blob: 2ac07127acb8862b96dc04c659ad5611c93fdb00 [file] [log] [blame]
 // Copyright (c) 2012 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. // // See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background. #include "crypto/p224.h" #include #include #include #include "base/sys_byteorder.h" namespace { using base::HostToNet32; using base::NetToHost32; // 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_t'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. using crypto::p224::FieldElement; // kP is the P224 prime. const FieldElement kP = { 1, 0, 0, 268431360, 268435455, 268435455, 268435455, 268435455, }; void Contract(FieldElement* inout); // IsZero returns 0xffffffff if a == 0 mod p and 0 otherwise. uint32_t IsZero(const FieldElement& a) { FieldElement minimal; memcpy(&minimal, &a, sizeof(minimal)); Contract(&minimal); uint32_t is_zero = 0, is_p = 0; for (unsigned i = 0; i < 8; i++) { is_zero |= minimal[i]; is_p |= minimal[i] - kP[i]; } // If either is_zero or is_p is 0, then we should return 1. is_zero |= is_zero >> 16; is_zero |= is_zero >> 8; is_zero |= is_zero >> 4; is_zero |= is_zero >> 2; is_zero |= is_zero >> 1; is_p |= is_p >> 16; is_p |= is_p >> 8; is_p |= is_p >> 4; is_p |= is_p >> 2; is_p |= is_p >> 1; // For is_zero and is_p, the LSB is 0 iff all the bits are zero. is_zero &= is_p & 1; is_zero = (~is_zero) << 31; is_zero = static_cast(is_zero) >> 31; return is_zero; } // Add computes *out = a+b // // a[i] + b[i] < 2**32 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_t kTwo31p3 = (1u << 31) + (1u << 3); static const uint32_t kTwo31m3 = (1u << 31) - (1u << 3); static const uint32_t kTwo31m15m3 = (1u << 31) - (1u << 15) - (1u << 3); // kZero31ModP is 0 mod p where bit 31 is set in all limbs so that we can // subtract smaller amounts without underflow. See the section "Subtraction" in // [1] for why. static const FieldElement kZero31ModP = { kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3, kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3 }; // Subtract computes *out = a-b // // a[i], b[i] < 2**30 // out[i] < 2**32 void Subtract(FieldElement* out, const FieldElement& a, const FieldElement& b) { for (int i = 0; i < 8; i++) { // See the section on "Subtraction" in [1] for details. (*out)[i] = a[i] + kZero31ModP[i] - b[i]; } } static const uint64_t kTwo63p35 = (1ull << 63) + (1ull << 35); static const uint64_t kTwo63m35 = (1ull << 63) - (1ull << 35); static const uint64_t kTwo63m35m19 = (1ull << 63) - (1ull << 35) - (1ull << 19); // kZero63ModP is 0 mod p where bit 63 is set in all limbs. See the section // "Subtraction" in [1] for why. static const uint64_t kZero63ModP[8] = { kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35, kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35, }; static const uint32_t kBottom28Bits = 0xfffffff; // LargeFieldElement also represents an element of the field. The limbs are // still spaced 28-bits apart and in little-endian order. So the limbs are at // 0, 28, 56, ..., 392 bits, each 64-bits wide. typedef uint64_t LargeFieldElement[15]; // ReduceLarge converts a LargeFieldElement to a FieldElement. // // in[i] < 2**62 void ReduceLarge(FieldElement* out, LargeFieldElement* inptr) { LargeFieldElement& in(*inptr); for (int i = 0; i < 8; i++) { in[i] += kZero63ModP[i]; } // Eliminate the coefficients at 2**224 and greater while maintaining the // same value mod p. for (int i = 14; i >= 8; i--) { in[i-8] -= in[i]; // reflection off the "+1" term of p. in[i-5] += (in[i] & 0xffff) << 12; // part of the "-2**96" reflection. in[i-4] += in[i] >> 16; // the rest of the "-2**96" reflection. } 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(in[i] & kBottom28Bits); } // Eliminate the term at 2*224 that we introduced while keeping the same // value mod p. in[0] -= in[8]; // reflection off the "+1" term of p. (*out)[3] += static_cast(in[8] & 0xffff) << 12; // "-2**96" term (*out)[4] += static_cast(in[8] >> 16); // rest of "-2**96" term // in[0] < 2**64 // out[3] < 2**29 // out[4] < 2**29 // out[1,2,5..7] < 2**28 (*out)[0] = static_cast(in[0] & kBottom28Bits); (*out)[1] += static_cast((in[0] >> 28) & kBottom28Bits); (*out)[2] += static_cast(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(a[i]) * static_cast(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_t r = static_cast(a[i]) * static_cast(a[j]); if (i == j) { tmp[i+j] += r; } else { tmp[i+j] += r << 1; } } } ReduceLarge(out, &tmp); } // Reduce reduces the coefficients of in_out to smaller bounds. // // On entry: a[i] < 2**31 + 2**30 // On exit: a[i] < 2**29 void Reduce(FieldElement* in_out) { FieldElement& a = *in_out; for (int i = 0; i < 7; i++) { a[i+1] += a[i] >> 28; a[i] &= kBottom28Bits; } uint32_t top = a[7] >> 28; a[7] &= kBottom28Bits; // top < 2**4 // Constant-time: mask = (top != 0) ? 0xffffffff : 0 uint32_t mask = top; mask |= mask >> 2; mask |= mask >> 1; mask <<= 31; mask = static_cast(static_cast(mask) >> 31); // Eliminate top while maintaining the same value mod p. a[0] -= top; a[3] += top << 12; // We may have just made a[0] negative but, if we did, then we must // have added something to a[3], thus it's > 2**12. Therefore we can // 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 by computing in**(2**224 - 2**96 - 1), i.e. // Fermat's little theorem. 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); } 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**29 // On exit, in[i] < 2**28 void Contract(FieldElement* inout) { FieldElement& out = *inout; // Reduce the coefficients to < 2**28. for (int i = 0; i < 7; i++) { out[i+1] += out[i] >> 28; out[i] &= kBottom28Bits; } uint32_t top = out[7] >> 28; out[7] &= kBottom28Bits; // Eliminate top while maintaining the same value mod p. out[0] -= top; 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_t mask = static_cast(static_cast(out[i]) >> 31); out[i] += (1 << 28) & mask; out[i+1] -= 1 & mask; } // We might have pushed out[3] over 2**28 so we perform another, partial // carry chain. for (int i = 3; i < 7; i++) { out[i+1] += out[i] >> 28; out[i] &= kBottom28Bits; } top = out[7] >> 28; out[7] &= kBottom28Bits; // Eliminate top while maintaining the same value mod p. out[0] -= top; out[3] += top << 12; // There are two cases to consider for out[3]: // 1) The first time that we eliminated top, we didn't push out[3] over // 2**28. In this case, the partial carry chain didn't change any values // and top is zero. // 2) We did push out[3] over 2**28 the first time that we eliminated top. // The first value of top was in [0..16), therefore, prior to eliminating // the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after // overflowing and being reduced by the second carry chain, out[3] <= // 0xf000. Thus it cannot have overflowed when we eliminated top for the // second time. // Again, we may just have made out[0] negative, so do the same carry down. // As before, if we made out[0] negative then we know that out[3] is // sufficiently positive. for (int i = 0; i < 3; i++) { uint32_t mask = static_cast(static_cast(out[i]) >> 31); out[i] += (1 << 28) & mask; out[i+1] -= 1 & mask; } // The value is < 2**224, but maybe greater than p. In order to reduce to a // unique, minimal value we see if the value is >= p and, if so, subtract p. // First we build a mask from the top four limbs, which must all be // equal to bottom28Bits if the whole value is >= p. If top_4_all_ones // ends up with any zero bits in the bottom 28 bits, then this wasn't // true. uint32_t top_4_all_ones = 0xffffffffu; for (int i = 4; i < 8; i++) { top_4_all_ones &= out[i]; } top_4_all_ones |= 0xf0000000; // Now we replicate any zero bits to all the bits in top_4_all_ones. top_4_all_ones &= top_4_all_ones >> 16; top_4_all_ones &= top_4_all_ones >> 8; top_4_all_ones &= top_4_all_ones >> 4; top_4_all_ones &= top_4_all_ones >> 2; top_4_all_ones &= top_4_all_ones >> 1; top_4_all_ones = static_cast(static_cast(top_4_all_ones << 31) >> 31); // Now we test whether the bottom three limbs are non-zero. uint32_t bottom_3_non_zero = out[0] | out[1] | out[2]; bottom_3_non_zero |= bottom_3_non_zero >> 16; bottom_3_non_zero |= bottom_3_non_zero >> 8; bottom_3_non_zero |= bottom_3_non_zero >> 4; bottom_3_non_zero |= bottom_3_non_zero >> 2; bottom_3_non_zero |= bottom_3_non_zero >> 1; bottom_3_non_zero = static_cast(static_cast(bottom_3_non_zero) >> 31); // Everything depends on the value of out[3]. // If it's > 0xffff000 and top_4_all_ones != 0 then the whole value is >= p // If it's = 0xffff000 and top_4_all_ones != 0 and bottom_3_non_zero != 0, // then the whole value is >= p // If it's < 0xffff000, then the whole value is < p uint32_t n = out[3] - 0xffff000; uint32_t out_3_equal = n; out_3_equal |= out_3_equal >> 16; out_3_equal |= out_3_equal >> 8; out_3_equal |= out_3_equal >> 4; out_3_equal |= out_3_equal >> 2; out_3_equal |= out_3_equal >> 1; out_3_equal = ~static_cast(static_cast(out_3_equal << 31) >> 31); // If out[3] > 0xffff000 then n's MSB will be zero. uint32_t out_3_gt = ~static_cast(static_cast(n << 31) >> 31); uint32_t mask = top_4_all_ones & ((out_3_equal & bottom_3_non_zero) | out_3_gt); 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. using crypto::p224::Point; // kB is parameter of the elliptic curve. const FieldElement kB = { 55967668, 11768882, 265861671, 185302395, 39211076, 180311059, 84673715, 188764328, }; void CopyConditional(Point* out, const Point& a, uint32_t mask); void DoubleJacobian(Point* out, const Point& a); // AddJacobian computes *out = a+b where a != b. void AddJacobian(Point *out, const Point& a, const Point& 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; uint32_t z1_is_zero = IsZero(a.z); uint32_t z2_is_zero = IsZero(b.z); // 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 Subtract(&h, u2, u1); Reduce(&h); uint32_t x_equal = IsZero(h); // I = (2*H)² for (int k = 0; k < 8; k++) { i[k] = h[k] << 1; } Reduce(&i); Square(&i, i); // J = H*I Mul(&j, h, i); // r = 2*(S2-S1) Subtract(&r, s2, s1); Reduce(&r); uint32_t y_equal = IsZero(r); if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { // The two input points are the same therefore we must use the dedicated // doubling function as the slope of the line is undefined. DoubleJacobian(out, a); return; } for (int k = 0; k < 8; k++) { r[k] <<= 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); Subtract(&out->z, z2z2, z1z1); Reduce(&out->z); Mul(&out->z, out->z, h); // X3 = r²-J-2*V for (int k = 0; k < 8; k++) { z1z1[k] = v[k] << 1; } Add(&z1z1, j, z1z1); Reduce(&z1z1); Square(&out->x, r); Subtract(&out->x, out->x, z1z1); Reduce(&out->x); // Y3 = r*(V-X3)-2*S1*J for (int k = 0; k < 8; k++) { s1[k] <<= 1; } Mul(&s1, s1, j); Subtract(&z1z1, v, out->x); Reduce(&z1z1); Mul(&z1z1, z1z1, r); Subtract(&out->y, z1z1, s1); Reduce(&out->y); CopyConditional(out, a, z2_is_zero); CopyConditional(out, b, z1_is_zero); } // DoubleJacobian computes *out = a+a. void DoubleJacobian(Point* out, const Point& 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); Subtract(&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); Subtract(&out->z, out->z, gamma); Reduce(&out->z); Subtract(&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); Subtract(&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); Subtract(&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); Subtract(&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(Point* out, const Point& a, uint32_t 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(Point* out, const Point& a, const uint8_t* scalar, size_t scalar_len) { memset(out, 0, sizeof(*out)); Point tmp; for (size_t i = 0; i < scalar_len; i++) { for (unsigned int bit_num = 0; bit_num < 8; bit_num++) { DoubleJacobian(out, *out); uint32_t bit = static_cast(static_cast( (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31)); AddJacobian(&tmp, a, *out); CopyConditional(out, tmp, 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_t* out, const uint32_t* in) { out[0] = NetToHost32(in[6]) & kBottom28Bits; out[1] = ((NetToHost32(in[5]) << 4) | (NetToHost32(in[6]) >> 28)) & kBottom28Bits; out[2] = ((NetToHost32(in[4]) << 8) | (NetToHost32(in[5]) >> 24)) & kBottom28Bits; out[3] = ((NetToHost32(in[3]) << 12) | (NetToHost32(in[4]) >> 20)) & kBottom28Bits; out[4] = ((NetToHost32(in[2]) << 16) | (NetToHost32(in[3]) >> 16)) & kBottom28Bits; out[5] = ((NetToHost32(in[1]) << 20) | (NetToHost32(in[2]) >> 12)) & kBottom28Bits; out[6] = ((NetToHost32(in[0]) << 24) | (NetToHost32(in[1]) >> 8)) & kBottom28Bits; out[7] = (NetToHost32(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_t* out, const uint32_t* in) { out[6] = HostToNet32((in[0] >> 0) | (in[1] << 28)); out[5] = HostToNet32((in[1] >> 4) | (in[2] << 24)); out[4] = HostToNet32((in[2] >> 8) | (in[3] << 20)); out[3] = HostToNet32((in[3] >> 12) | (in[4] << 16)); out[2] = HostToNet32((in[4] >> 16) | (in[5] << 12)); out[1] = HostToNet32((in[5] >> 20) | (in[6] << 8)); out[0] = HostToNet32((in[6] >> 24) | (in[7] << 4)); } } // anonymous namespace namespace crypto { namespace p224 { bool Point::SetFromString(base::StringPiece in) { if (in.size() != 2*28) return false; const uint32_t* inwords = reinterpret_cast(in.data()); Get224Bits(x, inwords); Get224Bits(y, inwords + 7); memset(&z, 0, sizeof(z)); z[0] = 1; // Check that the point is on the curve, i.e. that y² = x³ - 3x + b. FieldElement lhs; Square(&lhs, y); Contract(&lhs); FieldElement rhs; Square(&rhs, x); Mul(&rhs, x, rhs); FieldElement three_x; for (int i = 0; i < 8; i++) { three_x[i] = x[i] * 3; } Reduce(&three_x); Subtract(&rhs, rhs, three_x); Reduce(&rhs); ::Add(&rhs, rhs, kB); Contract(&rhs); return memcmp(&lhs, &rhs, sizeof(lhs)) == 0; } std::string Point::ToString() const { FieldElement zinv, zinv_sq, xx, yy; // If this is the point at infinity we return a string of all zeros. if (IsZero(this->z)) { static const char zeros[56] = {0}; return std::string(zeros, sizeof(zeros)); } Invert(&zinv, this->z); Square(&zinv_sq, zinv); Mul(&xx, x, zinv_sq); Mul(&zinv_sq, zinv_sq, zinv); Mul(&yy, y, zinv_sq); Contract(&xx); Contract(&yy); uint32_t outwords[14]; Put224Bits(outwords, xx); Put224Bits(outwords + 7, yy); return std::string(reinterpret_cast(outwords), sizeof(outwords)); } void ScalarMult(const Point& in, const uint8_t* scalar, Point* out) { ::ScalarMult(out, in, scalar, 28); } // kBasePoint is the base point (generator) of the elliptic curve group. static const Point 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 ScalarBaseMult(const uint8_t* scalar, Point* out) { ::ScalarMult(out, kBasePoint, scalar, 28); } void Add(const Point& a, const Point& b, Point* out) { AddJacobian(out, a, b); } void Negate(const Point& in, Point* 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); Subtract(&out->y, kP, y); Reduce(&out->y); memset(&out->z, 0, sizeof(out->z)); out->z[0] = 1; } } // namespace p224 } // namespace crypto