File: btcec.go

package info (click to toggle)
golang-github-btcsuite-btcd-btcec 0.0~git20161101.0.g8343278-1
  • links: PTS, VCS
  • area: main
  • in suites: buster, stretch
  • size: 1,488 kB
  • ctags: 216
  • sloc: makefile: 2
file content (956 lines) | stat: -rw-r--r-- 37,657 bytes parent folder | download | duplicates (2)
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
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
// Copyright 2013-2014 The btcsuite developers
// Use of this source code is governed by an ISC
// license that can be found in the LICENSE file.

package btcec

// References:
//   [SECG]: Recommended Elliptic Curve Domain Parameters
//     http://www.secg.org/sec2-v2.pdf
//
//   [GECC]: Guide to Elliptic Curve Cryptography (Hankerson, Menezes, Vanstone)

// This package operates, internally, on Jacobian coordinates. For a given
// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
// calculation can be performed within the transform (as in ScalarMult and
// ScalarBaseMult). But even for Add and Double, it's faster to apply and
// reverse the transform than to operate in affine coordinates.

import (
	"crypto/elliptic"
	"math/big"
	"sync"
)

var (
	// fieldOne is simply the integer 1 in field representation.  It is
	// used to avoid needing to create it multiple times during the internal
	// arithmetic.
	fieldOne = new(fieldVal).SetInt(1)
)

// KoblitzCurve supports a koblitz curve implementation that fits the ECC Curve
// interface from crypto/elliptic.
type KoblitzCurve struct {
	*elliptic.CurveParams
	q *big.Int
	H int // cofactor of the curve.

	// byteSize is simply the bit size / 8 and is provided for convenience
	// since it is calculated repeatedly.
	byteSize int

	// bytePoints
	bytePoints *[32][256][3]fieldVal

	// The next 6 values are used specifically for endomorphism
	// optimizations in ScalarMult.

	// lambda must fulfill lambda^3 = 1 mod N where N is the order of G.
	lambda *big.Int

	// beta must fulfill beta^3 = 1 mod P where P is the prime field of the
	// curve.
	beta *fieldVal

	// See the EndomorphismVectors in gensecp256k1.go to see how these are
	// derived.
	a1 *big.Int
	b1 *big.Int
	a2 *big.Int
	b2 *big.Int
}

// Params returns the parameters for the curve.
func (curve *KoblitzCurve) Params() *elliptic.CurveParams {
	return curve.CurveParams
}

// bigAffineToField takes an affine point (x, y) as big integers and converts
// it to an affine point as field values.
func (curve *KoblitzCurve) bigAffineToField(x, y *big.Int) (*fieldVal, *fieldVal) {
	x3, y3 := new(fieldVal), new(fieldVal)
	x3.SetByteSlice(x.Bytes())
	y3.SetByteSlice(y.Bytes())

	return x3, y3
}

// fieldJacobianToBigAffine takes a Jacobian point (x, y, z) as field values and
// converts it to an affine point as big integers.
func (curve *KoblitzCurve) fieldJacobianToBigAffine(x, y, z *fieldVal) (*big.Int, *big.Int) {
	// Inversions are expensive and both point addition and point doubling
	// are faster when working with points that have a z value of one.  So,
	// if the point needs to be converted to affine, go ahead and normalize
	// the point itself at the same time as the calculation is the same.
	var zInv, tempZ fieldVal
	zInv.Set(z).Inverse()   // zInv = Z^-1
	tempZ.SquareVal(&zInv)  // tempZ = Z^-2
	x.Mul(&tempZ)           // X = X/Z^2 (mag: 1)
	y.Mul(tempZ.Mul(&zInv)) // Y = Y/Z^3 (mag: 1)
	z.SetInt(1)             // Z = 1 (mag: 1)

	// Normalize the x and y values.
	x.Normalize()
	y.Normalize()

	// Convert the field values for the now affine point to big.Ints.
	x3, y3 := new(big.Int), new(big.Int)
	x3.SetBytes(x.Bytes()[:])
	y3.SetBytes(y.Bytes()[:])
	return x3, y3
}

// IsOnCurve returns boolean if the point (x,y) is on the curve.
// Part of the elliptic.Curve interface. This function differs from the
// crypto/elliptic algorithm since a = 0 not -3.
func (curve *KoblitzCurve) IsOnCurve(x, y *big.Int) bool {
	// Convert big ints to field values for faster arithmetic.
	fx, fy := curve.bigAffineToField(x, y)

	// Elliptic curve equation for secp256k1 is: y^2 = x^3 + 7
	y2 := new(fieldVal).SquareVal(fy).Normalize()
	result := new(fieldVal).SquareVal(fx).Mul(fx).AddInt(7).Normalize()
	return y2.Equals(result)
}

// addZ1AndZ2EqualsOne adds two Jacobian points that are already known to have
// z values of 1 and stores the result in (x3, y3, z3).  That is to say
// (x1, y1, 1) + (x2, y2, 1) = (x3, y3, z3).  It performs faster addition than
// the generic add routine since less arithmetic is needed due to the ability to
// avoid the z value multiplications.
func (curve *KoblitzCurve) addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
	// To compute the point addition efficiently, this implementation splits
	// the equation into intermediate elements which are used to minimize
	// the number of field multiplications using the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
	//
	// In particular it performs the calculations using the following:
	// H = X2-X1, HH = H^2, I = 4*HH, J = H*I, r = 2*(Y2-Y1), V = X1*I
	// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = 2*H
	//
	// This results in a cost of 4 field multiplications, 2 field squarings,
	// 6 field additions, and 5 integer multiplications.

	// When the x coordinates are the same for two points on the curve, the
	// y coordinates either must be the same, in which case it is point
	// doubling, or they are opposite and the result is the point at
	// infinity per the group law for elliptic curve cryptography.
	x1.Normalize()
	y1.Normalize()
	x2.Normalize()
	y2.Normalize()
	if x1.Equals(x2) {
		if y1.Equals(y2) {
			// Since x1 == x2 and y1 == y2, point doubling must be
			// done, otherwise the addition would end up dividing
			// by zero.
			curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
			return
		}

		// Since x1 == x2 and y1 == -y2, the sum is the point at
		// infinity per the group law.
		x3.SetInt(0)
		y3.SetInt(0)
		z3.SetInt(0)
		return
	}

	// Calculate X3, Y3, and Z3 according to the intermediate elements
	// breakdown above.
	var h, i, j, r, v fieldVal
	var negJ, neg2V, negX3 fieldVal
	h.Set(x1).Negate(1).Add(x2)                // H = X2-X1 (mag: 3)
	i.SquareVal(&h).MulInt(4)                  // I = 4*H^2 (mag: 4)
	j.Mul2(&h, &i)                             // J = H*I (mag: 1)
	r.Set(y1).Negate(1).Add(y2).MulInt(2)      // r = 2*(Y2-Y1) (mag: 6)
	v.Mul2(x1, &i)                             // V = X1*I (mag: 1)
	negJ.Set(&j).Negate(1)                     // negJ = -J (mag: 2)
	neg2V.Set(&v).MulInt(2).Negate(2)          // neg2V = -(2*V) (mag: 3)
	x3.Set(&r).Square().Add(&negJ).Add(&neg2V) // X3 = r^2-J-2*V (mag: 6)
	negX3.Set(x3).Negate(6)                    // negX3 = -X3 (mag: 7)
	j.Mul(y1).MulInt(2).Negate(2)              // J = -(2*Y1*J) (mag: 3)
	y3.Set(&v).Add(&negX3).Mul(&r).Add(&j)     // Y3 = r*(V-X3)-2*Y1*J (mag: 4)
	z3.Set(&h).MulInt(2)                       // Z3 = 2*H (mag: 6)

	// Normalize the resulting field values to a magnitude of 1 as needed.
	x3.Normalize()
	y3.Normalize()
	z3.Normalize()
}

// addZ1EqualsZ2 adds two Jacobian points that are already known to have the
// same z value and stores the result in (x3, y3, z3).  That is to say
// (x1, y1, z1) + (x2, y2, z1) = (x3, y3, z3).  It performs faster addition than
// the generic add routine since less arithmetic is needed due to the known
// equivalence.
func (curve *KoblitzCurve) addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
	// To compute the point addition efficiently, this implementation splits
	// the equation into intermediate elements which are used to minimize
	// the number of field multiplications using a slightly modified version
	// of the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-mmadd-2007-bl
	//
	// In particular it performs the calculations using the following:
	// A = X2-X1, B = A^2, C=Y2-Y1, D = C^2, E = X1*B, F = X2*B
	// X3 = D-E-F, Y3 = C*(E-X3)-Y1*(F-E), Z3 = Z1*A
	//
	// This results in a cost of 5 field multiplications, 2 field squarings,
	// 9 field additions, and 0 integer multiplications.

	// When the x coordinates are the same for two points on the curve, the
	// y coordinates either must be the same, in which case it is point
	// doubling, or they are opposite and the result is the point at
	// infinity per the group law for elliptic curve cryptography.
	x1.Normalize()
	y1.Normalize()
	x2.Normalize()
	y2.Normalize()
	if x1.Equals(x2) {
		if y1.Equals(y2) {
			// Since x1 == x2 and y1 == y2, point doubling must be
			// done, otherwise the addition would end up dividing
			// by zero.
			curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
			return
		}

		// Since x1 == x2 and y1 == -y2, the sum is the point at
		// infinity per the group law.
		x3.SetInt(0)
		y3.SetInt(0)
		z3.SetInt(0)
		return
	}

	// Calculate X3, Y3, and Z3 according to the intermediate elements
	// breakdown above.
	var a, b, c, d, e, f fieldVal
	var negX1, negY1, negE, negX3 fieldVal
	negX1.Set(x1).Negate(1)                // negX1 = -X1 (mag: 2)
	negY1.Set(y1).Negate(1)                // negY1 = -Y1 (mag: 2)
	a.Set(&negX1).Add(x2)                  // A = X2-X1 (mag: 3)
	b.SquareVal(&a)                        // B = A^2 (mag: 1)
	c.Set(&negY1).Add(y2)                  // C = Y2-Y1 (mag: 3)
	d.SquareVal(&c)                        // D = C^2 (mag: 1)
	e.Mul2(x1, &b)                         // E = X1*B (mag: 1)
	negE.Set(&e).Negate(1)                 // negE = -E (mag: 2)
	f.Mul2(x2, &b)                         // F = X2*B (mag: 1)
	x3.Add2(&e, &f).Negate(3).Add(&d)      // X3 = D-E-F (mag: 5)
	negX3.Set(x3).Negate(5).Normalize()    // negX3 = -X3 (mag: 1)
	y3.Set(y1).Mul(f.Add(&negE)).Negate(3) // Y3 = -(Y1*(F-E)) (mag: 4)
	y3.Add(e.Add(&negX3).Mul(&c))          // Y3 = C*(E-X3)+Y3 (mag: 5)
	z3.Mul2(z1, &a)                        // Z3 = Z1*A (mag: 1)

	// Normalize the resulting field values to a magnitude of 1 as needed.
	x3.Normalize()
	y3.Normalize()
}

// addZ2EqualsOne adds two Jacobian points when the second point is already
// known to have a z value of 1 (and the z value for the first point is not 1)
// and stores the result in (x3, y3, z3).  That is to say (x1, y1, z1) +
// (x2, y2, 1) = (x3, y3, z3).  It performs faster addition than the generic
// add routine since less arithmetic is needed due to the ability to avoid
// multiplications by the second point's z value.
func (curve *KoblitzCurve) addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3 *fieldVal) {
	// To compute the point addition efficiently, this implementation splits
	// the equation into intermediate elements which are used to minimize
	// the number of field multiplications using the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
	//
	// In particular it performs the calculations using the following:
	// Z1Z1 = Z1^2, U2 = X2*Z1Z1, S2 = Y2*Z1*Z1Z1, H = U2-X1, HH = H^2,
	// I = 4*HH, J = H*I, r = 2*(S2-Y1), V = X1*I
	// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*Y1*J, Z3 = (Z1+H)^2-Z1Z1-HH
	//
	// This results in a cost of 7 field multiplications, 4 field squarings,
	// 9 field additions, and 4 integer multiplications.

	// When the x coordinates are the same for two points on the curve, the
	// y coordinates either must be the same, in which case it is point
	// doubling, or they are opposite and the result is the point at
	// infinity per the group law for elliptic curve cryptography.  Since
	// any number of Jacobian coordinates can represent the same affine
	// point, the x and y values need to be converted to like terms.  Due to
	// the assumption made for this function that the second point has a z
	// value of 1 (z2=1), the first point is already "converted".
	var z1z1, u2, s2 fieldVal
	x1.Normalize()
	y1.Normalize()
	z1z1.SquareVal(z1)                        // Z1Z1 = Z1^2 (mag: 1)
	u2.Set(x2).Mul(&z1z1).Normalize()         // U2 = X2*Z1Z1 (mag: 1)
	s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
	if x1.Equals(&u2) {
		if y1.Equals(&s2) {
			// Since x1 == x2 and y1 == y2, point doubling must be
			// done, otherwise the addition would end up dividing
			// by zero.
			curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
			return
		}

		// Since x1 == x2 and y1 == -y2, the sum is the point at
		// infinity per the group law.
		x3.SetInt(0)
		y3.SetInt(0)
		z3.SetInt(0)
		return
	}

	// Calculate X3, Y3, and Z3 according to the intermediate elements
	// breakdown above.
	var h, hh, i, j, r, rr, v fieldVal
	var negX1, negY1, negX3 fieldVal
	negX1.Set(x1).Negate(1)                // negX1 = -X1 (mag: 2)
	h.Add2(&u2, &negX1)                    // H = U2-X1 (mag: 3)
	hh.SquareVal(&h)                       // HH = H^2 (mag: 1)
	i.Set(&hh).MulInt(4)                   // I = 4 * HH (mag: 4)
	j.Mul2(&h, &i)                         // J = H*I (mag: 1)
	negY1.Set(y1).Negate(1)                // negY1 = -Y1 (mag: 2)
	r.Set(&s2).Add(&negY1).MulInt(2)       // r = 2*(S2-Y1) (mag: 6)
	rr.SquareVal(&r)                       // rr = r^2 (mag: 1)
	v.Mul2(x1, &i)                         // V = X1*I (mag: 1)
	x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
	x3.Add(&rr)                            // X3 = r^2+X3 (mag: 5)
	negX3.Set(x3).Negate(5)                // negX3 = -X3 (mag: 6)
	y3.Set(y1).Mul(&j).MulInt(2).Negate(2) // Y3 = -(2*Y1*J) (mag: 3)
	y3.Add(v.Add(&negX3).Mul(&r))          // Y3 = r*(V-X3)+Y3 (mag: 4)
	z3.Add2(z1, &h).Square()               // Z3 = (Z1+H)^2 (mag: 1)
	z3.Add(z1z1.Add(&hh).Negate(2))        // Z3 = Z3-(Z1Z1+HH) (mag: 4)

	// Normalize the resulting field values to a magnitude of 1 as needed.
	x3.Normalize()
	y3.Normalize()
	z3.Normalize()
}

// addGeneric adds two Jacobian points (x1, y1, z1) and (x2, y2, z2) without any
// assumptions about the z values of the two points and stores the result in
// (x3, y3, z3).  That is to say (x1, y1, z1) + (x2, y2, z2) = (x3, y3, z3).  It
// is the slowest of the add routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
	// To compute the point addition efficiently, this implementation splits
	// the equation into intermediate elements which are used to minimize
	// the number of field multiplications using the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
	//
	// In particular it performs the calculations using the following:
	// Z1Z1 = Z1^2, Z2Z2 = Z2^2, U1 = X1*Z2Z2, U2 = X2*Z1Z1, S1 = Y1*Z2*Z2Z2
	// S2 = Y2*Z1*Z1Z1, H = U2-U1, I = (2*H)^2, J = H*I, r = 2*(S2-S1)
	// V = U1*I
	// X3 = r^2-J-2*V, Y3 = r*(V-X3)-2*S1*J, Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H
	//
	// This results in a cost of 11 field multiplications, 5 field squarings,
	// 9 field additions, and 4 integer multiplications.

	// When the x coordinates are the same for two points on the curve, the
	// y coordinates either must be the same, in which case it is point
	// doubling, or they are opposite and the result is the point at
	// infinity.  Since any number of Jacobian coordinates can represent the
	// same affine point, the x and y values need to be converted to like
	// terms.
	var z1z1, z2z2, u1, u2, s1, s2 fieldVal
	z1z1.SquareVal(z1)                        // Z1Z1 = Z1^2 (mag: 1)
	z2z2.SquareVal(z2)                        // Z2Z2 = Z2^2 (mag: 1)
	u1.Set(x1).Mul(&z2z2).Normalize()         // U1 = X1*Z2Z2 (mag: 1)
	u2.Set(x2).Mul(&z1z1).Normalize()         // U2 = X2*Z1Z1 (mag: 1)
	s1.Set(y1).Mul(&z2z2).Mul(z2).Normalize() // S1 = Y1*Z2*Z2Z2 (mag: 1)
	s2.Set(y2).Mul(&z1z1).Mul(z1).Normalize() // S2 = Y2*Z1*Z1Z1 (mag: 1)
	if u1.Equals(&u2) {
		if s1.Equals(&s2) {
			// Since x1 == x2 and y1 == y2, point doubling must be
			// done, otherwise the addition would end up dividing
			// by zero.
			curve.doubleJacobian(x1, y1, z1, x3, y3, z3)
			return
		}

		// Since x1 == x2 and y1 == -y2, the sum is the point at
		// infinity per the group law.
		x3.SetInt(0)
		y3.SetInt(0)
		z3.SetInt(0)
		return
	}

	// Calculate X3, Y3, and Z3 according to the intermediate elements
	// breakdown above.
	var h, i, j, r, rr, v fieldVal
	var negU1, negS1, negX3 fieldVal
	negU1.Set(&u1).Negate(1)               // negU1 = -U1 (mag: 2)
	h.Add2(&u2, &negU1)                    // H = U2-U1 (mag: 3)
	i.Set(&h).MulInt(2).Square()           // I = (2*H)^2 (mag: 2)
	j.Mul2(&h, &i)                         // J = H*I (mag: 1)
	negS1.Set(&s1).Negate(1)               // negS1 = -S1 (mag: 2)
	r.Set(&s2).Add(&negS1).MulInt(2)       // r = 2*(S2-S1) (mag: 6)
	rr.SquareVal(&r)                       // rr = r^2 (mag: 1)
	v.Mul2(&u1, &i)                        // V = U1*I (mag: 1)
	x3.Set(&v).MulInt(2).Add(&j).Negate(3) // X3 = -(J+2*V) (mag: 4)
	x3.Add(&rr)                            // X3 = r^2+X3 (mag: 5)
	negX3.Set(x3).Negate(5)                // negX3 = -X3 (mag: 6)
	y3.Mul2(&s1, &j).MulInt(2).Negate(2)   // Y3 = -(2*S1*J) (mag: 3)
	y3.Add(v.Add(&negX3).Mul(&r))          // Y3 = r*(V-X3)+Y3 (mag: 4)
	z3.Add2(z1, z2).Square()               // Z3 = (Z1+Z2)^2 (mag: 1)
	z3.Add(z1z1.Add(&z2z2).Negate(2))      // Z3 = Z3-(Z1Z1+Z2Z2) (mag: 4)
	z3.Mul(&h)                             // Z3 = Z3*H (mag: 1)

	// Normalize the resulting field values to a magnitude of 1 as needed.
	x3.Normalize()
	y3.Normalize()
}

// addJacobian adds the passed Jacobian points (x1, y1, z1) and (x2, y2, z2)
// together and stores the result in (x3, y3, z3).
func (curve *KoblitzCurve) addJacobian(x1, y1, z1, x2, y2, z2, x3, y3, z3 *fieldVal) {
	// A point at infinity is the identity according to the group law for
	// elliptic curve cryptography.  Thus, ∞ + P = P and P + ∞ = P.
	if (x1.IsZero() && y1.IsZero()) || z1.IsZero() {
		x3.Set(x2)
		y3.Set(y2)
		z3.Set(z2)
		return
	}
	if (x2.IsZero() && y2.IsZero()) || z2.IsZero() {
		x3.Set(x1)
		y3.Set(y1)
		z3.Set(z1)
		return
	}

	// Faster point addition can be achieved when certain assumptions are
	// met.  For example, when both points have the same z value, arithmetic
	// on the z values can be avoided.  This section thus checks for these
	// conditions and calls an appropriate add function which is accelerated
	// by using those assumptions.
	z1.Normalize()
	z2.Normalize()
	isZ1One := z1.Equals(fieldOne)
	isZ2One := z2.Equals(fieldOne)
	switch {
	case isZ1One && isZ2One:
		curve.addZ1AndZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
		return
	case z1.Equals(z2):
		curve.addZ1EqualsZ2(x1, y1, z1, x2, y2, x3, y3, z3)
		return
	case isZ2One:
		curve.addZ2EqualsOne(x1, y1, z1, x2, y2, x3, y3, z3)
		return
	}

	// None of the above assumptions are true, so fall back to generic
	// point addition.
	curve.addGeneric(x1, y1, z1, x2, y2, z2, x3, y3, z3)
}

// Add returns the sum of (x1,y1) and (x2,y2). Part of the elliptic.Curve
// interface.
func (curve *KoblitzCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
	// A point at infinity is the identity according to the group law for
	// elliptic curve cryptography.  Thus, ∞ + P = P and P + ∞ = P.
	if x1.Sign() == 0 && y1.Sign() == 0 {
		return x2, y2
	}
	if x2.Sign() == 0 && y2.Sign() == 0 {
		return x1, y1
	}

	// Convert the affine coordinates from big integers to field values
	// and do the point addition in Jacobian projective space.
	fx1, fy1 := curve.bigAffineToField(x1, y1)
	fx2, fy2 := curve.bigAffineToField(x2, y2)
	fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
	fOne := new(fieldVal).SetInt(1)
	curve.addJacobian(fx1, fy1, fOne, fx2, fy2, fOne, fx3, fy3, fz3)

	// Convert the Jacobian coordinate field values back to affine big
	// integers.
	return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}

// doubleZ1EqualsOne performs point doubling on the passed Jacobian point
// when the point is already known to have a z value of 1 and stores
// the result in (x3, y3, z3).  That is to say (x3, y3, z3) = 2*(x1, y1, 1).  It
// performs faster point doubling than the generic routine since less arithmetic
// is needed due to the ability to avoid multiplication by the z value.
func (curve *KoblitzCurve) doubleZ1EqualsOne(x1, y1, x3, y3, z3 *fieldVal) {
	// This function uses the assumptions that z1 is 1, thus the point
	// doubling formulas reduce to:
	//
	// X3 = (3*X1^2)^2 - 8*X1*Y1^2
	// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
	// Z3 = 2*Y1
	//
	// To compute the above efficiently, this implementation splits the
	// equation into intermediate elements which are used to minimize the
	// number of field multiplications in favor of field squarings which
	// are roughly 35% faster than field multiplications with the current
	// implementation at the time this was written.
	//
	// This uses a slightly modified version of the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-mdbl-2007-bl
	//
	// In particular it performs the calculations using the following:
	// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
	// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
	// Z3 = 2*Y1
	//
	// This results in a cost of 1 field multiplication, 5 field squarings,
	// 6 field additions, and 5 integer multiplications.
	var a, b, c, d, e, f fieldVal
	z3.Set(y1).MulInt(2)                     // Z3 = 2*Y1 (mag: 2)
	a.SquareVal(x1)                          // A = X1^2 (mag: 1)
	b.SquareVal(y1)                          // B = Y1^2 (mag: 1)
	c.SquareVal(&b)                          // C = B^2 (mag: 1)
	b.Add(x1).Square()                       // B = (X1+B)^2 (mag: 1)
	d.Set(&a).Add(&c).Negate(2)              // D = -(A+C) (mag: 3)
	d.Add(&b).MulInt(2)                      // D = 2*(B+D)(mag: 8)
	e.Set(&a).MulInt(3)                      // E = 3*A (mag: 3)
	f.SquareVal(&e)                          // F = E^2 (mag: 1)
	x3.Set(&d).MulInt(2).Negate(16)          // X3 = -(2*D) (mag: 17)
	x3.Add(&f)                               // X3 = F+X3 (mag: 18)
	f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
	y3.Set(&c).MulInt(8).Negate(8)           // Y3 = -(8*C) (mag: 9)
	y3.Add(f.Mul(&e))                        // Y3 = E*F+Y3 (mag: 10)

	// Normalize the field values back to a magnitude of 1.
	x3.Normalize()
	y3.Normalize()
	z3.Normalize()
}

// doubleGeneric performs point doubling on the passed Jacobian point without
// any assumptions about the z value and stores the result in (x3, y3, z3).
// That is to say (x3, y3, z3) = 2*(x1, y1, z1).  It is the slowest of the point
// doubling routines due to requiring the most arithmetic.
func (curve *KoblitzCurve) doubleGeneric(x1, y1, z1, x3, y3, z3 *fieldVal) {
	// Point doubling formula for Jacobian coordinates for the secp256k1
	// curve:
	// X3 = (3*X1^2)^2 - 8*X1*Y1^2
	// Y3 = (3*X1^2)*(4*X1*Y1^2 - X3) - 8*Y1^4
	// Z3 = 2*Y1*Z1
	//
	// To compute the above efficiently, this implementation splits the
	// equation into intermediate elements which are used to minimize the
	// number of field multiplications in favor of field squarings which
	// are roughly 35% faster than field multiplications with the current
	// implementation at the time this was written.
	//
	// This uses a slightly modified version of the method shown at:
	// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
	//
	// In particular it performs the calculations using the following:
	// A = X1^2, B = Y1^2, C = B^2, D = 2*((X1+B)^2-A-C)
	// E = 3*A, F = E^2, X3 = F-2*D, Y3 = E*(D-X3)-8*C
	// Z3 = 2*Y1*Z1
	//
	// This results in a cost of 1 field multiplication, 5 field squarings,
	// 6 field additions, and 5 integer multiplications.
	var a, b, c, d, e, f fieldVal
	z3.Mul2(y1, z1).MulInt(2)                // Z3 = 2*Y1*Z1 (mag: 2)
	a.SquareVal(x1)                          // A = X1^2 (mag: 1)
	b.SquareVal(y1)                          // B = Y1^2 (mag: 1)
	c.SquareVal(&b)                          // C = B^2 (mag: 1)
	b.Add(x1).Square()                       // B = (X1+B)^2 (mag: 1)
	d.Set(&a).Add(&c).Negate(2)              // D = -(A+C) (mag: 3)
	d.Add(&b).MulInt(2)                      // D = 2*(B+D)(mag: 8)
	e.Set(&a).MulInt(3)                      // E = 3*A (mag: 3)
	f.SquareVal(&e)                          // F = E^2 (mag: 1)
	x3.Set(&d).MulInt(2).Negate(16)          // X3 = -(2*D) (mag: 17)
	x3.Add(&f)                               // X3 = F+X3 (mag: 18)
	f.Set(x3).Negate(18).Add(&d).Normalize() // F = D-X3 (mag: 1)
	y3.Set(&c).MulInt(8).Negate(8)           // Y3 = -(8*C) (mag: 9)
	y3.Add(f.Mul(&e))                        // Y3 = E*F+Y3 (mag: 10)

	// Normalize the field values back to a magnitude of 1.
	x3.Normalize()
	y3.Normalize()
	z3.Normalize()
}

// doubleJacobian doubles the passed Jacobian point (x1, y1, z1) and stores the
// result in (x3, y3, z3).
func (curve *KoblitzCurve) doubleJacobian(x1, y1, z1, x3, y3, z3 *fieldVal) {
	// Doubling a point at infinity is still infinity.
	if y1.IsZero() || z1.IsZero() {
		x3.SetInt(0)
		y3.SetInt(0)
		z3.SetInt(0)
		return
	}

	// Slightly faster point doubling can be achieved when the z value is 1
	// by avoiding the multiplication on the z value.  This section calls
	// a point doubling function which is accelerated by using that
	// assumption when possible.
	if z1.Normalize().Equals(fieldOne) {
		curve.doubleZ1EqualsOne(x1, y1, x3, y3, z3)
		return
	}

	// Fall back to generic point doubling which works with arbitrary z
	// values.
	curve.doubleGeneric(x1, y1, z1, x3, y3, z3)
}

// Double returns 2*(x1,y1). Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
	if y1.Sign() == 0 {
		return new(big.Int), new(big.Int)
	}

	// Convert the affine coordinates from big integers to field values
	// and do the point doubling in Jacobian projective space.
	fx1, fy1 := curve.bigAffineToField(x1, y1)
	fx3, fy3, fz3 := new(fieldVal), new(fieldVal), new(fieldVal)
	fOne := new(fieldVal).SetInt(1)
	curve.doubleJacobian(fx1, fy1, fOne, fx3, fy3, fz3)

	// Convert the Jacobian coordinate field values back to affine big
	// integers.
	return curve.fieldJacobianToBigAffine(fx3, fy3, fz3)
}

// splitK returns a balanced length-two representation of k and their signs.
// This is algorithm 3.74 from [GECC].
//
// One thing of note about this algorithm is that no matter what c1 and c2 are,
// the final equation of k = k1 + k2 * lambda (mod n) will hold.  This is
// provable mathematically due to how a1/b1/a2/b2 are computed.
//
// c1 and c2 are chosen to minimize the max(k1,k2).
func (curve *KoblitzCurve) splitK(k []byte) ([]byte, []byte, int, int) {
	// All math here is done with big.Int, which is slow.
	// At some point, it might be useful to write something similar to
	// fieldVal but for N instead of P as the prime field if this ends up
	// being a bottleneck.
	bigIntK := new(big.Int)
	c1, c2 := new(big.Int), new(big.Int)
	tmp1, tmp2 := new(big.Int), new(big.Int)
	k1, k2 := new(big.Int), new(big.Int)

	bigIntK.SetBytes(k)
	// c1 = round(b2 * k / n) from step 4.
	// Rounding isn't really necessary and costs too much, hence skipped
	c1.Mul(curve.b2, bigIntK)
	c1.Div(c1, curve.N)
	// c2 = round(b1 * k / n) from step 4 (sign reversed to optimize one step)
	// Rounding isn't really necessary and costs too much, hence skipped
	c2.Mul(curve.b1, bigIntK)
	c2.Div(c2, curve.N)
	// k1 = k - c1 * a1 - c2 * a2 from step 5 (note c2's sign is reversed)
	tmp1.Mul(c1, curve.a1)
	tmp2.Mul(c2, curve.a2)
	k1.Sub(bigIntK, tmp1)
	k1.Add(k1, tmp2)
	// k2 = - c1 * b1 - c2 * b2 from step 5 (note c2's sign is reversed)
	tmp1.Mul(c1, curve.b1)
	tmp2.Mul(c2, curve.b2)
	k2.Sub(tmp2, tmp1)

	// Note Bytes() throws out the sign of k1 and k2. This matters
	// since k1 and/or k2 can be negative. Hence, we pass that
	// back separately.
	return k1.Bytes(), k2.Bytes(), k1.Sign(), k2.Sign()
}

// moduloReduce reduces k from more than 32 bytes to 32 bytes and under.  This
// is done by doing a simple modulo curve.N.  We can do this since G^N = 1 and
// thus any other valid point on the elliptic curve has the same order.
func (curve *KoblitzCurve) moduloReduce(k []byte) []byte {
	// Since the order of G is curve.N, we can use a much smaller number
	// by doing modulo curve.N
	if len(k) > curve.byteSize {
		// Reduce k by performing modulo curve.N.
		tmpK := new(big.Int).SetBytes(k)
		tmpK.Mod(tmpK, curve.N)
		return tmpK.Bytes()
	}

	return k
}

// NAF takes a positive integer k and returns the Non-Adjacent Form (NAF) as two
// byte slices.  The first is where 1s will be.  The second is where -1s will
// be.  NAF is convenient in that on average, only 1/3rd of its values are
// non-zero.  This is algorithm 3.30 from [GECC].
//
// Essentially, this makes it possible to minimize the number of operations
// since the resulting ints returned will be at least 50% 0s.
func NAF(k []byte) ([]byte, []byte) {
	// The essence of this algorithm is that whenever we have consecutive 1s
	// in the binary, we want to put a -1 in the lowest bit and get a bunch
	// of 0s up to the highest bit of consecutive 1s.  This is due to this
	// identity:
	// 2^n + 2^(n-1) + 2^(n-2) + ... + 2^(n-k) = 2^(n+1) - 2^(n-k)
	//
	// The algorithm thus may need to go 1 more bit than the length of the
	// bits we actually have, hence bits being 1 bit longer than was
	// necessary.  Since we need to know whether adding will cause a carry,
	// we go from right-to-left in this addition.
	var carry, curIsOne, nextIsOne bool
	// these default to zero
	retPos := make([]byte, len(k)+1)
	retNeg := make([]byte, len(k)+1)
	for i := len(k) - 1; i >= 0; i-- {
		curByte := k[i]
		for j := uint(0); j < 8; j++ {
			curIsOne = curByte&1 == 1
			if j == 7 {
				if i == 0 {
					nextIsOne = false
				} else {
					nextIsOne = k[i-1]&1 == 1
				}
			} else {
				nextIsOne = curByte&2 == 2
			}
			if carry {
				if curIsOne {
					// This bit is 1, so continue to carry
					// and don't need to do anything.
				} else {
					// We've hit a 0 after some number of
					// 1s.
					if nextIsOne {
						// Start carrying again since
						// a new sequence of 1s is
						// starting.
						retNeg[i+1] += 1 << j
					} else {
						// Stop carrying since 1s have
						// stopped.
						carry = false
						retPos[i+1] += 1 << j
					}
				}
			} else if curIsOne {
				if nextIsOne {
					// If this is the start of at least 2
					// consecutive 1s, set the current one
					// to -1 and start carrying.
					retNeg[i+1] += 1 << j
					carry = true
				} else {
					// This is a singleton, not consecutive
					// 1s.
					retPos[i+1] += 1 << j
				}
			}
			curByte >>= 1
		}
	}
	if carry {
		retPos[0] = 1
	}

	return retPos, retNeg
}

// ScalarMult returns k*(Bx, By) where k is a big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
	// Point Q = ∞ (point at infinity).
	qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)

	// Decompose K into k1 and k2 in order to halve the number of EC ops.
	// See Algorithm 3.74 in [GECC].
	k1, k2, signK1, signK2 := curve.splitK(curve.moduloReduce(k))

	// The main equation here to remember is:
	//   k * P = k1 * P + k2 * ϕ(P)
	//
	// P1 below is P in the equation, P2 below is ϕ(P) in the equation
	p1x, p1y := curve.bigAffineToField(Bx, By)
	p1yNeg := new(fieldVal).NegateVal(p1y, 1)
	p1z := new(fieldVal).SetInt(1)

	// NOTE: ϕ(x,y) = (βx,y).  The Jacobian z coordinate is 1, so this math
	// goes through.
	p2x := new(fieldVal).Mul2(p1x, curve.beta)
	p2y := new(fieldVal).Set(p1y)
	p2yNeg := new(fieldVal).NegateVal(p2y, 1)
	p2z := new(fieldVal).SetInt(1)

	// Flip the positive and negative values of the points as needed
	// depending on the signs of k1 and k2.  As mentioned in the equation
	// above, each of k1 and k2 are multiplied by the respective point.
	// Since -k * P is the same thing as k * -P, and the group law for
	// elliptic curves states that P(x, y) = -P(x, -y), it's faster and
	// simplifies the code to just make the point negative.
	if signK1 == -1 {
		p1y, p1yNeg = p1yNeg, p1y
	}
	if signK2 == -1 {
		p2y, p2yNeg = p2yNeg, p2y
	}

	// NAF versions of k1 and k2 should have a lot more zeros.
	//
	// The Pos version of the bytes contain the +1s and the Neg versions
	// contain the -1s.
	k1PosNAF, k1NegNAF := NAF(k1)
	k2PosNAF, k2NegNAF := NAF(k2)
	k1Len := len(k1PosNAF)
	k2Len := len(k2PosNAF)

	m := k1Len
	if m < k2Len {
		m = k2Len
	}

	// Add left-to-right using the NAF optimization.  See algorithm 3.77
	// from [GECC].  This should be faster overall since there will be a lot
	// more instances of 0, hence reducing the number of Jacobian additions
	// at the cost of 1 possible extra doubling.
	var k1BytePos, k1ByteNeg, k2BytePos, k2ByteNeg byte
	for i := 0; i < m; i++ {
		// Since we're going left-to-right, pad the front with 0s.
		if i < m-k1Len {
			k1BytePos = 0
			k1ByteNeg = 0
		} else {
			k1BytePos = k1PosNAF[i-(m-k1Len)]
			k1ByteNeg = k1NegNAF[i-(m-k1Len)]
		}
		if i < m-k2Len {
			k2BytePos = 0
			k2ByteNeg = 0
		} else {
			k2BytePos = k2PosNAF[i-(m-k2Len)]
			k2ByteNeg = k2NegNAF[i-(m-k2Len)]
		}

		for j := 7; j >= 0; j-- {
			// Q = 2 * Q
			curve.doubleJacobian(qx, qy, qz, qx, qy, qz)

			if k1BytePos&0x80 == 0x80 {
				curve.addJacobian(qx, qy, qz, p1x, p1y, p1z,
					qx, qy, qz)
			} else if k1ByteNeg&0x80 == 0x80 {
				curve.addJacobian(qx, qy, qz, p1x, p1yNeg, p1z,
					qx, qy, qz)
			}

			if k2BytePos&0x80 == 0x80 {
				curve.addJacobian(qx, qy, qz, p2x, p2y, p2z,
					qx, qy, qz)
			} else if k2ByteNeg&0x80 == 0x80 {
				curve.addJacobian(qx, qy, qz, p2x, p2yNeg, p2z,
					qx, qy, qz)
			}
			k1BytePos <<= 1
			k1ByteNeg <<= 1
			k2BytePos <<= 1
			k2ByteNeg <<= 1
		}
	}

	// Convert the Jacobian coordinate field values back to affine big.Ints.
	return curve.fieldJacobianToBigAffine(qx, qy, qz)
}

// ScalarBaseMult returns k*G where G is the base point of the group and k is a
// big endian integer.
// Part of the elliptic.Curve interface.
func (curve *KoblitzCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
	newK := curve.moduloReduce(k)
	diff := len(curve.bytePoints) - len(newK)

	// Point Q = ∞ (point at infinity).
	qx, qy, qz := new(fieldVal), new(fieldVal), new(fieldVal)

	// curve.bytePoints has all 256 byte points for each 8-bit window. The
	// strategy is to add up the byte points. This is best understood by
	// expressing k in base-256 which it already sort of is.
	// Each "digit" in the 8-bit window can be looked up using bytePoints
	// and added together.
	for i, byteVal := range newK {
		p := curve.bytePoints[diff+i][byteVal]
		curve.addJacobian(qx, qy, qz, &p[0], &p[1], &p[2], qx, qy, qz)
	}
	return curve.fieldJacobianToBigAffine(qx, qy, qz)
}

// QPlus1Div4 returns the Q+1/4 constant for the curve for use in calculating
// square roots via exponention.
func (curve *KoblitzCurve) QPlus1Div4() *big.Int {
	return curve.q
}

var initonce sync.Once
var secp256k1 KoblitzCurve

func initAll() {
	initS256()
}

// fromHex converts the passed hex string into a big integer pointer and will
// panic is there is an error.  This is only provided for the hard-coded
// constants so errors in the source code can bet detected. It will only (and
// must only) be called for initialization purposes.
func fromHex(s string) *big.Int {
	r, ok := new(big.Int).SetString(s, 16)
	if !ok {
		panic("invalid hex in source file: " + s)
	}
	return r
}

func initS256() {
	// Curve parameters taken from [SECG] section 2.4.1.
	secp256k1.CurveParams = new(elliptic.CurveParams)
	secp256k1.P = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")
	secp256k1.N = fromHex("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141")
	secp256k1.B = fromHex("0000000000000000000000000000000000000000000000000000000000000007")
	secp256k1.Gx = fromHex("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798")
	secp256k1.Gy = fromHex("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8")
	secp256k1.BitSize = 256
	secp256k1.H = 1
	secp256k1.q = new(big.Int).Div(new(big.Int).Add(secp256k1.P,
		big.NewInt(1)), big.NewInt(4))

	// Provided for convenience since this gets computed repeatedly.
	secp256k1.byteSize = secp256k1.BitSize / 8

	// Deserialize and set the pre-computed table used to accelerate scalar
	// base multiplication.  This is hard-coded data, so any errors are
	// panics because it means something is wrong in the source code.
	if err := loadS256BytePoints(); err != nil {
		panic(err)
	}

	// Next 6 constants are from Hal Finney's bitcointalk.org post:
	// https://bitcointalk.org/index.php?topic=3238.msg45565#msg45565
	// May he rest in peace.
	//
	// They have also been independently derived from the code in the
	// EndomorphismVectors function in gensecp256k1.go.
	secp256k1.lambda = fromHex("5363AD4CC05C30E0A5261C028812645A122E22EA20816678DF02967C1B23BD72")
	secp256k1.beta = new(fieldVal).SetHex("7AE96A2B657C07106E64479EAC3434E99CF0497512F58995C1396C28719501EE")
	secp256k1.a1 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
	secp256k1.b1 = fromHex("-E4437ED6010E88286F547FA90ABFE4C3")
	secp256k1.a2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
	secp256k1.b2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")

	// Alternatively, we can use the parameters below, however, they seem
	//  to be about 8% slower.
	// secp256k1.lambda = fromHex("AC9C52B33FA3CF1F5AD9E3FD77ED9BA4A880B9FC8EC739C2E0CFC810B51283CE")
	// secp256k1.beta = new(fieldVal).SetHex("851695D49A83F8EF919BB86153CBCB16630FB68AED0A766A3EC693D68E6AFA40")
	// secp256k1.a1 = fromHex("E4437ED6010E88286F547FA90ABFE4C3")
	// secp256k1.b1 = fromHex("-3086D221A7D46BCDE86C90E49284EB15")
	// secp256k1.a2 = fromHex("3086D221A7D46BCDE86C90E49284EB15")
	// secp256k1.b2 = fromHex("114CA50F7A8E2F3F657C1108D9D44CFD8")
}

// S256 returns a Curve which implements secp256k1.
func S256() *KoblitzCurve {
	initonce.Do(initAll)
	return &secp256k1
}