1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
use std::borrow::Cow;
use std::cmp;
use std::cmp::Ordering::{self, Less, Greater, Equal};
use std::iter::repeat;
use std::mem;
use traits;
use traits::{Zero, One};

use biguint::BigUint;

use bigint::Sign;
use bigint::Sign::{Minus, NoSign, Plus};

#[allow(non_snake_case)]
pub mod big_digit {
    /// A `BigDigit` is a `BigUint`'s composing element.
    pub type BigDigit = u32;

    /// A `DoubleBigDigit` is the internal type used to do the computations.  Its
    /// size is the double of the size of `BigDigit`.
    pub type DoubleBigDigit = u64;

    pub const ZERO_BIG_DIGIT: BigDigit = 0;

    // `DoubleBigDigit` size dependent
    pub const BITS: usize = 32;

    pub const BASE: DoubleBigDigit = 1 << BITS;
    const LO_MASK: DoubleBigDigit = (-1i32 as DoubleBigDigit) >> BITS;

    #[inline]
    fn get_hi(n: DoubleBigDigit) -> BigDigit {
        (n >> BITS) as BigDigit
    }
    #[inline]
    fn get_lo(n: DoubleBigDigit) -> BigDigit {
        (n & LO_MASK) as BigDigit
    }

    /// Split one `DoubleBigDigit` into two `BigDigit`s.
    #[inline]
    pub fn from_doublebigdigit(n: DoubleBigDigit) -> (BigDigit, BigDigit) {
        (get_hi(n), get_lo(n))
    }

    /// Join two `BigDigit`s into one `DoubleBigDigit`
    #[inline]
    pub fn to_doublebigdigit(hi: BigDigit, lo: BigDigit) -> DoubleBigDigit {
        (lo as DoubleBigDigit) | ((hi as DoubleBigDigit) << BITS)
    }
}

use big_digit::{BigDigit, DoubleBigDigit};

// Generic functions for add/subtract/multiply with carry/borrow:

// Add with carry:
#[inline]
fn adc(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
    let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) + (b as DoubleBigDigit) +
                                                  (*carry as DoubleBigDigit));

    *carry = hi;
    lo
}

// Subtract with borrow:
#[inline]
fn sbb(a: BigDigit, b: BigDigit, borrow: &mut BigDigit) -> BigDigit {
    let (hi, lo) = big_digit::from_doublebigdigit(big_digit::BASE + (a as DoubleBigDigit) -
                                                  (b as DoubleBigDigit) -
                                                  (*borrow as DoubleBigDigit));
    // hi * (base) + lo == 1*(base) + ai - bi - borrow
    // => ai - bi - borrow < 0 <=> hi == 0
    *borrow = (hi == 0) as BigDigit;
    lo
}

#[inline]
pub fn mac_with_carry(a: BigDigit, b: BigDigit, c: BigDigit, carry: &mut BigDigit) -> BigDigit {
    let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) +
                                                  (b as DoubleBigDigit) * (c as DoubleBigDigit) +
                                                  (*carry as DoubleBigDigit));
    *carry = hi;
    lo
}

#[inline]
pub fn mul_with_carry(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
    let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) * (b as DoubleBigDigit) +
                                                  (*carry as DoubleBigDigit));

    *carry = hi;
    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.normalize(), rem)
}

// Only for the Add impl:
#[must_use]
#[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
}

/// /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.");
}

pub fn sub2rev(a: &[BigDigit], b: &mut [BigDigit]) {
    debug_assert!(b.len() >= a.len());

    let mut borrow = 0;

    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);

    for (a, b) in a_lo.iter().zip(b_lo) {
        *b = sbb(*a, *b, &mut borrow);
    }

    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
fn mac_digit(acc: &mut [BigDigit], b: &[BigDigit], c: BigDigit) {
    if c == 0 {
        return;
    }

    let mut b_iter = b.iter();
    let mut carry = 0;

    for ai in acc.iter_mut() {
        if let Some(bi) = b_iter.next() {
            *ai = mac_with_carry(*ai, *bi, c, &mut carry);
        } else if carry != 0 {
            *ai = mac_with_carry(*ai, 0, c, &mut carry);
        } else {
            break;
        }
    }

    assert!(carry == 0);
}

/// 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)
    };

    // Karatsuba multiplication is slower than long multiplication for small x and y:
    //
    if x.len() <= 4 {
        for (i, xi) in x.iter().enumerate() {
            mac_digit(&mut acc[i..], y, *xi);
        }
    } else {
        /*
         * 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 = 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 = 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 = p.normalize();

                sub2(&mut acc[b..], &p.data[..]);
            },
            Minus   => {
                mac3(&mut acc[b..], &j0.data[..], &j1.data[..]);
            },
            NoSign  => (),
        }
    }
}

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.normalize()
}

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
}

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 == One::one() {
        return (u.clone(), Zero::zero());
    }

    // 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 = a.normalize();

        tmp = q0;
    }

    debug_assert!(a < b);

    (q.normalize(), 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_unit {
        0 => n.into_owned().data,
        _ => n.data[n_unit..].to_vec(),
    };

    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 {BigDigit, BigUint, BigInt};
    use Sign::Plus;
    use traits::Num;

    #[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);
    }
}