1// Copyright 2022 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5// Code generated by generate.go. DO NOT EDIT.
6
7package nistec
8
9import (
10	"crypto/internal/nistec/fiat"
11	"crypto/subtle"
12	"errors"
13	"sync"
14)
15
16// p521ElementLength is the length of an element of the base or scalar field,
17// which have the same bytes length for all NIST P curves.
18const p521ElementLength = 66
19
20// P521Point is a P521 point. The zero value is NOT valid.
21type P521Point struct {
22	// The point is represented in projective coordinates (X:Y:Z),
23	// where x = X/Z and y = Y/Z.
24	x, y, z *fiat.P521Element
25}
26
27// NewP521Point returns a new P521Point representing the point at infinity point.
28func NewP521Point() *P521Point {
29	return &P521Point{
30		x: new(fiat.P521Element),
31		y: new(fiat.P521Element).One(),
32		z: new(fiat.P521Element),
33	}
34}
35
36// SetGenerator sets p to the canonical generator and returns p.
37func (p *P521Point) SetGenerator() *P521Point {
38	p.x.SetBytes([]byte{0x0, 0xc6, 0x85, 0x8e, 0x6, 0xb7, 0x4, 0x4, 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x5, 0x3f, 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 0xbd, 0x66})
39	p.y.SetBytes([]byte{0x1, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, 0xc0, 0x4, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x1, 0x3f, 0xad, 0x7, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50})
40	p.z.One()
41	return p
42}
43
44// Set sets p = q and returns p.
45func (p *P521Point) Set(q *P521Point) *P521Point {
46	p.x.Set(q.x)
47	p.y.Set(q.y)
48	p.z.Set(q.z)
49	return p
50}
51
52// SetBytes sets p to the compressed, uncompressed, or infinity value encoded in
53// b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on
54// the curve, it returns nil and an error, and the receiver is unchanged.
55// Otherwise, it returns p.
56func (p *P521Point) SetBytes(b []byte) (*P521Point, error) {
57	switch {
58	// Point at infinity.
59	case len(b) == 1 && b[0] == 0:
60		return p.Set(NewP521Point()), nil
61
62	// Uncompressed form.
63	case len(b) == 1+2*p521ElementLength && b[0] == 4:
64		x, err := new(fiat.P521Element).SetBytes(b[1 : 1+p521ElementLength])
65		if err != nil {
66			return nil, err
67		}
68		y, err := new(fiat.P521Element).SetBytes(b[1+p521ElementLength:])
69		if err != nil {
70			return nil, err
71		}
72		if err := p521CheckOnCurve(x, y); err != nil {
73			return nil, err
74		}
75		p.x.Set(x)
76		p.y.Set(y)
77		p.z.One()
78		return p, nil
79
80	// Compressed form.
81	case len(b) == 1+p521ElementLength && (b[0] == 2 || b[0] == 3):
82		x, err := new(fiat.P521Element).SetBytes(b[1:])
83		if err != nil {
84			return nil, err
85		}
86
87		// y² = x³ - 3x + b
88		y := p521Polynomial(new(fiat.P521Element), x)
89		if !p521Sqrt(y, y) {
90			return nil, errors.New("invalid P521 compressed point encoding")
91		}
92
93		// Select the positive or negative root, as indicated by the least
94		// significant bit, based on the encoding type byte.
95		otherRoot := new(fiat.P521Element)
96		otherRoot.Sub(otherRoot, y)
97		cond := y.Bytes()[p521ElementLength-1]&1 ^ b[0]&1
98		y.Select(otherRoot, y, int(cond))
99
100		p.x.Set(x)
101		p.y.Set(y)
102		p.z.One()
103		return p, nil
104
105	default:
106		return nil, errors.New("invalid P521 point encoding")
107	}
108}
109
110var _p521B *fiat.P521Element
111var _p521BOnce sync.Once
112
113func p521B() *fiat.P521Element {
114	_p521BOnce.Do(func() {
115		_p521B, _ = new(fiat.P521Element).SetBytes([]byte{0x0, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 0x9, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 0xbf, 0x7, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x0})
116	})
117	return _p521B
118}
119
120// p521Polynomial sets y2 to x³ - 3x + b, and returns y2.
121func p521Polynomial(y2, x *fiat.P521Element) *fiat.P521Element {
122	y2.Square(x)
123	y2.Mul(y2, x)
124
125	threeX := new(fiat.P521Element).Add(x, x)
126	threeX.Add(threeX, x)
127	y2.Sub(y2, threeX)
128
129	return y2.Add(y2, p521B())
130}
131
132func p521CheckOnCurve(x, y *fiat.P521Element) error {
133	// y² = x³ - 3x + b
134	rhs := p521Polynomial(new(fiat.P521Element), x)
135	lhs := new(fiat.P521Element).Square(y)
136	if rhs.Equal(lhs) != 1 {
137		return errors.New("P521 point not on curve")
138	}
139	return nil
140}
141
142// Bytes returns the uncompressed or infinity encoding of p, as specified in
143// SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at
144// infinity is shorter than all other encodings.
145func (p *P521Point) Bytes() []byte {
146	// This function is outlined to make the allocations inline in the caller
147	// rather than happen on the heap.
148	var out [1 + 2*p521ElementLength]byte
149	return p.bytes(&out)
150}
151
152func (p *P521Point) bytes(out *[1 + 2*p521ElementLength]byte) []byte {
153	if p.z.IsZero() == 1 {
154		return append(out[:0], 0)
155	}
156
157	zinv := new(fiat.P521Element).Invert(p.z)
158	x := new(fiat.P521Element).Mul(p.x, zinv)
159	y := new(fiat.P521Element).Mul(p.y, zinv)
160
161	buf := append(out[:0], 4)
162	buf = append(buf, x.Bytes()...)
163	buf = append(buf, y.Bytes()...)
164	return buf
165}
166
167// BytesX returns the encoding of the x-coordinate of p, as specified in SEC 1,
168// Version 2.0, Section 2.3.5, or an error if p is the point at infinity.
169func (p *P521Point) BytesX() ([]byte, error) {
170	// This function is outlined to make the allocations inline in the caller
171	// rather than happen on the heap.
172	var out [p521ElementLength]byte
173	return p.bytesX(&out)
174}
175
176func (p *P521Point) bytesX(out *[p521ElementLength]byte) ([]byte, error) {
177	if p.z.IsZero() == 1 {
178		return nil, errors.New("P521 point is the point at infinity")
179	}
180
181	zinv := new(fiat.P521Element).Invert(p.z)
182	x := new(fiat.P521Element).Mul(p.x, zinv)
183
184	return append(out[:0], x.Bytes()...), nil
185}
186
187// BytesCompressed returns the compressed or infinity encoding of p, as
188// specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the
189// point at infinity is shorter than all other encodings.
190func (p *P521Point) BytesCompressed() []byte {
191	// This function is outlined to make the allocations inline in the caller
192	// rather than happen on the heap.
193	var out [1 + p521ElementLength]byte
194	return p.bytesCompressed(&out)
195}
196
197func (p *P521Point) bytesCompressed(out *[1 + p521ElementLength]byte) []byte {
198	if p.z.IsZero() == 1 {
199		return append(out[:0], 0)
200	}
201
202	zinv := new(fiat.P521Element).Invert(p.z)
203	x := new(fiat.P521Element).Mul(p.x, zinv)
204	y := new(fiat.P521Element).Mul(p.y, zinv)
205
206	// Encode the sign of the y coordinate (indicated by the least significant
207	// bit) as the encoding type (2 or 3).
208	buf := append(out[:0], 2)
209	buf[0] |= y.Bytes()[p521ElementLength-1] & 1
210	buf = append(buf, x.Bytes()...)
211	return buf
212}
213
214// Add sets q = p1 + p2, and returns q. The points may overlap.
215func (q *P521Point) Add(p1, p2 *P521Point) *P521Point {
216	// Complete addition formula for a = -3 from "Complete addition formulas for
217	// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
218
219	t0 := new(fiat.P521Element).Mul(p1.x, p2.x)  // t0 := X1 * X2
220	t1 := new(fiat.P521Element).Mul(p1.y, p2.y)  // t1 := Y1 * Y2
221	t2 := new(fiat.P521Element).Mul(p1.z, p2.z)  // t2 := Z1 * Z2
222	t3 := new(fiat.P521Element).Add(p1.x, p1.y)  // t3 := X1 + Y1
223	t4 := new(fiat.P521Element).Add(p2.x, p2.y)  // t4 := X2 + Y2
224	t3.Mul(t3, t4)                               // t3 := t3 * t4
225	t4.Add(t0, t1)                               // t4 := t0 + t1
226	t3.Sub(t3, t4)                               // t3 := t3 - t4
227	t4.Add(p1.y, p1.z)                           // t4 := Y1 + Z1
228	x3 := new(fiat.P521Element).Add(p2.y, p2.z)  // X3 := Y2 + Z2
229	t4.Mul(t4, x3)                               // t4 := t4 * X3
230	x3.Add(t1, t2)                               // X3 := t1 + t2
231	t4.Sub(t4, x3)                               // t4 := t4 - X3
232	x3.Add(p1.x, p1.z)                           // X3 := X1 + Z1
233	y3 := new(fiat.P521Element).Add(p2.x, p2.z)  // Y3 := X2 + Z2
234	x3.Mul(x3, y3)                               // X3 := X3 * Y3
235	y3.Add(t0, t2)                               // Y3 := t0 + t2
236	y3.Sub(x3, y3)                               // Y3 := X3 - Y3
237	z3 := new(fiat.P521Element).Mul(p521B(), t2) // Z3 := b * t2
238	x3.Sub(y3, z3)                               // X3 := Y3 - Z3
239	z3.Add(x3, x3)                               // Z3 := X3 + X3
240	x3.Add(x3, z3)                               // X3 := X3 + Z3
241	z3.Sub(t1, x3)                               // Z3 := t1 - X3
242	x3.Add(t1, x3)                               // X3 := t1 + X3
243	y3.Mul(p521B(), y3)                          // Y3 := b * Y3
244	t1.Add(t2, t2)                               // t1 := t2 + t2
245	t2.Add(t1, t2)                               // t2 := t1 + t2
246	y3.Sub(y3, t2)                               // Y3 := Y3 - t2
247	y3.Sub(y3, t0)                               // Y3 := Y3 - t0
248	t1.Add(y3, y3)                               // t1 := Y3 + Y3
249	y3.Add(t1, y3)                               // Y3 := t1 + Y3
250	t1.Add(t0, t0)                               // t1 := t0 + t0
251	t0.Add(t1, t0)                               // t0 := t1 + t0
252	t0.Sub(t0, t2)                               // t0 := t0 - t2
253	t1.Mul(t4, y3)                               // t1 := t4 * Y3
254	t2.Mul(t0, y3)                               // t2 := t0 * Y3
255	y3.Mul(x3, z3)                               // Y3 := X3 * Z3
256	y3.Add(y3, t2)                               // Y3 := Y3 + t2
257	x3.Mul(t3, x3)                               // X3 := t3 * X3
258	x3.Sub(x3, t1)                               // X3 := X3 - t1
259	z3.Mul(t4, z3)                               // Z3 := t4 * Z3
260	t1.Mul(t3, t0)                               // t1 := t3 * t0
261	z3.Add(z3, t1)                               // Z3 := Z3 + t1
262
263	q.x.Set(x3)
264	q.y.Set(y3)
265	q.z.Set(z3)
266	return q
267}
268
269// Double sets q = p + p, and returns q. The points may overlap.
270func (q *P521Point) Double(p *P521Point) *P521Point {
271	// Complete addition formula for a = -3 from "Complete addition formulas for
272	// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
273
274	t0 := new(fiat.P521Element).Square(p.x)      // t0 := X ^ 2
275	t1 := new(fiat.P521Element).Square(p.y)      // t1 := Y ^ 2
276	t2 := new(fiat.P521Element).Square(p.z)      // t2 := Z ^ 2
277	t3 := new(fiat.P521Element).Mul(p.x, p.y)    // t3 := X * Y
278	t3.Add(t3, t3)                               // t3 := t3 + t3
279	z3 := new(fiat.P521Element).Mul(p.x, p.z)    // Z3 := X * Z
280	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
281	y3 := new(fiat.P521Element).Mul(p521B(), t2) // Y3 := b * t2
282	y3.Sub(y3, z3)                               // Y3 := Y3 - Z3
283	x3 := new(fiat.P521Element).Add(y3, y3)      // X3 := Y3 + Y3
284	y3.Add(x3, y3)                               // Y3 := X3 + Y3
285	x3.Sub(t1, y3)                               // X3 := t1 - Y3
286	y3.Add(t1, y3)                               // Y3 := t1 + Y3
287	y3.Mul(x3, y3)                               // Y3 := X3 * Y3
288	x3.Mul(x3, t3)                               // X3 := X3 * t3
289	t3.Add(t2, t2)                               // t3 := t2 + t2
290	t2.Add(t2, t3)                               // t2 := t2 + t3
291	z3.Mul(p521B(), z3)                          // Z3 := b * Z3
292	z3.Sub(z3, t2)                               // Z3 := Z3 - t2
293	z3.Sub(z3, t0)                               // Z3 := Z3 - t0
294	t3.Add(z3, z3)                               // t3 := Z3 + Z3
295	z3.Add(z3, t3)                               // Z3 := Z3 + t3
296	t3.Add(t0, t0)                               // t3 := t0 + t0
297	t0.Add(t3, t0)                               // t0 := t3 + t0
298	t0.Sub(t0, t2)                               // t0 := t0 - t2
299	t0.Mul(t0, z3)                               // t0 := t0 * Z3
300	y3.Add(y3, t0)                               // Y3 := Y3 + t0
301	t0.Mul(p.y, p.z)                             // t0 := Y * Z
302	t0.Add(t0, t0)                               // t0 := t0 + t0
303	z3.Mul(t0, z3)                               // Z3 := t0 * Z3
304	x3.Sub(x3, z3)                               // X3 := X3 - Z3
305	z3.Mul(t0, t1)                               // Z3 := t0 * t1
306	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
307	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
308
309	q.x.Set(x3)
310	q.y.Set(y3)
311	q.z.Set(z3)
312	return q
313}
314
315// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
316func (q *P521Point) Select(p1, p2 *P521Point, cond int) *P521Point {
317	q.x.Select(p1.x, p2.x, cond)
318	q.y.Select(p1.y, p2.y, cond)
319	q.z.Select(p1.z, p2.z, cond)
320	return q
321}
322
323// A p521Table holds the first 15 multiples of a point at offset -1, so [1]P
324// is at table[0], [15]P is at table[14], and [0]P is implicitly the identity
325// point.
326type p521Table [15]*P521Point
327
328// Select selects the n-th multiple of the table base point into p. It works in
329// constant time by iterating over every entry of the table. n must be in [0, 15].
330func (table *p521Table) Select(p *P521Point, n uint8) {
331	if n >= 16 {
332		panic("nistec: internal error: p521Table called with out-of-bounds value")
333	}
334	p.Set(NewP521Point())
335	for i := uint8(1); i < 16; i++ {
336		cond := subtle.ConstantTimeByteEq(i, n)
337		p.Select(table[i-1], p, cond)
338	}
339}
340
341// ScalarMult sets p = scalar * q, and returns p.
342func (p *P521Point) ScalarMult(q *P521Point, scalar []byte) (*P521Point, error) {
343	// Compute a p521Table for the base point q. The explicit NewP521Point
344	// calls get inlined, letting the allocations live on the stack.
345	var table = p521Table{NewP521Point(), NewP521Point(), NewP521Point(),
346		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
347		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
348		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point()}
349	table[0].Set(q)
350	for i := 1; i < 15; i += 2 {
351		table[i].Double(table[i/2])
352		table[i+1].Add(table[i], q)
353	}
354
355	// Instead of doing the classic double-and-add chain, we do it with a
356	// four-bit window: we double four times, and then add [0-15]P.
357	t := NewP521Point()
358	p.Set(NewP521Point())
359	for i, byte := range scalar {
360		// No need to double on the first iteration, as p is the identity at
361		// this point, and [N]∞ = ∞.
362		if i != 0 {
363			p.Double(p)
364			p.Double(p)
365			p.Double(p)
366			p.Double(p)
367		}
368
369		windowValue := byte >> 4
370		table.Select(t, windowValue)
371		p.Add(p, t)
372
373		p.Double(p)
374		p.Double(p)
375		p.Double(p)
376		p.Double(p)
377
378		windowValue = byte & 0b1111
379		table.Select(t, windowValue)
380		p.Add(p, t)
381	}
382
383	return p, nil
384}
385
386var p521GeneratorTable *[p521ElementLength * 2]p521Table
387var p521GeneratorTableOnce sync.Once
388
389// generatorTable returns a sequence of p521Tables. The first table contains
390// multiples of G. Each successive table is the previous table doubled four
391// times.
392func (p *P521Point) generatorTable() *[p521ElementLength * 2]p521Table {
393	p521GeneratorTableOnce.Do(func() {
394		p521GeneratorTable = new([p521ElementLength * 2]p521Table)
395		base := NewP521Point().SetGenerator()
396		for i := 0; i < p521ElementLength*2; i++ {
397			p521GeneratorTable[i][0] = NewP521Point().Set(base)
398			for j := 1; j < 15; j++ {
399				p521GeneratorTable[i][j] = NewP521Point().Add(p521GeneratorTable[i][j-1], base)
400			}
401			base.Double(base)
402			base.Double(base)
403			base.Double(base)
404			base.Double(base)
405		}
406	})
407	return p521GeneratorTable
408}
409
410// ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and
411// returns p.
412func (p *P521Point) ScalarBaseMult(scalar []byte) (*P521Point, error) {
413	if len(scalar) != p521ElementLength {
414		return nil, errors.New("invalid scalar length")
415	}
416	tables := p.generatorTable()
417
418	// This is also a scalar multiplication with a four-bit window like in
419	// ScalarMult, but in this case the doublings are precomputed. The value
420	// [windowValue]G added at iteration k would normally get doubled
421	// (totIterations-k)×4 times, but with a larger precomputation we can
422	// instead add [2^((totIterations-k)×4)][windowValue]G and avoid the
423	// doublings between iterations.
424	t := NewP521Point()
425	p.Set(NewP521Point())
426	tableIndex := len(tables) - 1
427	for _, byte := range scalar {
428		windowValue := byte >> 4
429		tables[tableIndex].Select(t, windowValue)
430		p.Add(p, t)
431		tableIndex--
432
433		windowValue = byte & 0b1111
434		tables[tableIndex].Select(t, windowValue)
435		p.Add(p, t)
436		tableIndex--
437	}
438
439	return p, nil
440}
441
442// p521Sqrt sets e to a square root of x. If x is not a square, p521Sqrt returns
443// false and e is unchanged. e and x can overlap.
444func p521Sqrt(e, x *fiat.P521Element) (isSquare bool) {
445	candidate := new(fiat.P521Element)
446	p521SqrtCandidate(candidate, x)
447	square := new(fiat.P521Element).Square(candidate)
448	if square.Equal(x) != 1 {
449		return false
450	}
451	e.Set(candidate)
452	return true
453}
454
455// p521SqrtCandidate sets z to a square root candidate for x. z and x must not overlap.
456func p521SqrtCandidate(z, x *fiat.P521Element) {
457	// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
458	//
459	// The sequence of 0 multiplications and 519 squarings is derived from the
460	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
461	//
462	//	return  1 << 519
463	//
464
465	z.Square(x)
466	for s := 1; s < 519; s++ {
467		z.Square(z)
468	}
469}
470