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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 += DoubleBigDigit::from(a); *acc += DoubleBigDigit::from(b); 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 += SignedDoubleBigDigit::from(a); *acc -= SignedDoubleBigDigit::from(b); 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 += DoubleBigDigit::from(a); *acc += DoubleBigDigit::from(b) * DoubleBigDigit::from(c); 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 += DoubleBigDigit::from(a) * DoubleBigDigit::from(b); 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 = DoubleBigDigit::from(divisor); ((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) } pub fn rem_digit(a: &BigUint, b: BigDigit) -> BigDigit { let mut rem: DoubleBigDigit = 0; for &digit in a.data.iter().rev() { rem = (rem << big_digit::BITS) + DoubleBigDigit::from(digit); rem %= DoubleBigDigit::from(b); } rem as BigDigit } // 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 += &comp1 - &r4; 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(mut u: BigUint, mut d: BigUint) -> (BigUint, BigUint) { if d.is_zero() { panic!() } if u.is_zero() { return (Zero::zero(), Zero::zero()); } if d.data.len() == 1 { if d.data == [1] { return (u, Zero::zero()); } let (div, rem) = div_rem_digit(u, d.data[0]); // reuse d d.data.clear(); d += rem; return (div, d); } // Required or the q_len calculation below can underflow: match u.cmp(&d) { Less => return (Zero::zero(), u), Equal => { u.set_one(); return (u, 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 (q, r) = if shift == 0 { // no need to clone d div_rem_core(u, &d) } else { div_rem_core(u << shift, &(d << shift)) }; // renormalize the remainder (q, r >> shift) } pub fn div_rem_ref(u: &BigUint, d: &BigUint) -> (BigUint, BigUint) { if d.is_zero() { panic!() } if u.is_zero() { return (Zero::zero(), Zero::zero()); } if d.data.len() == 1 { if d.data == [1] { return (u.clone(), Zero::zero()); } 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 (q, r) = if shift == 0 { // no need to clone d div_rem_core(u.clone(), d) } else { div_rem_core(u << shift, &(d << shift)) }; // renormalize the remainder (q, r >> shift) } /// an implementation of Knuth, TAOCP vol 2 section 4.3, algorithm D /// /// # Correctness /// /// This function requires the following conditions to run correctly and/or effectively /// /// - `a > b` /// - `d.data.len() > 1` /// - `d.data.last().unwrap().leading_zeros() == 0` fn div_rem_core(mut a: BigUint, b: &BigUint) -> (BigUint, BigUint) { // 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 -= one; 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) } /// 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; } } 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); } }