crypto: add simple P224 implementation.

crypto/p224.cc

+// Copyright (c) 2011 The Chromium Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+// This is an implementation of the P224 elliptic curve group. It's written to
+// be short and simple rather than fast, although it's still constant-time.
+
+#include <string.h>
+#include <arpa/inet.h>
+
+#include "crypto/p224.h"
+
+namespace {
+
+// Field element functions.
+//
+// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1.
+//
+// Field elements are represented by a FieldElement, which is a typedef to an
+// array of 8 uint32's. The value of a FieldElement, a, is:
+//   a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7]
+//
+// Using 28-bit limbs means that there's only 4 bits of headroom, which is less
+// than we would really like. But it has the useful feature that we hit 2**224
+// exactly, making the reflections during a reduce much nicer.
+
+typedef crypto::P224::FieldElement FieldElement;

[Wez, 2011/11/02 23:46:06]

nit: Why not just "using crypto::P224::FieldElement"?

[agl, 2011/11/03 17:20:49]

Doesn't work I'm afraid:
../../crypto/p224.cc:35: error: 'crypto::P224' is not a namespace

+
+// Add computes *out = a+b
+//
+// Bounds on a and b are such that the sum of each corresponding limb of a and
+// b mustn't exceed 2**32.

[Wez, 2011/11/02 23:46:06]

Express this similarly to the comment on Sub(), for consistency.

[agl, 2011/11/03 17:20:49]

Done.

+void Add(FieldElement* out, const FieldElement& a, const FieldElement& b) {
+  for (int i = 0; i < 8; i++) {
+    (*out)[i] = a[i] + b[i];
+  }
+}
+
+static const uint32 kTwo31p3 = (1u<<31) + (1u<<3);
+static const uint32 kTwo31m3 = (1u<<31) - (1u<<3);
+static const uint32 kTwo31m15m3 = (1u<<31) - (1u<<15) - (1u<<3);
+// kZero31ModP is 0 mod p where bit 31 is set in all limbs.
+static const FieldElement kZero31ModP = {
+  kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3,
+  kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3

[Wez, 2011/11/02 23:46:06]

It would be great to clarify with _why_ we want to have bit 32 set in all limbs,
and why the mod p causes these slightly strange-looking "reflection" terms.

[agl, 2011/11/03 17:20:49]

That could go on for a while so I've cited hthe subtraction section of
http://www.imperialviolet.org/2010/12/04/ecc.html

[Wez, 2011/11/03 20:03:46]

What I had in mind was a sentence explaining that setting bit 31 ensures that we
can subtract a 28-bit limb without underflow, and that because we're working mod
p, we can express zero as a multiple of p, chosen to have that property, but
perhaps I'm over-simplifying there.  Your reference sounds good!

[agl, 2011/11/04 20:13:22]

Have updated the comment.

+};
+
+// Sub computes *out = a-b
+//
+// a[i], b[i] < 2**30
+// out[i] < 2**32
+void Sub(FieldElement* out, const FieldElement& a, const FieldElement& b) {

[Wez, 2011/11/02 23:46:06]

nit: Sub -> Subtract, according to the style guide.

[agl, 2011/11/03 17:20:49]

Done.

+  for (int i = 0; i < 8; i++) {
+    (*out)[i] = a[i] + kZero31ModP[i] - b[i];

[Wez, 2011/11/02 23:46:06]

It looks like the "zero" constant here is being used to handle underflow
gracefully, in the expectation that the result will be subsequently be reduced
back to a value meeting the input criteria, so I think there should be a comment
either on this function, or perhaps along with the introductory text about field
element functions, to explain that property.

[agl, 2011/11/03 17:20:49]

Have also cited the same section in ecc.html.

+  }
+}
+
+static const uint64 kTwo63p35 = (1ull<<63) + (1ull<<35);
+static const uint64 kTwo63m35 = (1ull<<63) - (1ull<<35);
+static const uint64 kTwo63m35m19 = (1ull<<63) - (1ull<<35) - (1ull<<19);
+// kZero31ModP is 0 mod p where bit 63 is set in all limbs.

[Wez, 2011/11/02 23:46:06]

kZero31ModP -> kZero63ModP

[agl, 2011/11/03 17:20:49]

Done.

+static const uint64 kZero63ModP[8] = {
+  kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35,
+  kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35,
+};
+
+static const uint32 kBottom12Bits = 0xfff;

[Wez, 2011/11/02 23:46:06]

You define this but never use it?

[agl, 2011/11/03 17:20:49]

Removed.

+static const uint32 kBottom28Bits = 0xfffffff;
+
+// LargeFieldElement also represents an element of the field. The limbs are
+// still spaced 28-bits apart and in little-endian order.

[Wez, 2011/11/02 23:46:06]

Since LargeFieldElement still represents 28-bits with each limb, but has twice
as many limbs, for coefficients up to 448-bit, in effect?

nit: I first mis-interpreted this as meaning that the limbs were packed in
28-bit chunks into this array, which was confusing (and wrong).  Is there any
clearer wording you can think of?

[agl, 2011/11/03 17:20:49]

It only has coefficients up to 392 bits, but it's 64-bits wide so it stretches
to 448, yes.
I've tried: "So the limbs are at 0, 28, 56, ..., 392 bits, each 64-bits wide."

[Wez, 2011/11/03 20:03:46]

Would it be correct to say that each limb "represents" 28-bits, despite now
being 64-bits?  If so then I think that would be clear enough.

[agl, 2011/11/04 20:13:22]

"represents"? Not really I'm afraid.

+typedef uint64 LargeFieldElement[15];
+
+// ReduceLarge converts a LargeFieldElement to a FieldElement.
+//
+// in[i] < 2**62
+void ReduceLarge(FieldElement* out, LargeFieldElement& in) {

[Wez, 2011/11/02 23:46:06]

You're modifying LargeFieldElement, so it should really be pass-by-pointer, I
think.

[agl, 2011/11/03 17:20:49]

Done.

+  for (int i = 0; i < 8; i++) {
+    in[i] += kZero63ModP[i];
+  }
+
+  // Eliminate the coefficients at 2**224 and greater.

[Wez, 2011/11/02 23:46:06]

Explain that you're using a mod p operation to reduce to an equivalent, while
eliminating the coefficients?

[agl, 2011/11/03 17:20:49]

Done.

+  for (int i = 14; i >= 8; i--) {
+    in[i-8] -= in[i];
+    in[i-5] += (in[i] & 0xffff) << 12;
+    in[i-4] += in[i] >> 16;

[Wez, 2011/11/02 23:46:06]

nit: It would help laymen like me to have each of these terms commented to
indicate the term in p of which they are a reflection.

[agl, 2011/11/03 17:20:49]

Done.

+  }
+  in[8] = 0;
+  // in[0..8] < 2**64
+
+  // As the values become small enough, we start to store them in |out| and use
+  // 32-bit operations.
+  for (int i = 1; i < 8; i++) {
+    in[i+1] += in[i] >> 28;
+    (*out)[i] = static_cast<uint32>(in[i] & kBottom28Bits);
+  }
+  in[0] -= in[8];
+  (*out)[3] += static_cast<uint32>(in[8] & 0xffff) << 12;
+  (*out)[4] += static_cast<uint32>(in[8] >> 16);

[Wez, 2011/11/02 23:46:06]

This looks like another mod p operation?

[agl, 2011/11/03 17:20:49]

Done.

+  // in[0] < 2**64
+  // out[3] < 2**29
+  // out[4] < 2**29
+  // out[1,2,5..7] < 2**28
+
+  (*out)[0] = static_cast<uint32>(in[0] & kBottom28Bits);
+  (*out)[1] += static_cast<uint32>((in[0] >> 28) & kBottom28Bits);
+  (*out)[2] += static_cast<uint32>(in[0] >> 56);
+  // out[0] < 2**28
+  // out[1..4] < 2**29
+  // out[5..7] < 2**28
+}
+
+// Mul computes *out = a*b
+//
+// a[i] < 2**29, b[i] < 2**30 (or vice versa)
+// out[i] < 2**29
+void Mul(FieldElement* out, const FieldElement& a, const FieldElement& b) {
+  LargeFieldElement tmp;
+  memset(&tmp, 0, sizeof(tmp));
+
+  for (int i = 0; i < 8; i++) {
+    for (int j = 0; j < 8; j++) {
+      tmp[i+j] += static_cast<uint64>(a[i]) * static_cast<uint64>(b[j]);
+    }
+  }
+
+  ReduceLarge(out, tmp);
+}
+
+// Square computes *out = a*a
+//
+// a[i] < 2**29
+// out[i] < 2**29
+void Square(FieldElement* out, const FieldElement& a) {
+  LargeFieldElement tmp;
+  memset(&tmp, 0, sizeof(tmp));
+
+  for (int i = 0; i < 8; i++) {
+    for (int j = 0; j <= i; j++) {
+      uint64 r = static_cast<uint64>(a[i]) * static_cast<uint64>(a[j]);
+      if (i == j) {
+        tmp[i+j] += r;
+      } else {
+        tmp[i+j] += r << 1;
+      }
+    }
+  }
+
+  ReduceLarge(out, tmp);
+}
+
+// Reduce reduces the coefficients of a to smaller bounds.
+//
+// On entry: a[i] < 2**31 + 2**30
+// On exit: a[i] < 2**29
+void Reduce(FieldElement* in) {

[Wez, 2011/11/02 23:46:06]

Either rename in to in_out, or update the comment.

[agl, 2011/11/03 17:20:49]

Done.

+  FieldElement& a = *in;
+
+  for (int i = 0; i < 7; i++) {
+    a[i+1] += a[i] >> 28;
+    a[i] &= kBottom28Bits;
+  }
+  uint32 top = a[7] >> 28;
+  a[7] &= kBottom28Bits;
+
+  // top < 2**4
+  uint32 mask = top;
+  mask |= mask >> 2;
+  mask |= mask >> 1;
+  mask <<= 31;
+  mask = static_cast<uint32>(static_cast<int32>(mask) >> 31);
+  // Mask is all ones if top != 0, all zero otherwise

[Wez, 2011/11/02 23:46:06]

I think you're doing this to ensure constant-time execution, in which case a
comment along the lines of:

// Constant-time: mask = (top != 0) ? 0xffffffff : 0

would help clarify.

[agl, 2011/11/03 17:20:49]

Done.

+
+  a[0] -= top;
+  a[3] += top << 12;

[Wez, 2011/11/02 23:46:06]

This comes, again, as a result of folding things down mod p, which the 2**96 and
1 terms?

[agl, 2011/11/03 17:20:49]

Yes. Commented.

+
+  // We may have just made a[0] negative but, if we did, then we must
+  // have added something to a[3], this it's > 2**12. Therefore we can

[Wez, 2011/11/02 23:46:06]

typo: this -> thus?

[agl, 2011/11/03 17:20:49]

Done.

+  // carry down to a[0].
+  a[3] -= 1 & mask;
+  a[2] += mask & ((1<<28) - 1);
+  a[1] += mask & ((1<<28) - 1);
+  a[0] += mask & (1<<28);
+}
+
+// Invert calcuates *out = in^-1 using Fermat's little theorem.

[Wez, 2011/11/02 23:46:06]

Suggest indicating as part of this comment that the target we are aiming for is
|in| ** (2**224 - 2**96 - 1), so that the comments alongside the operations are
more obvious.

[agl, 2011/11/03 17:20:49]

Done.

+void Invert(FieldElement* out, const FieldElement& in) {
+  FieldElement f1, f2, f3, f4;
+
+  Square(&f1, in);                      // 2
+  Mul(&f1, f1, in);                     // 2**2 - 1
+  Square(&f1, f1);                      // 2**3 - 2
+  Mul(&f1, f1, in);                     // 2**3 - 1
+  Square(&f2, f1);                      // 2**4 - 2
+  Square(&f2, f2);                      // 2**5 - 4
+  Square(&f2, f2);                      // 2**6 - 8
+  Mul(&f1, f1, f2);                     // 2**6 - 1
+  Square(&f2, f1);                      // 2**7 - 2
+  for (int i = 0; i < 5; i++) {         // 2**12 - 2**6
+          Square(&f2, f2);

[Wez, 2011/11/02 23:46:06]

I like the exuberance of your indentation, but the style guide doesn't.

[agl, 2011/11/03 17:20:49]

Done.

+  }
+  Mul(&f2, f2, f1);                     // 2**12 - 1
+  Square(&f3, f2);                      // 2**13 - 2
+  for (int i = 0; i < 11; i++) {        // 2**24 - 2**12
+          Square(&f3, f3);
+  }
+  Mul(&f2, f3, f2);                     // 2**24 - 1
+  Square(&f3, f2);                      // 2**25 - 2
+  for (int i = 0; i < 23; i++) {        // 2**48 - 2**24
+          Square(&f3, f3);
+  }
+  Mul(&f3, f3, f2);                     // 2**48 - 1
+  Square(&f4, f3);                      // 2**49 - 2
+  for (int i = 0; i < 47; i++) {        // 2**96 - 2**48
+          Square(&f4, f4);
+  }
+  Mul(&f3, f3, f4);                     // 2**96 - 1
+  Square(&f4, f3);                      // 2**97 - 2
+  for (int i = 0; i < 23; i++) {        // 2**120 - 2**24
+          Square(&f4, f4);
+  }
+  Mul(&f2, f4, f2);                     // 2**120 - 1
+  for (int i = 0; i < 6; i++) {         // 2**126 - 2**6
+          Square(&f2, f2);
+  }
+  Mul(&f1, f1, f2);                     // 2**126 - 1
+  Square(&f1, f1);                      // 2**127 - 2
+  Mul(&f1, f1, in);                     // 2**127 - 1
+  for (int i = 0; i < 97; i++) {        // 2**224 - 2**97
+          Square(&f1, f1);
+  }
+  Mul(out, f1, f3);                     // 2**224 - 2**96 - 1
+}
+
+// Contract converts a FieldElement to its minimal, distinguished form.
+//
+// On entry, in[i] < 2**32
+// On exit, in[i] < 2**28
+void Contract(FieldElement* inout) {
+  FieldElement& out = *inout;
+
+  for (int i = 0; i < 7; i++) {

[Wez, 2011/11/02 23:46:06]

Again, a comment on this block indicating that we're just dealing with the
"overflow" terms here...

[agl, 2011/11/03 17:20:49]

Done.

+    out[i+1] += out[i] >> 28;
+    out[i] &= kBottom28Bits;
+  }
+  uint32 top = out[7] >> 28;
+  out[7] &= kBottom28Bits;
+
+  out[0] -= top;

[Wez, 2011/11/02 23:46:06]

... and a comment to indicate that we're then going mod p, which reflects top
into the 2**96 & 1 terms here, would aid readability.

[agl, 2011/11/03 17:20:49]

Done.

+  out[3] += top << 12;
+
+  // We may just have made out[0] negative. So we carry down. If we made
+  // out[0] negative then we know that out[3] is sufficiently positive
+  // because we just added to it.
+  for (int i = 0; i < 3; i++) {
+    uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31);
+    out[i] += (1 << 28) & mask;
+    out[i+1] -= 1 & mask;
+  }
+
+  // Now we see if the value is >= p and, if so, subtract p.

[Wez, 2011/11/02 23:46:06]

For what purpose?  Is the value in the range 0-2p, so that if v>p then
subtracting p would get us to 0-p?
Again, clarifying that this implementation is non-obvious because it's
constant-time would help laymen like me.

[agl, 2011/11/03 17:20:49]

Done.

+
+  // First we build a mask from the top four limbs, which must all be
+  // equal to bottom28Bits if the whole value is >= p. If top4AllOnes
+  // ends up with any zero bits in the bottom 28 bits, then this wasn't
+  // true.
+  uint32 top4AllOnes = 0xffffffffu;
+  for (int i = 4; i < 8; i++) {
+    top4AllOnes &= (out[i] & kBottom28Bits) - 1;
+  }
+  top4AllOnes |= 0xf0000000;
+  // Now we replicate any zero bits to all the bits in top4AllOnes.
+  top4AllOnes &= top4AllOnes >> 16;
+  top4AllOnes &= top4AllOnes >> 8;
+  top4AllOnes &= top4AllOnes >> 4;
+  top4AllOnes &= top4AllOnes >> 2;
+  top4AllOnes &= top4AllOnes >> 1;
+  top4AllOnes =
+      static_cast<uint32>(static_cast<int32>(top4AllOnes << 31) >> 31);
+
+  // Now we test whether the bottom three limbs are non-zero.
+  uint32 bottom3NonZero = out[0] | out[1] | out[2];
+  bottom3NonZero |= bottom3NonZero >> 16;
+  bottom3NonZero |= bottom3NonZero >> 8;
+  bottom3NonZero |= bottom3NonZero >> 4;
+  bottom3NonZero |= bottom3NonZero >> 2;
+  bottom3NonZero |= bottom3NonZero >> 1;
+  bottom3NonZero =
+      static_cast<uint32>(static_cast<int32>(bottom3NonZero << 31) >> 31);
+
+  // Everything depends on the value of out[3].
+  //    If it's > 0xffff000 and top4AllOnes != 0 then the whole value is >= p
+  //    If it's = 0xffff000 and top4AllOnes != 0 and bottom3NonZero != 0,
+  //      then the whole value is >= p
+  //    If it's < 0xffff000, then the whole value is < p
+  uint32 n = out[3] - 0xffff000;
+  uint32 out3Equal = n;
+  out3Equal |= out3Equal >> 16;
+  out3Equal |= out3Equal >> 8;
+  out3Equal |= out3Equal >> 4;
+  out3Equal |= out3Equal >> 2;
+  out3Equal |= out3Equal >> 1;
+  out3Equal =
+      ~static_cast<uint32>(static_cast<int32>(out3Equal << 31) >> 31);
+
+  // If out[3] > 0xffff000 then n's MSB will be zero.
+  uint32 out3GT = ~static_cast<uint32>(static_cast<int32>(n << 31) >> 31);
+
+  uint32 mask = top4AllOnes & ((out3Equal & bottom3NonZero) | out3GT);
+  out[0] -= 1 & mask;
+  out[3] -= 0xffff000 & mask;
+  out[4] -= 0xfffffff & mask;
+  out[5] -= 0xfffffff & mask;
+  out[6] -= 0xfffffff & mask;
+  out[7] -= 0xfffffff & mask;
+}
+
+
+// Group element functions.
+//
+// These functions deal with group elements. The group is an elliptic curve
+// group with a = -3 defined in FIPS 186-3, section D.2.2.
+
+typedef crypto::P224::InternalPoint GroupElement;
+
+// kP is the P224 prime.
+const FieldElement kP = {
+  1, 0, 0, 268431360,
+  268435455, 268435455, 268435455, 268435455,
+};
+
+// kB is parameter of the elliptic curve.
+const FieldElement kB = {
+  55967668, 11768882, 265861671, 185302395,
+  39211076, 180311059, 84673715, 188764328,
+};
+
+// AddJacobian computes *out = a+b where a != b.

[Wez, 2011/11/02 23:46:06]

nit: The body of this function may be easier to read if you leave a line between
the code chunks implementing each operation, e.g:

// Z1Z1 = Z1²
Square(&z1z1, a.z);

// Z2Z2 = Z2²
Square(&z2z2, b.z);

// U1 = X1*Z2Z2
Mul(&u1, a.x, z2z2);

[agl, 2011/11/03 17:20:49]

Done.

+void AddJacobian(GroupElement *out,
+                 const GroupElement& a,
+                 const GroupElement& b) {
+  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
+  FieldElement z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v;
+
+  // Z1Z1 = Z1²
+  Square(&z1z1, a.z);
+  // Z2Z2 = Z2²
+  Square(&z2z2, b.z);
+  // U1 = X1*Z2Z2
+  Mul(&u1, a.x, z2z2);
+  // U2 = X2*Z1Z1
+  Mul(&u2, b.x, z1z1);
+  // S1 = Y1*Z2*Z2Z2
+  Mul(&s1, b.z, z2z2);
+  Mul(&s1, a.y, s1);
+  // S2 = Y2*Z1*Z1Z1
+  Mul(&s2, a.z, z1z1);
+  Mul(&s2, b.y, s2);
+  // H = U2-U1
+  Sub(&h, u2, u1);
+  Reduce(&h);
+  // I = (2*H)²
+  for (int j = 0; j < 8; j++) {
+    i[j] = h[j] << 1;
+  }
+  Reduce(&i);
+  Square(&i, i);
+  // J = H*I
+  Mul(&j, h, i);
+  // r = 2*(S2-S1)
+  Sub(&r, s2, s1);
+  Reduce(&r);
+  for (int i = 0; i < 8; i++) {
+    r[i] <<= 1;
+  }
+  Reduce(&r);
+  // V = U1*I
+  Mul(&v, u1, i);
+  // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H
+  Add(&z1z1, z1z1, z2z2);
+  Add(&z2z2, a.z, b.z);
+  Reduce(&z2z2);
+  Square(&z2z2, z2z2);
+  Sub(&out->z, z2z2, z1z1);
+  Reduce(&out->z);
+  Mul(&out->z, out->z, h);
+  // X3 = r²-J-2*V
+  for (int i = 0; i < 8; i++) {
+    z1z1[i] = v[i] << 1;
+  }
+  Add(&z1z1, j, z1z1);
+  Reduce(&z1z1);
+  Square(&out->x, r);
+  Sub(&out->x, out->x, z1z1);
+  Reduce(&out->x);
+  // Y3 = r*(V-X3)-2*S1*J
+  for (int i = 0; i < 8; i++) {
+    s1[i] <<= 1;
+  }
+  Mul(&s1, s1, j);
+  Sub(&z1z1, v, out->x);
+  Reduce(&z1z1);
+  Mul(&z1z1, z1z1, r);
+  Sub(&out->y, z1z1, s1);
+  Reduce(&out->y);
+}
+
+// DoubleJacobian computes *out = a+a.
+void DoubleJacobian(GroupElement* out, const GroupElement& a) {
+  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
+  FieldElement delta, gamma, beta, alpha, t;
+
+  Square(&delta, a.z);
+  Square(&gamma, a.y);
+  Mul(&beta, a.x, gamma);
+
+  // alpha = 3*(X1-delta)*(X1+delta)
+  Add(&t, a.x, delta);
+  for (int i = 0; i < 8; i++) {
+          t[i] += t[i] << 1;
+  }
+  Reduce(&t);
+  Sub(&alpha, a.x, delta);
+  Reduce(&alpha);
+  Mul(&alpha, alpha, t);
+
+  // Z3 = (Y1+Z1)²-gamma-delta
+  Add(&out->z, a.y, a.z);
+  Reduce(&out->z);
+  Square(&out->z, out->z);
+  Sub(&out->z, out->z, gamma);
+  Reduce(&out->z);
+  Sub(&out->z, out->z, delta);
+  Reduce(&out->z);
+
+  // X3 = alpha²-8*beta
+  for (int i = 0; i < 8; i++) {
+          delta[i] = beta[i] << 3;
+  }
+  Reduce(&delta);
+  Square(&out->x, alpha);
+  Sub(&out->x, out->x, delta);
+  Reduce(&out->x);
+
+  // Y3 = alpha*(4*beta-X3)-8*gamma²
+  for (int i = 0; i < 8; i++) {
+          beta[i] <<= 2;
+  }
+  Reduce(&beta);
+  Sub(&beta, beta, out->x);
+  Reduce(&beta);
+  Square(&gamma, gamma);
+  for (int i = 0; i < 8; i++) {
+          gamma[i] <<= 3;
+  }
+  Reduce(&gamma);
+  Mul(&out->y, alpha, beta);
+  Sub(&out->y, out->y, gamma);
+  Reduce(&out->y);
+}
+
+// CopyConditional sets *out=a if mask is 0xffffffff. mask must be either 0 of
+// 0xffffffff.
+void CopyConditional(GroupElement* out,
+                     const GroupElement& a,
+                     uint32 mask) {
+  for (int i = 0; i < 8; i++) {
+    out->x[i] ^= mask & (a.x[i] ^ out->x[i]);
+    out->y[i] ^= mask & (a.y[i] ^ out->y[i]);
+    out->z[i] ^= mask & (a.z[i] ^ out->z[i]);
+  }
+}
+
+// ScalarMult calculates *out = a*scalar where scalar is a big-endian number of
+// length scalar_len and != 0.
+void ScalarMult(GroupElement* out, const GroupElement& a,
+                const uint8* scalar, size_t scalar_len) {
+  memset(out, 0, sizeof(*out));
+  GroupElement tmp;
+
+  uint32 first_bit = 0xffffffff;
+  for (size_t i = 0; i < scalar_len; i++) {
+    for (unsigned int bit_num = 0; bit_num < 8; bit_num++) {
+      DoubleJacobian(out, *out);
+      uint32 bit = static_cast<uint32>(static_cast<int32>(
+          (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31));
+      AddJacobian(&tmp, a, *out);
+      CopyConditional(out, a, first_bit & bit);
+      CopyConditional(out, tmp, ~first_bit & bit);
+      first_bit = first_bit & ~bit;
+    }
+  }
+}
+
+// Get224Bits reads 7 words from in and scatters their contents in
+// little-endian form into 8 words at out, 28 bits per output word.
+void Get224Bits(uint32* out, const uint32* in) {
+  out[0] = ntohl(in[6]) & kBottom28Bits;
+  out[1] = ((ntohl(in[5]) << 4) | (ntohl(in[6]) >> 28)) & kBottom28Bits;
+  out[2] = ((ntohl(in[4]) << 8) | (ntohl(in[5]) >> 24)) & kBottom28Bits;
+  out[3] = ((ntohl(in[3]) << 12) | (ntohl(in[4]) >> 20)) & kBottom28Bits;
+  out[4] = ((ntohl(in[2]) << 16) | (ntohl(in[3]) >> 16)) & kBottom28Bits;
+  out[5] = ((ntohl(in[1]) << 20) | (ntohl(in[2]) >> 12)) & kBottom28Bits;
+  out[6] = ((ntohl(in[0]) << 24) | (ntohl(in[1]) >> 8)) & kBottom28Bits;
+  out[7] = (ntohl(in[0]) >> 4) & kBottom28Bits;
+}
+
+// Put224Bits performs the inverse operation to Get224Bits: taking 28 bits from
+// each of 8 input words and writing them in big-endian order to 7 words at
+// out.
+void Put224Bits(uint32* out, const uint32* in) {
+  out[6] = htonl((in[0] >> 0) | (in[1] << 28));
+  out[5] = htonl((in[1] >> 4) | (in[2] << 24));
+  out[4] = htonl((in[2] >> 8) | (in[3] << 20));
+  out[3] = htonl((in[3] >> 12) | (in[4] << 16));
+  out[2] = htonl((in[4] >> 16) | (in[5] << 12));
+  out[1] = htonl((in[5] >> 20) | (in[6] << 8));
+  out[0] = htonl((in[6] >> 24) | (in[7] << 4));
+}
+
+} // anonymous namespace
+
+namespace crypto {
+
+bool P224::ToInternal(const ExternalPoint& in, InternalPoint* out) {
+  const uint32* inwords = reinterpret_cast<const uint32*>(in.affine);
+  Get224Bits(out->x, inwords);
+  Get224Bits(out->y, inwords + 7);
+  memset(&out->z, 0, sizeof(out->z));
+  out->z[0] = 1;
+
+  // Check that the point is on the curve, i.e. that y² = x³ - 3x + b.
+  FieldElement lhs;
+  Square(&lhs, out->y);
+  Contract(&lhs);
+
+  FieldElement rhs;
+  Square(&rhs, out->x);
+  Mul(&rhs, out->x, rhs);
+
+  FieldElement three_x;
+  for (int i = 0; i < 8; i++) {
+    three_x[i] = out->x[i] * 3;
+  }
+  Reduce(&three_x);
+  Sub(&rhs, rhs, three_x);
+  Reduce(&rhs);
+
+  ::Add(&rhs, rhs, kB);
+  Contract(&rhs);
+  return memcmp(&lhs, &rhs, sizeof(lhs)) == 0;
+}
+
+void P224::ToExternal(const InternalPoint& in, ExternalPoint* out) {
+  FieldElement zinv, zinv_sq, x, y;
+
+  Invert(&zinv, in.z);
+  Square(&zinv_sq, zinv);
+  Mul(&x, in.x, zinv_sq);
+  Mul(&zinv_sq, zinv_sq, zinv);
+  Mul(&y, in.y, zinv_sq);
+
+  Contract(&x);
+  Contract(&y);
+
+  uint32* outwords = reinterpret_cast<uint32*>(out->affine);
+  Put224Bits(outwords, x);
+  Put224Bits(outwords + 7, y);
+}
+
+void P224::ScalarMult(const InternalPoint& in,
+                      const uint8* scalar,
+                      InternalPoint* out) {
+  ::ScalarMult(out, in, scalar, 28);
+}
+
+static const P224::InternalPoint kBasePoint = {
+  {22813985, 52956513, 34677300, 203240812,
+   12143107, 133374265, 225162431, 191946955},
+  {83918388, 223877528, 122119236, 123340192,
+   266784067, 263504429, 146143011, 198407736},
+  {1, 0, 0, 0, 0, 0, 0, 0},
+};
+
+void P224::ScalarBaseMult(const uint8* scalar, InternalPoint* out) {
+  ::ScalarMult(out, kBasePoint, scalar, 28);
+}
+
+void P224::Add(const InternalPoint& a, const InternalPoint& b,
+               InternalPoint* out) {
+  AddJacobian(out, a, b);
+}
+
+void P224::Negate(const InternalPoint& in, InternalPoint* out) {
+  // Guide to elliptic curve cryptography, page 89 suggests that (X : X+Y : Z)
+  // is the negative in Jacobian coordinates, but it doesn't actually appear to
+  // be true in testing so this performs the negation in affine coordinates.
+  FieldElement zinv, zinv_sq, y;
+  Invert(&zinv, in.z);
+  Square(&zinv_sq, zinv);
+  Mul(&out->x, in.x, zinv_sq);
+  Mul(&zinv_sq, zinv_sq, zinv);
+  Mul(&y, in.y, zinv_sq);
+
+  Sub(&out->y, kP, y);
+  Reduce(&out->y);
+
+  memset(&out->z, 0, sizeof(out->z));
+  out->z[0] = 1;
+}
+
+}  // namespace crypto