-
Notifications
You must be signed in to change notification settings - Fork 58
/
mod.go
470 lines (340 loc) · 14.4 KB
/
mod.go
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
// uint256: Fixed size 256-bit math library
// Copyright 2021 uint256 Authors
// SPDX-License-Identifier: BSD-3-Clause
package uint256
import (
"math/bits"
)
// Reciprocal computes a 320-bit value representing 1/m
//
// Notes:
// - specialized for m[3] != 0, hence limited to 2^192 <= m < 2^256
// - returns zero if m[3] == 0
// - starts with a 32-bit division, refines with newton-raphson iterations
func Reciprocal(m *Int) (mu [5]uint64) {
if m[3] == 0 {
return mu
}
s := bits.LeadingZeros64(m[3]) // Replace with leadingZeros(m) for general case
p := 255 - s // floor(log_2(m)), m>0
// 0 or a power of 2?
// Check if at least one bit is set in m[2], m[1] or m[0],
// or at least two bits in m[3]
if m[0] | m[1] | m[2] | (m[3] & (m[3]-1)) == 0 {
mu[4] = ^uint64(0) >> uint(p & 63)
mu[3] = ^uint64(0)
mu[2] = ^uint64(0)
mu[1] = ^uint64(0)
mu[0] = ^uint64(0)
return mu
}
// Maximise division precision by left-aligning divisor
var (
y Int // left-aligned copy of m
r0 uint32 // estimate of 2^31/y
)
y.Lsh(m, uint(s)) // 1/2 < y < 1
// Extract most significant 32 bits
yh := uint32(y[3] >> 32)
if yh == 0x80000000 { // Avoid overflow in division
r0 = 0xffffffff
} else {
r0, _ = bits.Div32(0x80000000, 0, yh)
}
// First iteration: 32 -> 64
t1 := uint64(r0) // 2^31/y
t1 *= t1 // 2^62/y^2
t1, _ = bits.Mul64(t1, y[3]) // 2^62/y^2 * 2^64/y / 2^64 = 2^62/y
r1 := uint64(r0) << 32 // 2^63/y
r1 -= t1 // 2^63/y - 2^62/y = 2^62/y
r1 *= 2 // 2^63/y
if (r1 | (y[3]<<1)) == 0 {
r1 = ^uint64(0)
}
// Second iteration: 64 -> 128
// square: 2^126/y^2
a2h, a2l := bits.Mul64(r1, r1)
// multiply by y: e2h:e2l:b2h = 2^126/y^2 * 2^128/y / 2^128 = 2^126/y
b2h, _ := bits.Mul64(a2l, y[2])
c2h, c2l := bits.Mul64(a2l, y[3])
d2h, d2l := bits.Mul64(a2h, y[2])
e2h, e2l := bits.Mul64(a2h, y[3])
b2h, c := bits.Add64(b2h, c2l, 0)
e2l, c = bits.Add64(e2l, c2h, c)
e2h, _ = bits.Add64(e2h, 0, c)
_, c = bits.Add64(b2h, d2l, 0)
e2l, c = bits.Add64(e2l, d2h, c)
e2h, _ = bits.Add64(e2h, 0, c)
// subtract: t2h:t2l = 2^127/y - 2^126/y = 2^126/y
t2l, b := bits.Sub64( 0, e2l, 0)
t2h, _ := bits.Sub64(r1, e2h, b)
// double: r2h:r2l = 2^127/y
r2l, c := bits.Add64(t2l, t2l, 0)
r2h, _ := bits.Add64(t2h, t2h, c)
if (r2h | r2l | (y[3]<<1)) == 0 {
r2h = ^uint64(0)
r2l = ^uint64(0)
}
// Third iteration: 128 -> 192
// square r2 (keep 256 bits): 2^190/y^2
a3h, a3l := bits.Mul64(r2l, r2l)
b3h, b3l := bits.Mul64(r2l, r2h)
c3h, c3l := bits.Mul64(r2h, r2h)
a3h, c = bits.Add64(a3h, b3l, 0)
c3l, c = bits.Add64(c3l, b3h, c)
c3h, _ = bits.Add64(c3h, 0, c)
a3h, c = bits.Add64(a3h, b3l, 0)
c3l, c = bits.Add64(c3l, b3h, c)
c3h, _ = bits.Add64(c3h, 0, c)
// multiply by y: q = 2^190/y^2 * 2^192/y / 2^192 = 2^190/y
x0 := a3l
x1 := a3h
x2 := c3l
x3 := c3h
var q0, q1, q2, q3, q4, t0 uint64
q0, _ = bits.Mul64(x2, y[0])
q1, t0 = bits.Mul64(x3, y[0]); q0, c = bits.Add64(q0, t0, 0); q1, _ = bits.Add64(q1, 0, c)
t1, _ = bits.Mul64(x1, y[1]); q0, c = bits.Add64(q0, t1, 0)
q2, t0 = bits.Mul64(x3, y[1]); q1, c = bits.Add64(q1, t0, c); q2, _ = bits.Add64(q2, 0, c)
t1, t0 = bits.Mul64(x2, y[1]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c); q2, _ = bits.Add64(q2, 0, c)
t1, t0 = bits.Mul64(x1, y[2]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c)
q3, t0 = bits.Mul64(x3, y[2]); q2, c = bits.Add64(q2, t0, c); q3, _ = bits.Add64(q3, 0, c)
t1, _ = bits.Mul64(x0, y[2]); q0, c = bits.Add64(q0, t1, 0)
t1, t0 = bits.Mul64(x2, y[2]); q1, c = bits.Add64(q1, t0, c); q2, c = bits.Add64(q2, t1, c); q3, _ = bits.Add64(q3, 0, c)
t1, t0 = bits.Mul64(x1, y[3]); q1, c = bits.Add64(q1, t0, 0); q2, c = bits.Add64(q2, t1, c)
q4, t0 = bits.Mul64(x3, y[3]); q3, c = bits.Add64(q3, t0, c); q4, _ = bits.Add64(q4, 0, c)
t1, t0 = bits.Mul64(x0, y[3]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c)
t1, t0 = bits.Mul64(x2, y[3]); q2, c = bits.Add64(q2, t0, c); q3, c = bits.Add64(q3, t1, c); q4, _ = bits.Add64(q4, 0, c)
// subtract: t3 = 2^191/y - 2^190/y = 2^190/y
_, b = bits.Sub64( 0, q0, 0)
_, b = bits.Sub64( 0, q1, b)
t3l, b := bits.Sub64( 0, q2, b)
t3m, b := bits.Sub64(r2l, q3, b)
t3h, _ := bits.Sub64(r2h, q4, b)
// double: r3 = 2^191/y
r3l, c := bits.Add64(t3l, t3l, 0)
r3m, c := bits.Add64(t3m, t3m, c)
r3h, _ := bits.Add64(t3h, t3h, c)
// Fourth iteration: 192 -> 320
// square r3
a4h, a4l := bits.Mul64(r3l, r3l)
b4h, b4l := bits.Mul64(r3l, r3m)
c4h, c4l := bits.Mul64(r3l, r3h)
d4h, d4l := bits.Mul64(r3m, r3m)
e4h, e4l := bits.Mul64(r3m, r3h)
f4h, f4l := bits.Mul64(r3h, r3h)
b4h, c = bits.Add64(b4h, c4l, 0)
e4l, c = bits.Add64(e4l, c4h, c)
e4h, _ = bits.Add64(e4h, 0, c)
a4h, c = bits.Add64(a4h, b4l, 0)
d4l, c = bits.Add64(d4l, b4h, c)
d4h, c = bits.Add64(d4h, e4l, c)
f4l, c = bits.Add64(f4l, e4h, c)
f4h, _ = bits.Add64(f4h, 0, c)
a4h, c = bits.Add64(a4h, b4l, 0)
d4l, c = bits.Add64(d4l, b4h, c)
d4h, c = bits.Add64(d4h, e4l, c)
f4l, c = bits.Add64(f4l, e4h, c)
f4h, _ = bits.Add64(f4h, 0, c)
// multiply by y
x1, x0 = bits.Mul64(d4h, y[0])
x3, x2 = bits.Mul64(f4h, y[0])
t1, t0 = bits.Mul64(f4l, y[0]); x1, c = bits.Add64(x1, t0, 0); x2, c = bits.Add64(x2, t1, c)
x3, _ = bits.Add64(x3, 0, c)
t1, t0 = bits.Mul64(d4h, y[1]); x1, c = bits.Add64(x1, t0, 0); x2, c = bits.Add64(x2, t1, c)
x4, t0 := bits.Mul64(f4h, y[1]); x3, c = bits.Add64(x3, t0, c); x4, _ = bits.Add64(x4, 0, c)
t1, t0 = bits.Mul64(d4l, y[1]); x0, c = bits.Add64(x0, t0, 0); x1, c = bits.Add64(x1, t1, c)
t1, t0 = bits.Mul64(f4l, y[1]); x2, c = bits.Add64(x2, t0, c); x3, c = bits.Add64(x3, t1, c)
x4, _ = bits.Add64(x4, 0, c)
t1, t0 = bits.Mul64(a4h, y[2]); x0, c = bits.Add64(x0, t0, 0); x1, c = bits.Add64(x1, t1, c)
t1, t0 = bits.Mul64(d4h, y[2]); x2, c = bits.Add64(x2, t0, c); x3, c = bits.Add64(x3, t1, c)
x5, t0 := bits.Mul64(f4h, y[2]); x4, c = bits.Add64(x4, t0, c); x5, _ = bits.Add64(x5, 0, c)
t1, t0 = bits.Mul64(d4l, y[2]); x1, c = bits.Add64(x1, t0, 0); x2, c = bits.Add64(x2, t1, c)
t1, t0 = bits.Mul64(f4l, y[2]); x3, c = bits.Add64(x3, t0, c); x4, c = bits.Add64(x4, t1, c)
x5, _ = bits.Add64(x5, 0, c)
t1, t0 = bits.Mul64(a4h, y[3]); x1, c = bits.Add64(x1, t0, 0); x2, c = bits.Add64(x2, t1, c)
t1, t0 = bits.Mul64(d4h, y[3]); x3, c = bits.Add64(x3, t0, c); x4, c = bits.Add64(x4, t1, c)
x6, t0 := bits.Mul64(f4h, y[3]); x5, c = bits.Add64(x5, t0, c); x6, _ = bits.Add64(x6, 0, c)
t1, t0 = bits.Mul64(a4l, y[3]); x0, c = bits.Add64(x0, t0, 0); x1, c = bits.Add64(x1, t1, c)
t1, t0 = bits.Mul64(d4l, y[3]); x2, c = bits.Add64(x2, t0, c); x3, c = bits.Add64(x3, t1, c)
t1, t0 = bits.Mul64(f4l, y[3]); x4, c = bits.Add64(x4, t0, c); x5, c = bits.Add64(x5, t1, c)
x6, _ = bits.Add64(x6, 0, c)
// subtract
_, b = bits.Sub64( 0, x0, 0)
_, b = bits.Sub64( 0, x1, b)
r4l, b := bits.Sub64( 0, x2, b)
r4k, b := bits.Sub64( 0, x3, b)
r4j, b := bits.Sub64(r3l, x4, b)
r4i, b := bits.Sub64(r3m, x5, b)
r4h, _ := bits.Sub64(r3h, x6, b)
// Multiply candidate for 1/4y by y, with full precision
x0 = r4l
x1 = r4k
x2 = r4j
x3 = r4i
x4 = r4h
q1, q0 = bits.Mul64(x0, y[0])
q3, q2 = bits.Mul64(x2, y[0])
q5, q4 := bits.Mul64(x4, y[0])
t1, t0 = bits.Mul64(x1, y[0]); q1, c = bits.Add64(q1, t0, 0); q2, c = bits.Add64(q2, t1, c)
t1, t0 = bits.Mul64(x3, y[0]); q3, c = bits.Add64(q3, t0, c); q4, c = bits.Add64(q4, t1, c); q5, _ = bits.Add64(q5, 0, c)
t1, t0 = bits.Mul64(x0, y[1]); q1, c = bits.Add64(q1, t0, 0); q2, c = bits.Add64(q2, t1, c)
t1, t0 = bits.Mul64(x2, y[1]); q3, c = bits.Add64(q3, t0, c); q4, c = bits.Add64(q4, t1, c)
q6, t0 := bits.Mul64(x4, y[1]); q5, c = bits.Add64(q5, t0, c); q6, _ = bits.Add64(q6, 0, c)
t1, t0 = bits.Mul64(x1, y[1]); q2, c = bits.Add64(q2, t0, 0); q3, c = bits.Add64(q3, t1, c)
t1, t0 = bits.Mul64(x3, y[1]); q4, c = bits.Add64(q4, t0, c); q5, c = bits.Add64(q5, t1, c); q6, _ = bits.Add64(q6, 0, c)
t1, t0 = bits.Mul64(x0, y[2]); q2, c = bits.Add64(q2, t0, 0); q3, c = bits.Add64(q3, t1, c)
t1, t0 = bits.Mul64(x2, y[2]); q4, c = bits.Add64(q4, t0, c); q5, c = bits.Add64(q5, t1, c)
q7, t0 := bits.Mul64(x4, y[2]); q6, c = bits.Add64(q6, t0, c); q7, _ = bits.Add64(q7, 0, c)
t1, t0 = bits.Mul64(x1, y[2]); q3, c = bits.Add64(q3, t0, 0); q4, c = bits.Add64(q4, t1, c)
t1, t0 = bits.Mul64(x3, y[2]); q5, c = bits.Add64(q5, t0, c); q6, c = bits.Add64(q6, t1, c); q7, _ = bits.Add64(q7, 0, c)
t1, t0 = bits.Mul64(x0, y[3]); q3, c = bits.Add64(q3, t0, 0); q4, c = bits.Add64(q4, t1, c)
t1, t0 = bits.Mul64(x2, y[3]); q5, c = bits.Add64(q5, t0, c); q6, c = bits.Add64(q6, t1, c)
q8, t0 := bits.Mul64(x4, y[3]); q7, c = bits.Add64(q7, t0, c); q8, _ = bits.Add64(q8, 0, c)
t1, t0 = bits.Mul64(x1, y[3]); q4, c = bits.Add64(q4, t0, 0); q5, c = bits.Add64(q5, t1, c)
t1, t0 = bits.Mul64(x3, y[3]); q6, c = bits.Add64(q6, t0, c); q7, c = bits.Add64(q7, t1, c); q8, _ = bits.Add64(q8, 0, c)
// Final adjustment
// subtract q from 1/4
_, b = bits.Sub64(0, q0, 0)
_, b = bits.Sub64(0, q1, b)
_, b = bits.Sub64(0, q2, b)
_, b = bits.Sub64(0, q3, b)
_, b = bits.Sub64(0, q4, b)
_, b = bits.Sub64(0, q5, b)
_, b = bits.Sub64(0, q6, b)
_, b = bits.Sub64(0, q7, b)
_, b = bits.Sub64(uint64(1) << 62, q8, b)
// decrement the result
x0, t := bits.Sub64(r4l, 1, 0)
x1, t = bits.Sub64(r4k, 0, t)
x2, t = bits.Sub64(r4j, 0, t)
x3, t = bits.Sub64(r4i, 0, t)
x4, _ = bits.Sub64(r4h, 0, t)
// commit the decrement if the subtraction underflowed (reciprocal was too large)
if b != 0 {
r4h, r4i, r4j, r4k, r4l = x4, x3, x2, x1, x0
}
// Shift to correct bit alignment, truncating excess bits
p = (p & 63) - 1
x0, c = bits.Add64(r4l, r4l, 0)
x1, c = bits.Add64(r4k, r4k, c)
x2, c = bits.Add64(r4j, r4j, c)
x3, c = bits.Add64(r4i, r4i, c)
x4, _ = bits.Add64(r4h, r4h, c)
if p < 0 {
r4h, r4i, r4j, r4k, r4l = x4, x3, x2, x1, x0
p = 0 // avoid negative shift below
}
{
r := uint(p) // right shift
l := uint(64 - r) // left shift
x0 = (r4l >> r) | (r4k << l)
x1 = (r4k >> r) | (r4j << l)
x2 = (r4j >> r) | (r4i << l)
x3 = (r4i >> r) | (r4h << l)
x4 = (r4h >> r)
}
if p > 0 {
r4h, r4i, r4j, r4k, r4l = x4, x3, x2, x1, x0
}
mu[0] = r4l
mu[1] = r4k
mu[2] = r4j
mu[3] = r4i
mu[4] = r4h
return mu
}
// reduce4 computes the least non-negative residue of x modulo m
//
// requires a four-word modulus (m[3] != 0) and its inverse (mu)
func (z *Int) reduce4(x *[8]uint64, m *Int, mu *[5]uint64) *Int {
// NB: Most variable names in the comments match the pseudocode for
// Barrett reduction in the Handbook of Applied Cryptography.
// q1 = x/2^192
x0 := x[3]
x1 := x[4]
x2 := x[5]
x3 := x[6]
x4 := x[7]
// q2 = q1 * mu; q3 = q2 / 2^320
var q0, q1, q2, q3, q4, q5, t0, t1, c uint64
q0, _ = bits.Mul64(x3, mu[0])
q1, t0 = bits.Mul64(x4, mu[0]); q0, c = bits.Add64(q0, t0, 0); q1, _ = bits.Add64(q1, 0, c)
t1, _ = bits.Mul64(x2, mu[1]); q0, c = bits.Add64(q0, t1, 0)
q2, t0 = bits.Mul64(x4, mu[1]); q1, c = bits.Add64(q1, t0, c); q2, _ = bits.Add64(q2, 0, c)
t1, t0 = bits.Mul64(x3, mu[1]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c); q2, _ = bits.Add64(q2, 0, c)
t1, t0 = bits.Mul64(x2, mu[2]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c)
q3, t0 = bits.Mul64(x4, mu[2]); q2, c = bits.Add64(q2, t0, c); q3, _ = bits.Add64(q3, 0, c)
t1, _ = bits.Mul64(x1, mu[2]); q0, c = bits.Add64(q0, t1, 0)
t1, t0 = bits.Mul64(x3, mu[2]); q1, c = bits.Add64(q1, t0, c); q2, c = bits.Add64(q2, t1, c); q3, _ = bits.Add64(q3, 0, c)
t1, _ = bits.Mul64(x0, mu[3]); q0, c = bits.Add64(q0, t1, 0)
t1, t0 = bits.Mul64(x2, mu[3]); q1, c = bits.Add64(q1, t0, c); q2, c = bits.Add64(q2, t1, c)
q4, t0 = bits.Mul64(x4, mu[3]); q3, c = bits.Add64(q3, t0, c); q4, _ = bits.Add64(q4, 0, c)
t1, t0 = bits.Mul64(x1, mu[3]); q0, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c)
t1, t0 = bits.Mul64(x3, mu[3]); q2, c = bits.Add64(q2, t0, c); q3, c = bits.Add64(q3, t1, c); q4, _ = bits.Add64(q4, 0, c)
t1, t0 = bits.Mul64(x0, mu[4]); _, c = bits.Add64(q0, t0, 0); q1, c = bits.Add64(q1, t1, c)
t1, t0 = bits.Mul64(x2, mu[4]); q2, c = bits.Add64(q2, t0, c); q3, c = bits.Add64(q3, t1, c)
q5, t0 = bits.Mul64(x4, mu[4]); q4, c = bits.Add64(q4, t0, c); q5, _ = bits.Add64(q5, 0, c)
t1, t0 = bits.Mul64(x1, mu[4]); q1, c = bits.Add64(q1, t0, 0); q2, c = bits.Add64(q2, t1, c)
t1, t0 = bits.Mul64(x3, mu[4]); q3, c = bits.Add64(q3, t0, c); q4, c = bits.Add64(q4, t1, c); q5, _ = bits.Add64(q5, 0, c)
// Drop the fractional part of q3
q0 = q1
q1 = q2
q2 = q3
q3 = q4
q4 = q5
// r1 = x mod 2^320
x0 = x[0]
x1 = x[1]
x2 = x[2]
x3 = x[3]
x4 = x[4]
// r2 = q3 * m mod 2^320
var r0, r1, r2, r3, r4 uint64
r4, r3 = bits.Mul64(q0, m[3])
_, t0 = bits.Mul64(q1, m[3]); r4, _ = bits.Add64(r4, t0, 0)
t1, r2 = bits.Mul64(q0, m[2]); r3, c = bits.Add64(r3, t1, 0)
_, t0 = bits.Mul64(q2, m[2]); r4, _ = bits.Add64(r4, t0, c)
t1, t0 = bits.Mul64(q1, m[2]); r3, c = bits.Add64(r3, t0, 0); r4, _ = bits.Add64(r4, t1, c)
t1, r1 = bits.Mul64(q0, m[1]); r2, c = bits.Add64(r2, t1, 0)
t1, t0 = bits.Mul64(q2, m[1]); r3, c = bits.Add64(r3, t0, c); r4, _ = bits.Add64(r4, t1, c)
t1, t0 = bits.Mul64(q1, m[1]); r2, c = bits.Add64(r2, t0, 0); r3, c = bits.Add64(r3, t1, c)
_, t0 = bits.Mul64(q3, m[1]); r4, _ = bits.Add64(r4, t0, c)
t1, r0 = bits.Mul64(q0, m[0]); r1, c = bits.Add64(r1, t1, 0)
t1, t0 = bits.Mul64(q2, m[0]); r2, c = bits.Add64(r2, t0, c); r3, c = bits.Add64(r3, t1, c)
_, t0 = bits.Mul64(q4, m[0]); r4, _ = bits.Add64(r4, t0, c)
t1, t0 = bits.Mul64(q1, m[0]); r1, c = bits.Add64(r1, t0, 0); r2, c = bits.Add64(r2, t1, c)
t1, t0 = bits.Mul64(q3, m[0]); r3, c = bits.Add64(r3, t0, c); r4, _ = bits.Add64(r4, t1, c)
// r = r1 - r2
var b uint64
r0, b = bits.Sub64(x0, r0, 0)
r1, b = bits.Sub64(x1, r1, b)
r2, b = bits.Sub64(x2, r2, b)
r3, b = bits.Sub64(x3, r3, b)
r4, b = bits.Sub64(x4, r4, b)
// if r<0 then r+=m
if b != 0 {
r0, c = bits.Add64(r0, m[0], 0)
r1, c = bits.Add64(r1, m[1], c)
r2, c = bits.Add64(r2, m[2], c)
r3, c = bits.Add64(r3, m[3], c)
r4, _ = bits.Add64(r4, 0, c)
}
// while (r>=m) r-=m
for {
// q = r - m
q0, b = bits.Sub64(r0, m[0], 0)
q1, b = bits.Sub64(r1, m[1], b)
q2, b = bits.Sub64(r2, m[2], b)
q3, b = bits.Sub64(r3, m[3], b)
q4, b = bits.Sub64(r4, 0, b)
// if borrow break
if b != 0 {
break
}
// r = q
r4, r3, r2, r1, r0 = q4, q3, q2, q1, q0
}
z[3], z[2], z[1], z[0] = r3, r2, r1, r0
return z
}