Permalink
714 lines (609 sloc) 21.4 KB
use std::borrow::Cow;
use std::cmp;
use std::cmp::Ordering::{self, Equal, Greater, Less};
use std::iter::repeat;
use std::mem;
use traits;
use traits::{One, Zero};
use biguint::BigUint;
use bigint::BigInt;
use bigint::Sign;
use bigint::Sign::{Minus, NoSign, Plus};
use big_digit::{self, BigDigit, DoubleBigDigit, SignedDoubleBigDigit};
// Generic functions for add/subtract/multiply with carry/borrow:
// Add with carry:
#[inline]
fn adc(a: BigDigit, b: BigDigit, acc: &mut DoubleBigDigit) -> BigDigit {
*acc += a as DoubleBigDigit;
*acc += b as DoubleBigDigit;
let lo = *acc as BigDigit;
*acc >>= big_digit::BITS;
lo
}
// Subtract with borrow:
#[inline]
fn sbb(a: BigDigit, b: BigDigit, acc: &mut SignedDoubleBigDigit) -> BigDigit {
*acc += a as SignedDoubleBigDigit;
*acc -= b as SignedDoubleBigDigit;
let lo = *acc as BigDigit;
*acc >>= big_digit::BITS;
lo
}
#[inline]
pub fn mac_with_carry(a: BigDigit, b: BigDigit, c: BigDigit, acc: &mut DoubleBigDigit) -> BigDigit {
*acc += a as DoubleBigDigit;
*acc += (b as DoubleBigDigit) * (c as DoubleBigDigit);
let lo = *acc as BigDigit;
*acc >>= big_digit::BITS;
lo
}
#[inline]
pub fn mul_with_carry(a: BigDigit, b: BigDigit, acc: &mut DoubleBigDigit) -> BigDigit {
*acc += (a as DoubleBigDigit) * (b as DoubleBigDigit);
let lo = *acc as BigDigit;
*acc >>= big_digit::BITS;
lo
}
/// Divide a two digit numerator by a one digit divisor, returns quotient and remainder:
///
/// Note: the caller must ensure that both the quotient and remainder will fit into a single digit.
/// This is _not_ true for an arbitrary numerator/denominator.
///
/// (This function also matches what the x86 divide instruction does).
#[inline]
fn div_wide(hi: BigDigit, lo: BigDigit, divisor: BigDigit) -> (BigDigit, BigDigit) {
debug_assert!(hi < divisor);
let lhs = big_digit::to_doublebigdigit(hi, lo);
let rhs = divisor as DoubleBigDigit;
((lhs / rhs) as BigDigit, (lhs % rhs) as BigDigit)
}
pub fn div_rem_digit(mut a: BigUint, b: BigDigit) -> (BigUint, BigDigit) {
let mut rem = 0;
for d in a.data.iter_mut().rev() {
let (q, r) = div_wide(rem, *d, b);
*d = q;
rem = r;
}
(a.normalized(), rem)
}
// Only for the Add impl:
#[inline]
pub fn __add2(a: &mut [BigDigit], b: &[BigDigit]) -> BigDigit {
debug_assert!(a.len() >= b.len());
let mut carry = 0;
let (a_lo, a_hi) = a.split_at_mut(b.len());
for (a, b) in a_lo.iter_mut().zip(b) {
*a = adc(*a, *b, &mut carry);
}
if carry != 0 {
for a in a_hi {
*a = adc(*a, 0, &mut carry);
if carry == 0 {
break;
}
}
}
carry as BigDigit
}
/// /Two argument addition of raw slices:
/// a += b
///
/// The caller _must_ ensure that a is big enough to store the result - typically this means
/// resizing a to max(a.len(), b.len()) + 1, to fit a possible carry.
pub fn add2(a: &mut [BigDigit], b: &[BigDigit]) {
let carry = __add2(a, b);
debug_assert!(carry == 0);
}
pub fn sub2(a: &mut [BigDigit], b: &[BigDigit]) {
let mut borrow = 0;
let len = cmp::min(a.len(), b.len());
let (a_lo, a_hi) = a.split_at_mut(len);
let (b_lo, b_hi) = b.split_at(len);
for (a, b) in a_lo.iter_mut().zip(b_lo) {
*a = sbb(*a, *b, &mut borrow);
}
if borrow != 0 {
for a in a_hi {
*a = sbb(*a, 0, &mut borrow);
if borrow == 0 {
break;
}
}
}
// note: we're _required_ to fail on underflow
assert!(
borrow == 0 && b_hi.iter().all(|x| *x == 0),
"Cannot subtract b from a because b is larger than a."
);
}
// Only for the Sub impl. `a` and `b` must have same length.
#[inline]
pub fn __sub2rev(a: &[BigDigit], b: &mut [BigDigit]) -> BigDigit {
debug_assert!(b.len() == a.len());
let mut borrow = 0;
for (ai, bi) in a.iter().zip(b) {
*bi = sbb(*ai, *bi, &mut borrow);
}
borrow as BigDigit
}
pub fn sub2rev(a: &[BigDigit], b: &mut [BigDigit]) {
debug_assert!(b.len() >= a.len());
let len = cmp::min(a.len(), b.len());
let (a_lo, a_hi) = a.split_at(len);
let (b_lo, b_hi) = b.split_at_mut(len);
let borrow = __sub2rev(a_lo, b_lo);
assert!(a_hi.is_empty());
// note: we're _required_ to fail on underflow
assert!(
borrow == 0 && b_hi.iter().all(|x| *x == 0),
"Cannot subtract b from a because b is larger than a."
);
}
pub fn sub_sign(a: &[BigDigit], b: &[BigDigit]) -> (Sign, BigUint) {
// Normalize:
let a = &a[..a.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
let b = &b[..b.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
match cmp_slice(a, b) {
Greater => {
let mut a = a.to_vec();
sub2(&mut a, b);
(Plus, BigUint::new(a))
}
Less => {
let mut b = b.to_vec();
sub2(&mut b, a);
(Minus, BigUint::new(b))
}
_ => (NoSign, Zero::zero()),
}
}
/// Three argument multiply accumulate:
/// acc += b * c
pub fn mac_digit(acc: &mut [BigDigit], b: &[BigDigit], c: BigDigit) {
if c == 0 {
return;
}
let mut carry = 0;
let (a_lo, a_hi) = acc.split_at_mut(b.len());
for (a, &b) in a_lo.iter_mut().zip(b) {
*a = mac_with_carry(*a, b, c, &mut carry);
}
let mut a = a_hi.iter_mut();
while carry != 0 {
let a = a.next().expect("carry overflow during multiplication!");
*a = adc(*a, 0, &mut carry);
}
}
/// Three argument multiply accumulate:
/// acc += b * c
fn mac3(acc: &mut [BigDigit], b: &[BigDigit], c: &[BigDigit]) {
let (x, y) = if b.len() < c.len() { (b, c) } else { (c, b) };
// We use three algorithms for different input sizes.
//
// - For small inputs, long multiplication is fastest.
// - Next we use Karatsuba multiplication (Toom-2), which we have optimized
// to avoid unnecessary allocations for intermediate values.
// - For the largest inputs we use Toom-3, which better optimizes the
// number of operations, but uses more temporary allocations.
//
// The thresholds are somewhat arbitrary, chosen by evaluating the results
// of `cargo bench --bench bigint multiply`.
if x.len() <= 32 {
// Long multiplication:
for (i, xi) in x.iter().enumerate() {
mac_digit(&mut acc[i..], y, *xi);
}
} else if x.len() <= 256 {
/*
* Karatsuba multiplication:
*
* The idea is that we break x and y up into two smaller numbers that each have about half
* as many digits, like so (note that multiplying by b is just a shift):
*
* x = x0 + x1 * b
* y = y0 + y1 * b
*
* With some algebra, we can compute x * y with three smaller products, where the inputs to
* each of the smaller products have only about half as many digits as x and y:
*
* x * y = (x0 + x1 * b) * (y0 + y1 * b)
*
* x * y = x0 * y0
* + x0 * y1 * b
* + x1 * y0 * b
* + x1 * y1 * b^2
*
* Let p0 = x0 * y0 and p2 = x1 * y1:
*
* x * y = p0
* + (x0 * y1 + x1 * y0) * b
* + p2 * b^2
*
* The real trick is that middle term:
*
* x0 * y1 + x1 * y0
*
* = x0 * y1 + x1 * y0 - p0 + p0 - p2 + p2
*
* = x0 * y1 + x1 * y0 - x0 * y0 - x1 * y1 + p0 + p2
*
* Now we complete the square:
*
* = -(x0 * y0 - x0 * y1 - x1 * y0 + x1 * y1) + p0 + p2
*
* = -((x1 - x0) * (y1 - y0)) + p0 + p2
*
* Let p1 = (x1 - x0) * (y1 - y0), and substitute back into our original formula:
*
* x * y = p0
* + (p0 + p2 - p1) * b
* + p2 * b^2
*
* Where the three intermediate products are:
*
* p0 = x0 * y0
* p1 = (x1 - x0) * (y1 - y0)
* p2 = x1 * y1
*
* In doing the computation, we take great care to avoid unnecessary temporary variables
* (since creating a BigUint requires a heap allocation): thus, we rearrange the formula a
* bit so we can use the same temporary variable for all the intermediate products:
*
* x * y = p2 * b^2 + p2 * b
* + p0 * b + p0
* - p1 * b
*
* The other trick we use is instead of doing explicit shifts, we slice acc at the
* appropriate offset when doing the add.
*/
/*
* When x is smaller than y, it's significantly faster to pick b such that x is split in
* half, not y:
*/
let b = x.len() / 2;
let (x0, x1) = x.split_at(b);
let (y0, y1) = y.split_at(b);
/*
* We reuse the same BigUint for all the intermediate multiplies and have to size p
* appropriately here: x1.len() >= x0.len and y1.len() >= y0.len():
*/
let len = x1.len() + y1.len() + 1;
let mut p = BigUint { data: vec![0; len] };
// p2 = x1 * y1
mac3(&mut p.data[..], x1, y1);
// Not required, but the adds go faster if we drop any unneeded 0s from the end:
p.normalize();
add2(&mut acc[b..], &p.data[..]);
add2(&mut acc[b * 2..], &p.data[..]);
// Zero out p before the next multiply:
p.data.truncate(0);
p.data.extend(repeat(0).take(len));
// p0 = x0 * y0
mac3(&mut p.data[..], x0, y0);
p.normalize();
add2(&mut acc[..], &p.data[..]);
add2(&mut acc[b..], &p.data[..]);
// p1 = (x1 - x0) * (y1 - y0)
// We do this one last, since it may be negative and acc can't ever be negative:
let (j0_sign, j0) = sub_sign(x1, x0);
let (j1_sign, j1) = sub_sign(y1, y0);
match j0_sign * j1_sign {
Plus => {
p.data.truncate(0);
p.data.extend(repeat(0).take(len));
mac3(&mut p.data[..], &j0.data[..], &j1.data[..]);
p.normalize();
sub2(&mut acc[b..], &p.data[..]);
}
Minus => {
mac3(&mut acc[b..], &j0.data[..], &j1.data[..]);
}
NoSign => (),
}
} else {
// Toom-3 multiplication:
//
// Toom-3 is like Karatsuba above, but dividing the inputs into three parts.
// Both are instances of Toom-Cook, using `k=3` and `k=2` respectively.
//
// The general idea is to treat the large integers digits as
// polynomials of a certain degree and determine the coefficients/digits
// of the product of the two via interpolation of the polynomial product.
let i = y.len() / 3 + 1;
let x0_len = cmp::min(x.len(), i);
let x1_len = cmp::min(x.len() - x0_len, i);
let y0_len = i;
let y1_len = cmp::min(y.len() - y0_len, i);
// Break x and y into three parts, representating an order two polynomial.
// t is chosen to be the size of a digit so we can use faster shifts
// in place of multiplications.
//
// x(t) = x2*t^2 + x1*t + x0
let x0 = BigInt::from_slice(Plus, &x[..x0_len]);
let x1 = BigInt::from_slice(Plus, &x[x0_len..x0_len + x1_len]);
let x2 = BigInt::from_slice(Plus, &x[x0_len + x1_len..]);
// y(t) = y2*t^2 + y1*t + y0
let y0 = BigInt::from_slice(Plus, &y[..y0_len]);
let y1 = BigInt::from_slice(Plus, &y[y0_len..y0_len + y1_len]);
let y2 = BigInt::from_slice(Plus, &y[y0_len + y1_len..]);
// Let w(t) = x(t) * y(t)
//
// This gives us the following order-4 polynomial.
//
// w(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
//
// We need to find the coefficients w4, w3, w2, w1 and w0. Instead
// of simply multiplying the x and y in total, we can evaluate w
// at 5 points. An n-degree polynomial is uniquely identified by (n + 1)
// points.
//
// It is arbitrary as to what points we evaluate w at but we use the
// following.
//
// w(t) at t = 0, 1, -1, -2 and inf
//
// The values for w(t) in terms of x(t)*y(t) at these points are:
//
// let a = w(0) = x0 * y0
// let b = w(1) = (x2 + x1 + x0) * (y2 + y1 + y0)
// let c = w(-1) = (x2 - x1 + x0) * (y2 - y1 + y0)
// let d = w(-2) = (4*x2 - 2*x1 + x0) * (4*y2 - 2*y1 + y0)
// let e = w(inf) = x2 * y2 as t -> inf
// x0 + x2, avoiding temporaries
let p = &x0 + &x2;
// y0 + y2, avoiding temporaries
let q = &y0 + &y2;
// x2 - x1 + x0, avoiding temporaries
let p2 = &p - &x1;
// y2 - y1 + y0, avoiding temporaries
let q2 = &q - &y1;
// w(0)
let r0 = &x0 * &y0;
// w(inf)
let r4 = &x2 * &y2;
// w(1)
let r1 = (p + x1) * (q + y1);
// w(-1)
let r2 = &p2 * &q2;
// w(-2)
let r3 = ((p2 + x2) * 2 - x0) * ((q2 + y2) * 2 - y0);
// Evaluating these points gives us the following system of linear equations.
//
// 0 0 0 0 1 | a
// 1 1 1 1 1 | b
// 1 -1 1 -1 1 | c
// 16 -8 4 -2 1 | d
// 1 0 0 0 0 | e
//
// The solved equation (after gaussian elimination or similar)
// in terms of its coefficients:
//
// w0 = w(0)
// w1 = w(0)/2 + w(1)/3 - w(-1) + w(2)/6 - 2*w(inf)
// w2 = -w(0) + w(1)/2 + w(-1)/2 - w(inf)
// w3 = -w(0)/2 + w(1)/6 + w(-1)/2 - w(1)/6
// w4 = w(inf)
//
// This particular sequence is given by Bodrato and is an interpolation
// of the above equations.
let mut comp3: BigInt = (r3 - &r1) / 3;
let mut comp1: BigInt = (r1 - &r2) / 2;
let mut comp2: BigInt = r2 - &r0;
comp3 = (&comp2 - comp3) / 2 + &r4 * 2;
comp2 = comp2 + &comp1 - &r4;
comp1 = comp1 - &comp3;
// Recomposition. The coefficients of the polynomial are now known.
//
// Evaluate at w(t) where t is our given base to get the result.
let result = r0
+ (comp1 << 32 * i)
+ (comp2 << 2 * 32 * i)
+ (comp3 << 3 * 32 * i)
+ (r4 << 4 * 32 * i);
let result_pos = result.to_biguint().unwrap();
add2(&mut acc[..], &result_pos.data);
}
}
pub fn mul3(x: &[BigDigit], y: &[BigDigit]) -> BigUint {
let len = x.len() + y.len() + 1;
let mut prod = BigUint { data: vec![0; len] };
mac3(&mut prod.data[..], x, y);
prod.normalized()
}
pub fn scalar_mul(a: &mut [BigDigit], b: BigDigit) -> BigDigit {
let mut carry = 0;
for a in a.iter_mut() {
*a = mul_with_carry(*a, b, &mut carry);
}
carry as BigDigit
}
pub fn div_rem(u: &BigUint, d: &BigUint) -> (BigUint, BigUint) {
if d.is_zero() {
panic!()
}
if u.is_zero() {
return (Zero::zero(), Zero::zero());
}
if d.data == [1] {
return (u.clone(), Zero::zero());
}
if d.data.len() == 1 {
let (div, rem) = div_rem_digit(u.clone(), d.data[0]);
return (div, rem.into());
}
// Required or the q_len calculation below can underflow:
match u.cmp(d) {
Less => return (Zero::zero(), u.clone()),
Equal => return (One::one(), Zero::zero()),
Greater => {} // Do nothing
}
// This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:
//
// First, normalize the arguments so the highest bit in the highest digit of the divisor is
// set: the main loop uses the highest digit of the divisor for generating guesses, so we
// want it to be the largest number we can efficiently divide by.
//
let shift = d.data.last().unwrap().leading_zeros() as usize;
let mut a = u << shift;
let b = d << shift;
// The algorithm works by incrementally calculating "guesses", q0, for part of the
// remainder. Once we have any number q0 such that q0 * b <= a, we can set
//
// q += q0
// a -= q0 * b
//
// and then iterate until a < b. Then, (q, a) will be our desired quotient and remainder.
//
// q0, our guess, is calculated by dividing the last few digits of a by the last digit of b
// - this should give us a guess that is "close" to the actual quotient, but is possibly
// greater than the actual quotient. If q0 * b > a, we simply use iterated subtraction
// until we have a guess such that q0 * b <= a.
//
let bn = *b.data.last().unwrap();
let q_len = a.data.len() - b.data.len() + 1;
let mut q = BigUint {
data: vec![0; q_len],
};
// We reuse the same temporary to avoid hitting the allocator in our inner loop - this is
// sized to hold a0 (in the common case; if a particular digit of the quotient is zero a0
// can be bigger).
//
let mut tmp = BigUint {
data: Vec::with_capacity(2),
};
for j in (0..q_len).rev() {
/*
* When calculating our next guess q0, we don't need to consider the digits below j
* + b.data.len() - 1: we're guessing digit j of the quotient (i.e. q0 << j) from
* digit bn of the divisor (i.e. bn << (b.data.len() - 1) - so the product of those
* two numbers will be zero in all digits up to (j + b.data.len() - 1).
*/
let offset = j + b.data.len() - 1;
if offset >= a.data.len() {
continue;
}
/* just avoiding a heap allocation: */
let mut a0 = tmp;
a0.data.truncate(0);
a0.data.extend(a.data[offset..].iter().cloned());
/*
* q0 << j * big_digit::BITS is our actual quotient estimate - we do the shifts
* implicitly at the end, when adding and subtracting to a and q. Not only do we
* save the cost of the shifts, the rest of the arithmetic gets to work with
* smaller numbers.
*/
let (mut q0, _) = div_rem_digit(a0, bn);
let mut prod = &b * &q0;
while cmp_slice(&prod.data[..], &a.data[j..]) == Greater {
let one: BigUint = One::one();
q0 = q0 - one;
prod = prod - &b;
}
add2(&mut q.data[j..], &q0.data[..]);
sub2(&mut a.data[j..], &prod.data[..]);
a.normalize();
tmp = q0;
}
debug_assert!(a < b);
(q.normalized(), a >> shift)
}
/// Find last set bit
/// fls(0) == 0, fls(u32::MAX) == 32
pub fn fls<T: traits::PrimInt>(v: T) -> usize {
mem::size_of::<T>() * 8 - v.leading_zeros() as usize
}
pub fn ilog2<T: traits::PrimInt>(v: T) -> usize {
fls(v) - 1
}
#[inline]
pub fn biguint_shl(n: Cow<BigUint>, bits: usize) -> BigUint {
let n_unit = bits / big_digit::BITS;
let mut data = match n_unit {
0 => n.into_owned().data,
_ => {
let len = n_unit + n.data.len() + 1;
let mut data = Vec::with_capacity(len);
data.extend(repeat(0).take(n_unit));
data.extend(n.data.iter().cloned());
data
}
};
let n_bits = bits % big_digit::BITS;
if n_bits > 0 {
let mut carry = 0;
for elem in data[n_unit..].iter_mut() {
let new_carry = *elem >> (big_digit::BITS - n_bits);
*elem = (*elem << n_bits) | carry;
carry = new_carry;
}
if carry != 0 {
data.push(carry);
}
}
BigUint::new(data)
}
#[inline]
pub fn biguint_shr(n: Cow<BigUint>, bits: usize) -> BigUint {
let n_unit = bits / big_digit::BITS;
if n_unit >= n.data.len() {
return Zero::zero();
}
let mut data = match n {
Cow::Borrowed(n) => n.data[n_unit..].to_vec(),
Cow::Owned(mut n) => {
n.data.drain(..n_unit);
n.data
}
};
let n_bits = bits % big_digit::BITS;
if n_bits > 0 {
let mut borrow = 0;
for elem in data.iter_mut().rev() {
let new_borrow = *elem << (big_digit::BITS - n_bits);
*elem = (*elem >> n_bits) | borrow;
borrow = new_borrow;
}
}
BigUint::new(data)
}
pub fn cmp_slice(a: &[BigDigit], b: &[BigDigit]) -> Ordering {
debug_assert!(a.last() != Some(&0));
debug_assert!(b.last() != Some(&0));
let (a_len, b_len) = (a.len(), b.len());
if a_len < b_len {
return Less;
}
if a_len > b_len {
return Greater;
}
for (&ai, &bi) in a.iter().rev().zip(b.iter().rev()) {
if ai < bi {
return Less;
}
if ai > bi {
return Greater;
}
}
return Equal;
}
#[cfg(test)]
mod algorithm_tests {
use big_digit::BigDigit;
use traits::Num;
use Sign::Plus;
use {BigInt, BigUint};
#[test]
fn test_sub_sign() {
use super::sub_sign;
fn sub_sign_i(a: &[BigDigit], b: &[BigDigit]) -> BigInt {
let (sign, val) = sub_sign(a, b);
BigInt::from_biguint(sign, val)
}
let a = BigUint::from_str_radix("265252859812191058636308480000000", 10).unwrap();
let b = BigUint::from_str_radix("26525285981219105863630848000000", 10).unwrap();
let a_i = BigInt::from_biguint(Plus, a.clone());
let b_i = BigInt::from_biguint(Plus, b.clone());
assert_eq!(sub_sign_i(&a.data[..], &b.data[..]), &a_i - &b_i);
assert_eq!(sub_sign_i(&b.data[..], &a.data[..]), &b_i - &a_i);
}
}