an "Almost Montgomery Multiplication": |
| /// x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result |
| /// z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m. |
| </span><span class="attribute">#[allow(clippy::many_single_char_names)] |
| </span><span class="kw">fn </span>montgomery(x: <span class="kw-2">&</span>BigUint, y: <span class="kw-2">&</span>BigUint, m: <span class="kw-2">&</span>BigUint, k: BigDigit, n: usize) -> 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 && y.data.len() == n && m.data.len() == n, |
| <span class="string">"{:?} {:?} {:?} {}"</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">&mut </span>z.data[i..n + i], <span class="kw-2">&</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">&mut </span>z.data[i..n + i], <span class="kw-2">&</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 < c2 || cy < 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">&mut </span>first, <span class="kw-2">&</span>second, <span class="kw-2">&</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">&mut </span>[BigDigit], x: <span class="kw-2">&</span>[BigDigit], y: BigDigit) -> 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">&mut </span>[BigDigit], x: <span class="kw-2">&</span>[BigDigit], y: <span class="kw-2">&</span>[BigDigit]) -> 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 "Hacker's Delight", section 2-12 (overflow detection) |
| </span>c = ((yi & !xi) | ((yi | !xi) & zi)) >> (big_digit::BITS - <span class="number">1</span>) |
| } |
| |
| c |
| } |
| |
| <span class="doccomment">/// z1<<_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) -> (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 < x || yc < y { <span class="number">1 </span>} <span class="kw">else </span>{ <span class="number">0 </span>}; |
| |
| (z1, z0) |
| } |
| |
| <span class="doccomment">/// z1 << _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) -> (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 >> 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">&</span>BigUint, y: <span class="kw-2">&</span>BigUint, m: <span class="kw-2">&</span>BigUint) -> BigUint { |
| <span class="macro">assert!</span>(m.data[<span class="number">0</span>] & <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 >= m as long as len(x) == len(m). |
| </span><span class="kw">if </span>x.data.len() > num_words { |
| x %= m; |
| <span class="comment">// Note: now len(x) <= numWords, not guaranteed ==. |
| </span>} |
| <span class="kw">if </span>x.data.len() < 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() < 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><< n); |
| powers.push(montgomery(<span class="kw-2">&</span>one, <span class="kw-2">&</span>rr, m, mr.n0inv, num_words)); |
| powers.push(montgomery(<span class="kw-2">&</span>x, <span class="kw-2">&</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><< n { |
| <span class="kw">let </span>r = montgomery(<span class="kw-2">&</span>powers[i - <span class="number">1</span>], <span class="kw-2">&</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 < 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">&</span>z, <span class="kw-2">&</span>z, m, mr.n0inv, num_words); |
| z = montgomery(<span class="kw-2">&</span>zz, <span class="kw-2">&</span>zz, m, mr.n0inv, num_words); |
| zz = montgomery(<span class="kw-2">&</span>z, <span class="kw-2">&</span>z, m, mr.n0inv, num_words); |
| z = montgomery(<span class="kw-2">&</span>zz, <span class="kw-2">&</span>zz, m, mr.n0inv, num_words); |
| } |
| zz = montgomery( |
| <span class="kw-2">&</span>z, |
| <span class="kw-2">&</span>powers[(yi >> (big_digit::BITS - n)) <span class="kw">as </span>usize], |
| m, |
| mr.n0inv, |
| num_words, |
| ); |
| mem::swap(<span class="kw-2">&mut </span>z, <span class="kw-2">&mut </span>zz); |
| yi <<= n; |
| j += n; |
| } |
| } |
| |
| <span class="comment">// convert to regular number |
| </span>zz = montgomery(<span class="kw-2">&</span>z, <span class="kw-2">&</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 >= <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 >= <span class="kw-2">*</span>m { |
| zz %= m; |
| } |
| } |
| |
| zz.normalize(); |
| zz |
| } |
| </code></pre></div> |
| </section></div></main><div id="rustdoc-vars" data-root-path="../../../" data-current-crate="num_bigint" data-themes="ayu,dark,light" data-resource-suffix="" data-rustdoc-version="1.66.0-nightly (5c8bff74b 2022-10-21)" ></div></body></html> |