1// Copyright (c) 2012 The Chromium Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style license that can be
3// found in the LICENSE file.
4
5// This is an implementation of the P224 elliptic curve group. It's written to
6// be short and simple rather than fast, although it's still constant-time.
7//
8// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background.
9
10#include "crypto/p224.h"
11
12#include <stddef.h>
13#include <stdint.h>
14#include <string.h>
15
16#include "base/sys_byteorder.h"
17
18namespace {
19
20using base::HostToNet32;
21using base::NetToHost32;
22
23// Field element functions.
24//
25// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1.
26//
27// Field elements are represented by a FieldElement, which is a typedef to an
28// array of 8 uint32_t's. The value of a FieldElement, a, is:
29//   a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7]
30//
31// Using 28-bit limbs means that there's only 4 bits of headroom, which is less
32// than we would really like. But it has the useful feature that we hit 2**224
33// exactly, making the reflections during a reduce much nicer.
34
35using crypto::p224::FieldElement;
36
37// kP is the P224 prime.
38const FieldElement kP = {
39  1, 0, 0, 268431360,
40  268435455, 268435455, 268435455, 268435455,
41};
42
43void Contract(FieldElement* inout);
44
45// IsZero returns 0xffffffff if a == 0 mod p and 0 otherwise.
46uint32_t IsZero(const FieldElement& a) {
47  FieldElement minimal;
48  memcpy(&minimal, &a, sizeof(minimal));
49  Contract(&minimal);
50
51  uint32_t is_zero = 0, is_p = 0;
52  for (unsigned i = 0; i < 8; i++) {
53    is_zero |= minimal[i];
54    is_p |= minimal[i] - kP[i];
55  }
56
57  // If either is_zero or is_p is 0, then we should return 1.
58  is_zero |= is_zero >> 16;
59  is_zero |= is_zero >> 8;
60  is_zero |= is_zero >> 4;
61  is_zero |= is_zero >> 2;
62  is_zero |= is_zero >> 1;
63
64  is_p |= is_p >> 16;
65  is_p |= is_p >> 8;
66  is_p |= is_p >> 4;
67  is_p |= is_p >> 2;
68  is_p |= is_p >> 1;
69
70  // For is_zero and is_p, the LSB is 0 iff all the bits are zero.
71  is_zero &= is_p & 1;
72  is_zero = (~is_zero) << 31;
73  is_zero = static_cast<int32_t>(is_zero) >> 31;
74  return is_zero;
75}
76
77// Add computes *out = a+b
78//
79// a[i] + b[i] < 2**32
80void Add(FieldElement* out, const FieldElement& a, const FieldElement& b) {
81  for (int i = 0; i < 8; i++) {
82    (*out)[i] = a[i] + b[i];
83  }
84}
85
86static const uint32_t kTwo31p3 = (1u << 31) + (1u << 3);
87static const uint32_t kTwo31m3 = (1u << 31) - (1u << 3);
88static const uint32_t kTwo31m15m3 = (1u << 31) - (1u << 15) - (1u << 3);
89// kZero31ModP is 0 mod p where bit 31 is set in all limbs so that we can
90// subtract smaller amounts without underflow. See the section "Subtraction" in
91// [1] for why.
92static const FieldElement kZero31ModP = {
93  kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3,
94  kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3
95};
96
97// Subtract computes *out = a-b
98//
99// a[i], b[i] < 2**30
100// out[i] < 2**32
101void Subtract(FieldElement* out, const FieldElement& a, const FieldElement& b) {
102  for (int i = 0; i < 8; i++) {
103    // See the section on "Subtraction" in [1] for details.
104    (*out)[i] = a[i] + kZero31ModP[i] - b[i];
105  }
106}
107
108static const uint64_t kTwo63p35 = (1ull << 63) + (1ull << 35);
109static const uint64_t kTwo63m35 = (1ull << 63) - (1ull << 35);
110static const uint64_t kTwo63m35m19 = (1ull << 63) - (1ull << 35) - (1ull << 19);
111// kZero63ModP is 0 mod p where bit 63 is set in all limbs. See the section
112// "Subtraction" in [1] for why.
113static const uint64_t kZero63ModP[8] = {
114    kTwo63p35,    kTwo63m35, kTwo63m35, kTwo63m35,
115    kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35,
116};
117
118static const uint32_t kBottom28Bits = 0xfffffff;
119
120// LargeFieldElement also represents an element of the field. The limbs are
121// still spaced 28-bits apart and in little-endian order. So the limbs are at
122// 0, 28, 56, ..., 392 bits, each 64-bits wide.
123typedef uint64_t LargeFieldElement[15];
124
125// ReduceLarge converts a LargeFieldElement to a FieldElement.
126//
127// in[i] < 2**62
128void ReduceLarge(FieldElement* out, LargeFieldElement* inptr) {
129  LargeFieldElement& in(*inptr);
130
131  for (int i = 0; i < 8; i++) {
132    in[i] += kZero63ModP[i];
133  }
134
135  // Eliminate the coefficients at 2**224 and greater while maintaining the
136  // same value mod p.
137  for (int i = 14; i >= 8; i--) {
138    in[i-8] -= in[i];  // reflection off the "+1" term of p.
139    in[i-5] += (in[i] & 0xffff) << 12;  // part of the "-2**96" reflection.
140    in[i-4] += in[i] >> 16;  // the rest of the "-2**96" reflection.
141  }
142  in[8] = 0;
143  // in[0..8] < 2**64
144
145  // As the values become small enough, we start to store them in |out| and use
146  // 32-bit operations.
147  for (int i = 1; i < 8; i++) {
148    in[i+1] += in[i] >> 28;
149    (*out)[i] = static_cast<uint32_t>(in[i] & kBottom28Bits);
150  }
151  // Eliminate the term at 2*224 that we introduced while keeping the same
152  // value mod p.
153  in[0] -= in[8];  // reflection off the "+1" term of p.
154  (*out)[3] += static_cast<uint32_t>(in[8] & 0xffff) << 12;  // "-2**96" term
155  (*out)[4] += static_cast<uint32_t>(in[8] >> 16);  // rest of "-2**96" term
156  // in[0] < 2**64
157  // out[3] < 2**29
158  // out[4] < 2**29
159  // out[1,2,5..7] < 2**28
160
161  (*out)[0] = static_cast<uint32_t>(in[0] & kBottom28Bits);
162  (*out)[1] += static_cast<uint32_t>((in[0] >> 28) & kBottom28Bits);
163  (*out)[2] += static_cast<uint32_t>(in[0] >> 56);
164  // out[0] < 2**28
165  // out[1..4] < 2**29
166  // out[5..7] < 2**28
167}
168
169// Mul computes *out = a*b
170//
171// a[i] < 2**29, b[i] < 2**30 (or vice versa)
172// out[i] < 2**29
173void Mul(FieldElement* out, const FieldElement& a, const FieldElement& b) {
174  LargeFieldElement tmp;
175  memset(&tmp, 0, sizeof(tmp));
176
177  for (int i = 0; i < 8; i++) {
178    for (int j = 0; j < 8; j++) {
179      tmp[i + j] += static_cast<uint64_t>(a[i]) * static_cast<uint64_t>(b[j]);
180    }
181  }
182
183  ReduceLarge(out, &tmp);
184}
185
186// Square computes *out = a*a
187//
188// a[i] < 2**29
189// out[i] < 2**29
190void Square(FieldElement* out, const FieldElement& a) {
191  LargeFieldElement tmp;
192  memset(&tmp, 0, sizeof(tmp));
193
194  for (int i = 0; i < 8; i++) {
195    for (int j = 0; j <= i; j++) {
196      uint64_t r = static_cast<uint64_t>(a[i]) * static_cast<uint64_t>(a[j]);
197      if (i == j) {
198        tmp[i+j] += r;
199      } else {
200        tmp[i+j] += r << 1;
201      }
202    }
203  }
204
205  ReduceLarge(out, &tmp);
206}
207
208// Reduce reduces the coefficients of in_out to smaller bounds.
209//
210// On entry: a[i] < 2**31 + 2**30
211// On exit: a[i] < 2**29
212void Reduce(FieldElement* in_out) {
213  FieldElement& a = *in_out;
214
215  for (int i = 0; i < 7; i++) {
216    a[i+1] += a[i] >> 28;
217    a[i] &= kBottom28Bits;
218  }
219  uint32_t top = a[7] >> 28;
220  a[7] &= kBottom28Bits;
221
222  // top < 2**4
223  // Constant-time: mask = (top != 0) ? 0xffffffff : 0
224  uint32_t mask = top;
225  mask |= mask >> 2;
226  mask |= mask >> 1;
227  mask <<= 31;
228  mask = static_cast<uint32_t>(static_cast<int32_t>(mask) >> 31);
229
230  // Eliminate top while maintaining the same value mod p.
231  a[0] -= top;
232  a[3] += top << 12;
233
234  // We may have just made a[0] negative but, if we did, then we must
235  // have added something to a[3], thus it's > 2**12. Therefore we can
236  // carry down to a[0].
237  a[3] -= 1 & mask;
238  a[2] += mask & ((1<<28) - 1);
239  a[1] += mask & ((1<<28) - 1);
240  a[0] += mask & (1<<28);
241}
242
243// Invert calcuates *out = in**-1 by computing in**(2**224 - 2**96 - 1), i.e.
244// Fermat's little theorem.
245void Invert(FieldElement* out, const FieldElement& in) {
246  FieldElement f1, f2, f3, f4;
247
248  Square(&f1, in);                        // 2
249  Mul(&f1, f1, in);                       // 2**2 - 1
250  Square(&f1, f1);                        // 2**3 - 2
251  Mul(&f1, f1, in);                       // 2**3 - 1
252  Square(&f2, f1);                        // 2**4 - 2
253  Square(&f2, f2);                        // 2**5 - 4
254  Square(&f2, f2);                        // 2**6 - 8
255  Mul(&f1, f1, f2);                       // 2**6 - 1
256  Square(&f2, f1);                        // 2**7 - 2
257  for (int i = 0; i < 5; i++) {           // 2**12 - 2**6
258    Square(&f2, f2);
259  }
260  Mul(&f2, f2, f1);                       // 2**12 - 1
261  Square(&f3, f2);                        // 2**13 - 2
262  for (int i = 0; i < 11; i++) {          // 2**24 - 2**12
263    Square(&f3, f3);
264  }
265  Mul(&f2, f3, f2);                       // 2**24 - 1
266  Square(&f3, f2);                        // 2**25 - 2
267  for (int i = 0; i < 23; i++) {          // 2**48 - 2**24
268    Square(&f3, f3);
269  }
270  Mul(&f3, f3, f2);                       // 2**48 - 1
271  Square(&f4, f3);                        // 2**49 - 2
272  for (int i = 0; i < 47; i++) {          // 2**96 - 2**48
273    Square(&f4, f4);
274  }
275  Mul(&f3, f3, f4);                       // 2**96 - 1
276  Square(&f4, f3);                        // 2**97 - 2
277  for (int i = 0; i < 23; i++) {          // 2**120 - 2**24
278    Square(&f4, f4);
279  }
280  Mul(&f2, f4, f2);                       // 2**120 - 1
281  for (int i = 0; i < 6; i++) {           // 2**126 - 2**6
282    Square(&f2, f2);
283  }
284  Mul(&f1, f1, f2);                       // 2**126 - 1
285  Square(&f1, f1);                        // 2**127 - 2
286  Mul(&f1, f1, in);                       // 2**127 - 1
287  for (int i = 0; i < 97; i++) {          // 2**224 - 2**97
288    Square(&f1, f1);
289  }
290  Mul(out, f1, f3);                       // 2**224 - 2**96 - 1
291}
292
293// Contract converts a FieldElement to its minimal, distinguished form.
294//
295// On entry, in[i] < 2**29
296// On exit, in[i] < 2**28
297void Contract(FieldElement* inout) {
298  FieldElement& out = *inout;
299
300  // Reduce the coefficients to < 2**28.
301  for (int i = 0; i < 7; i++) {
302    out[i+1] += out[i] >> 28;
303    out[i] &= kBottom28Bits;
304  }
305  uint32_t top = out[7] >> 28;
306  out[7] &= kBottom28Bits;
307
308  // Eliminate top while maintaining the same value mod p.
309  out[0] -= top;
310  out[3] += top << 12;
311
312  // We may just have made out[0] negative. So we carry down. If we made
313  // out[0] negative then we know that out[3] is sufficiently positive
314  // because we just added to it.
315  for (int i = 0; i < 3; i++) {
316    uint32_t mask = static_cast<uint32_t>(static_cast<int32_t>(out[i]) >> 31);
317    out[i] += (1 << 28) & mask;
318    out[i+1] -= 1 & mask;
319  }
320
321  // We might have pushed out[3] over 2**28 so we perform another, partial
322  // carry chain.
323  for (int i = 3; i < 7; i++) {
324    out[i+1] += out[i] >> 28;
325    out[i] &= kBottom28Bits;
326  }
327  top = out[7] >> 28;
328  out[7] &= kBottom28Bits;
329
330  // Eliminate top while maintaining the same value mod p.
331  out[0] -= top;
332  out[3] += top << 12;
333
334  // There are two cases to consider for out[3]:
335  //   1) The first time that we eliminated top, we didn't push out[3] over
336  //      2**28. In this case, the partial carry chain didn't change any values
337  //      and top is zero.
338  //   2) We did push out[3] over 2**28 the first time that we eliminated top.
339  //      The first value of top was in [0..16), therefore, prior to eliminating
340  //      the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after
341  //      overflowing and being reduced by the second carry chain, out[3] <=
342  //      0xf000. Thus it cannot have overflowed when we eliminated top for the
343  //      second time.
344
345  // Again, we may just have made out[0] negative, so do the same carry down.
346  // As before, if we made out[0] negative then we know that out[3] is
347  // sufficiently positive.
348  for (int i = 0; i < 3; i++) {
349    uint32_t mask = static_cast<uint32_t>(static_cast<int32_t>(out[i]) >> 31);
350    out[i] += (1 << 28) & mask;
351    out[i+1] -= 1 & mask;
352  }
353
354  // The value is < 2**224, but maybe greater than p. In order to reduce to a
355  // unique, minimal value we see if the value is >= p and, if so, subtract p.
356
357  // First we build a mask from the top four limbs, which must all be
358  // equal to bottom28Bits if the whole value is >= p. If top_4_all_ones
359  // ends up with any zero bits in the bottom 28 bits, then this wasn't
360  // true.
361  uint32_t top_4_all_ones = 0xffffffffu;
362  for (int i = 4; i < 8; i++) {
363    top_4_all_ones &= out[i];
364  }
365  top_4_all_ones |= 0xf0000000;
366  // Now we replicate any zero bits to all the bits in top_4_all_ones.
367  top_4_all_ones &= top_4_all_ones >> 16;
368  top_4_all_ones &= top_4_all_ones >> 8;
369  top_4_all_ones &= top_4_all_ones >> 4;
370  top_4_all_ones &= top_4_all_ones >> 2;
371  top_4_all_ones &= top_4_all_ones >> 1;
372  top_4_all_ones =
373      static_cast<uint32_t>(static_cast<int32_t>(top_4_all_ones << 31) >> 31);
374
375  // Now we test whether the bottom three limbs are non-zero.
376  uint32_t bottom_3_non_zero = out[0] | out[1] | out[2];
377  bottom_3_non_zero |= bottom_3_non_zero >> 16;
378  bottom_3_non_zero |= bottom_3_non_zero >> 8;
379  bottom_3_non_zero |= bottom_3_non_zero >> 4;
380  bottom_3_non_zero |= bottom_3_non_zero >> 2;
381  bottom_3_non_zero |= bottom_3_non_zero >> 1;
382  bottom_3_non_zero =
383      static_cast<uint32_t>(static_cast<int32_t>(bottom_3_non_zero) >> 31);
384
385  // Everything depends on the value of out[3].
386  //    If it's > 0xffff000 and top_4_all_ones != 0 then the whole value is >= p
387  //    If it's = 0xffff000 and top_4_all_ones != 0 and bottom_3_non_zero != 0,
388  //      then the whole value is >= p
389  //    If it's < 0xffff000, then the whole value is < p
390  uint32_t n = out[3] - 0xffff000;
391  uint32_t out_3_equal = n;
392  out_3_equal |= out_3_equal >> 16;
393  out_3_equal |= out_3_equal >> 8;
394  out_3_equal |= out_3_equal >> 4;
395  out_3_equal |= out_3_equal >> 2;
396  out_3_equal |= out_3_equal >> 1;
397  out_3_equal =
398      ~static_cast<uint32_t>(static_cast<int32_t>(out_3_equal << 31) >> 31);
399
400  // If out[3] > 0xffff000 then n's MSB will be zero.
401  uint32_t out_3_gt =
402      ~static_cast<uint32_t>(static_cast<int32_t>(n << 31) >> 31);
403
404  uint32_t mask =
405      top_4_all_ones & ((out_3_equal & bottom_3_non_zero) | out_3_gt);
406  out[0] -= 1 & mask;
407  out[3] -= 0xffff000 & mask;
408  out[4] -= 0xfffffff & mask;
409  out[5] -= 0xfffffff & mask;
410  out[6] -= 0xfffffff & mask;
411  out[7] -= 0xfffffff & mask;
412}
413
414
415// Group element functions.
416//
417// These functions deal with group elements. The group is an elliptic curve
418// group with a = -3 defined in FIPS 186-3, section D.2.2.
419
420using crypto::p224::Point;
421
422// kB is parameter of the elliptic curve.
423const FieldElement kB = {
424  55967668, 11768882, 265861671, 185302395,
425  39211076, 180311059, 84673715, 188764328,
426};
427
428void CopyConditional(Point* out, const Point& a, uint32_t mask);
429void DoubleJacobian(Point* out, const Point& a);
430
431// AddJacobian computes *out = a+b where a != b.
432void AddJacobian(Point *out,
433                 const Point& a,
434                 const Point& b) {
435  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
436  FieldElement z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v;
437
438  uint32_t z1_is_zero = IsZero(a.z);
439  uint32_t z2_is_zero = IsZero(b.z);
440
441  // Z1Z1 = Z1²
442  Square(&z1z1, a.z);
443
444  // Z2Z2 = Z2²
445  Square(&z2z2, b.z);
446
447  // U1 = X1*Z2Z2
448  Mul(&u1, a.x, z2z2);
449
450  // U2 = X2*Z1Z1
451  Mul(&u2, b.x, z1z1);
452
453  // S1 = Y1*Z2*Z2Z2
454  Mul(&s1, b.z, z2z2);
455  Mul(&s1, a.y, s1);
456
457  // S2 = Y2*Z1*Z1Z1
458  Mul(&s2, a.z, z1z1);
459  Mul(&s2, b.y, s2);
460
461  // H = U2-U1
462  Subtract(&h, u2, u1);
463  Reduce(&h);
464  uint32_t x_equal = IsZero(h);
465
466  // I = (2*H)²
467  for (int k = 0; k < 8; k++) {
468    i[k] = h[k] << 1;
469  }
470  Reduce(&i);
471  Square(&i, i);
472
473  // J = H*I
474  Mul(&j, h, i);
475  // r = 2*(S2-S1)
476  Subtract(&r, s2, s1);
477  Reduce(&r);
478  uint32_t y_equal = IsZero(r);
479
480  if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
481    // The two input points are the same therefore we must use the dedicated
482    // doubling function as the slope of the line is undefined.
483    DoubleJacobian(out, a);
484    return;
485  }
486
487  for (int k = 0; k < 8; k++) {
488    r[k] <<= 1;
489  }
490  Reduce(&r);
491
492  // V = U1*I
493  Mul(&v, u1, i);
494
495  // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H
496  Add(&z1z1, z1z1, z2z2);
497  Add(&z2z2, a.z, b.z);
498  Reduce(&z2z2);
499  Square(&z2z2, z2z2);
500  Subtract(&out->z, z2z2, z1z1);
501  Reduce(&out->z);
502  Mul(&out->z, out->z, h);
503
504  // X3 = r²-J-2*V
505  for (int k = 0; k < 8; k++) {
506    z1z1[k] = v[k] << 1;
507  }
508  Add(&z1z1, j, z1z1);
509  Reduce(&z1z1);
510  Square(&out->x, r);
511  Subtract(&out->x, out->x, z1z1);
512  Reduce(&out->x);
513
514  // Y3 = r*(V-X3)-2*S1*J
515  for (int k = 0; k < 8; k++) {
516    s1[k] <<= 1;
517  }
518  Mul(&s1, s1, j);
519  Subtract(&z1z1, v, out->x);
520  Reduce(&z1z1);
521  Mul(&z1z1, z1z1, r);
522  Subtract(&out->y, z1z1, s1);
523  Reduce(&out->y);
524
525  CopyConditional(out, a, z2_is_zero);
526  CopyConditional(out, b, z1_is_zero);
527}
528
529// DoubleJacobian computes *out = a+a.
530void DoubleJacobian(Point* out, const Point& a) {
531  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
532  FieldElement delta, gamma, beta, alpha, t;
533
534  Square(&delta, a.z);
535  Square(&gamma, a.y);
536  Mul(&beta, a.x, gamma);
537
538  // alpha = 3*(X1-delta)*(X1+delta)
539  Add(&t, a.x, delta);
540  for (int i = 0; i < 8; i++) {
541          t[i] += t[i] << 1;
542  }
543  Reduce(&t);
544  Subtract(&alpha, a.x, delta);
545  Reduce(&alpha);
546  Mul(&alpha, alpha, t);
547
548  // Z3 = (Y1+Z1)²-gamma-delta
549  Add(&out->z, a.y, a.z);
550  Reduce(&out->z);
551  Square(&out->z, out->z);
552  Subtract(&out->z, out->z, gamma);
553  Reduce(&out->z);
554  Subtract(&out->z, out->z, delta);
555  Reduce(&out->z);
556
557  // X3 = alpha²-8*beta
558  for (int i = 0; i < 8; i++) {
559          delta[i] = beta[i] << 3;
560  }
561  Reduce(&delta);
562  Square(&out->x, alpha);
563  Subtract(&out->x, out->x, delta);
564  Reduce(&out->x);
565
566  // Y3 = alpha*(4*beta-X3)-8*gamma²
567  for (int i = 0; i < 8; i++) {
568          beta[i] <<= 2;
569  }
570  Reduce(&beta);
571  Subtract(&beta, beta, out->x);
572  Reduce(&beta);
573  Square(&gamma, gamma);
574  for (int i = 0; i < 8; i++) {
575          gamma[i] <<= 3;
576  }
577  Reduce(&gamma);
578  Mul(&out->y, alpha, beta);
579  Subtract(&out->y, out->y, gamma);
580  Reduce(&out->y);
581}
582
583// CopyConditional sets *out=a if mask is 0xffffffff. mask must be either 0 of
584// 0xffffffff.
585void CopyConditional(Point* out, const Point& a, uint32_t mask) {
586  for (int i = 0; i < 8; i++) {
587    out->x[i] ^= mask & (a.x[i] ^ out->x[i]);
588    out->y[i] ^= mask & (a.y[i] ^ out->y[i]);
589    out->z[i] ^= mask & (a.z[i] ^ out->z[i]);
590  }
591}
592
593// ScalarMult calculates *out = a*scalar where scalar is a big-endian number of
594// length scalar_len and != 0.
595void ScalarMult(Point* out,
596                const Point& a,
597                const uint8_t* scalar,
598                size_t scalar_len) {
599  memset(out, 0, sizeof(*out));
600  Point tmp;
601
602  for (size_t i = 0; i < scalar_len; i++) {
603    for (unsigned int bit_num = 0; bit_num < 8; bit_num++) {
604      DoubleJacobian(out, *out);
605      uint32_t bit = static_cast<uint32_t>(static_cast<int32_t>(
606          (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31));
607      AddJacobian(&tmp, a, *out);
608      CopyConditional(out, tmp, bit);
609    }
610  }
611}
612
613// Get224Bits reads 7 words from in and scatters their contents in
614// little-endian form into 8 words at out, 28 bits per output word.
615void Get224Bits(uint32_t* out, const uint32_t* in) {
616  out[0] = NetToHost32(in[6]) & kBottom28Bits;
617  out[1] = ((NetToHost32(in[5]) << 4) |
618            (NetToHost32(in[6]) >> 28)) & kBottom28Bits;
619  out[2] = ((NetToHost32(in[4]) << 8) |
620            (NetToHost32(in[5]) >> 24)) & kBottom28Bits;
621  out[3] = ((NetToHost32(in[3]) << 12) |
622            (NetToHost32(in[4]) >> 20)) & kBottom28Bits;
623  out[4] = ((NetToHost32(in[2]) << 16) |
624            (NetToHost32(in[3]) >> 16)) & kBottom28Bits;
625  out[5] = ((NetToHost32(in[1]) << 20) |
626            (NetToHost32(in[2]) >> 12)) & kBottom28Bits;
627  out[6] = ((NetToHost32(in[0]) << 24) |
628            (NetToHost32(in[1]) >> 8)) & kBottom28Bits;
629  out[7] = (NetToHost32(in[0]) >> 4) & kBottom28Bits;
630}
631
632// Put224Bits performs the inverse operation to Get224Bits: taking 28 bits from
633// each of 8 input words and writing them in big-endian order to 7 words at
634// out.
635void Put224Bits(uint32_t* out, const uint32_t* in) {
636  out[6] = HostToNet32((in[0] >> 0) | (in[1] << 28));
637  out[5] = HostToNet32((in[1] >> 4) | (in[2] << 24));
638  out[4] = HostToNet32((in[2] >> 8) | (in[3] << 20));
639  out[3] = HostToNet32((in[3] >> 12) | (in[4] << 16));
640  out[2] = HostToNet32((in[4] >> 16) | (in[5] << 12));
641  out[1] = HostToNet32((in[5] >> 20) | (in[6] << 8));
642  out[0] = HostToNet32((in[6] >> 24) | (in[7] << 4));
643}
644
645}  // anonymous namespace
646
647namespace crypto {
648
649namespace p224 {
650
651bool Point::SetFromString(const base::StringPiece& in) {
652  if (in.size() != 2*28)
653    return false;
654  const uint32_t* inwords = reinterpret_cast<const uint32_t*>(in.data());
655  Get224Bits(x, inwords);
656  Get224Bits(y, inwords + 7);
657  memset(&z, 0, sizeof(z));
658  z[0] = 1;
659
660  // Check that the point is on the curve, i.e. that y² = x³ - 3x + b.
661  FieldElement lhs;
662  Square(&lhs, y);
663  Contract(&lhs);
664
665  FieldElement rhs;
666  Square(&rhs, x);
667  Mul(&rhs, x, rhs);
668
669  FieldElement three_x;
670  for (int i = 0; i < 8; i++) {
671    three_x[i] = x[i] * 3;
672  }
673  Reduce(&three_x);
674  Subtract(&rhs, rhs, three_x);
675  Reduce(&rhs);
676
677  ::Add(&rhs, rhs, kB);
678  Contract(&rhs);
679  return memcmp(&lhs, &rhs, sizeof(lhs)) == 0;
680}
681
682std::string Point::ToString() const {
683  FieldElement zinv, zinv_sq, xx, yy;
684
685  // If this is the point at infinity we return a string of all zeros.
686  if (IsZero(this->z)) {
687    static const char zeros[56] = {0};
688    return std::string(zeros, sizeof(zeros));
689  }
690
691  Invert(&zinv, this->z);
692  Square(&zinv_sq, zinv);
693  Mul(&xx, x, zinv_sq);
694  Mul(&zinv_sq, zinv_sq, zinv);
695  Mul(&yy, y, zinv_sq);
696
697  Contract(&xx);
698  Contract(&yy);
699
700  uint32_t outwords[14];
701  Put224Bits(outwords, xx);
702  Put224Bits(outwords + 7, yy);
703  return std::string(reinterpret_cast<const char*>(outwords), sizeof(outwords));
704}
705
706void ScalarMult(const Point& in, const uint8_t* scalar, Point* out) {
707  ::ScalarMult(out, in, scalar, 28);
708}
709
710// kBasePoint is the base point (generator) of the elliptic curve group.
711static const Point kBasePoint = {
712  {22813985, 52956513, 34677300, 203240812,
713   12143107, 133374265, 225162431, 191946955},
714  {83918388, 223877528, 122119236, 123340192,
715   266784067, 263504429, 146143011, 198407736},
716  {1, 0, 0, 0, 0, 0, 0, 0},
717};
718
719void ScalarBaseMult(const uint8_t* scalar, Point* out) {
720  ::ScalarMult(out, kBasePoint, scalar, 28);
721}
722
723void Add(const Point& a, const Point& b, Point* out) {
724  AddJacobian(out, a, b);
725}
726
727void Negate(const Point& in, Point* out) {
728  // Guide to elliptic curve cryptography, page 89 suggests that (X : X+Y : Z)
729  // is the negative in Jacobian coordinates, but it doesn't actually appear to
730  // be true in testing so this performs the negation in affine coordinates.
731  FieldElement zinv, zinv_sq, y;
732  Invert(&zinv, in.z);
733  Square(&zinv_sq, zinv);
734  Mul(&out->x, in.x, zinv_sq);
735  Mul(&zinv_sq, zinv_sq, zinv);
736  Mul(&y, in.y, zinv_sq);
737
738  Subtract(&out->y, kP, y);
739  Reduce(&out->y);
740
741  memset(&out->z, 0, sizeof(out->z));
742  out->z[0] = 1;
743}
744
745}  // namespace p224
746
747}  // namespace crypto
748