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crypto/elliptic: split up P-256 field and group ops
This makes Gerrit recognize the rename of the field implementation and facilitates the review. No code changes. For #52182 Change-Id: I827004e175db1ae2fcdf17d0f586ff21503d27e3 Reviewed-on: https://go-review.googlesource.com/c/go/+/390754 Reviewed-by: Ian Lance Taylor <iant@google.com> Reviewed-by: Russ Cox <rsc@golang.org> Reviewed-by: Roland Shoemaker <roland@golang.org> Run-TryBot: Filippo Valsorda <filippo@golang.org> Auto-Submit: Filippo Valsorda <filippo@golang.org> TryBot-Result: Gopher Robot <gobot@golang.org>
This commit is contained in:
Filippo Valsorda
Gopher Robot
parent
f0c0e0f255
commit
0b5218cf4e
@ -57,38 +57,6 @@ func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int)
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return p256ToAffine(&x1, &y1, &z1)
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}
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// Field elements are represented as nine, unsigned 32-bit words.
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//
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// The value of a field element is:
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// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
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//
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// That is, each limb is alternately 29 or 28-bits wide in little-endian
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// order.
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//
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// This means that a field element hits 2**257, rather than 2**256 as we would
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// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
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// problems when multiplying as terms end up one bit short of a limb which
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// would require much bit-shifting to correct.
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//
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// Finally, the values stored in a field element are in Montgomery form. So the
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// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
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// 2**257.
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const (
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p256Limbs = 9
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bottom29Bits = 0x1fffffff
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)
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var (
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// p256One is the number 1 as a field element.
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p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
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p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
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// p256P is the prime modulus as a field element.
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p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
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// p2562P is the twice prime modulus as a field element.
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p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
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)
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// p256Precomputed contains precomputed values to aid the calculation of scalar
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// multiples of the base point, G. It's actually two, equal length, tables
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// concatenated.
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@ -181,613 +149,6 @@ var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{
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0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
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}
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// Field element operations:
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const bottom28Bits = 0xfffffff
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// nonZeroToAllOnes returns:
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//
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// 0xffffffff for 0 < x <= 2**31
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// 0 for x == 0 or x > 2**31.
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func nonZeroToAllOnes(x uint32) uint32 {
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return ((x - 1) >> 31) - 1
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}
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// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
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// which is a term at 2**257.
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//
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// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
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// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
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func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
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carry_mask := nonZeroToAllOnes(carry)
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inout[0] += carry << 1
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inout[3] += 0x10000000 & carry_mask
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// carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
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// previous line therefore this doesn't underflow.
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inout[3] -= carry << 11
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inout[4] += (0x20000000 - 1) & carry_mask
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inout[5] += (0x10000000 - 1) & carry_mask
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inout[6] += (0x20000000 - 1) & carry_mask
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inout[6] -= carry << 22
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// This may underflow if carry is non-zero but, if so, we'll fix it in the
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// next line.
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inout[7] -= 1 & carry_mask
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inout[7] += carry << 25
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}
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// p256Sum sets out = in+in2.
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//
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// On entry: in[i]+in2[i] must not overflow a 32-bit word.
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// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
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func p256Sum(out, in, in2 *[p256Limbs]uint32) {
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carry := uint32(0)
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for i := 0; ; i++ {
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out[i] = in[i] + in2[i]
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out[i] += carry
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carry = out[i] >> 29
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out[i] &= bottom29Bits
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i++
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if i == p256Limbs {
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break
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}
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out[i] = in[i] + in2[i]
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out[i] += carry
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carry = out[i] >> 28
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out[i] &= bottom28Bits
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}
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p256ReduceCarry(out, carry)
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}
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const (
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two30m2 = 1<<30 - 1<<2
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two30p13m2 = 1<<30 + 1<<13 - 1<<2
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two31m2 = 1<<31 - 1<<2
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two31m3 = 1<<31 - 1<<3
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two31p24m2 = 1<<31 + 1<<24 - 1<<2
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two30m27m2 = 1<<30 - 1<<27 - 1<<2
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)
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// p256Zero31 is 0 mod p.
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var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
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// p256Diff sets out = in-in2.
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//
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// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
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// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
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// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
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func p256Diff(out, in, in2 *[p256Limbs]uint32) {
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var carry uint32
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for i := 0; ; i++ {
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out[i] = in[i] - in2[i]
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out[i] += p256Zero31[i]
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out[i] += carry
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carry = out[i] >> 29
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out[i] &= bottom29Bits
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i++
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if i == p256Limbs {
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break
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}
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out[i] = in[i] - in2[i]
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out[i] += p256Zero31[i]
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out[i] += carry
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carry = out[i] >> 28
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out[i] &= bottom28Bits
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}
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p256ReduceCarry(out, carry)
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}
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// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
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// the same 29,28,... bit positions as a field element.
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//
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// The values in field elements are in Montgomery form: x*R mod p where R =
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// 2**257. Since we just multiplied two Montgomery values together, the result
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// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
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// in Montgomery form.
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//
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// On entry: tmp[i] < 2**64.
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// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
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func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
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// The following table may be helpful when reading this code:
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//
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// Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
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// Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
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// Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
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// (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
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var tmp2 [18]uint32
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var carry, x, xMask uint32
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// tmp contains 64-bit words with the same 29,28,29-bit positions as a
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// field element. So the top of an element of tmp might overlap with
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// another element two positions down. The following loop eliminates
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// this overlap.
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tmp2[0] = uint32(tmp[0]) & bottom29Bits
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tmp2[1] = uint32(tmp[0]) >> 29
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tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
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tmp2[1] += uint32(tmp[1]) & bottom28Bits
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carry = tmp2[1] >> 28
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tmp2[1] &= bottom28Bits
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for i := 2; i < 17; i++ {
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tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
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tmp2[i] += (uint32(tmp[i-1])) >> 28
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tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
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tmp2[i] += uint32(tmp[i]) & bottom29Bits
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tmp2[i] += carry
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carry = tmp2[i] >> 29
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tmp2[i] &= bottom29Bits
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i++
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if i == 17 {
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break
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}
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tmp2[i] = uint32(tmp[i-2]>>32) >> 25
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tmp2[i] += uint32(tmp[i-1]) >> 29
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tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
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tmp2[i] += uint32(tmp[i]) & bottom28Bits
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tmp2[i] += carry
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carry = tmp2[i] >> 28
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tmp2[i] &= bottom28Bits
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}
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tmp2[17] = uint32(tmp[15]>>32) >> 25
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tmp2[17] += uint32(tmp[16]) >> 29
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tmp2[17] += uint32(tmp[16]>>32) << 3
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tmp2[17] += carry
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// Montgomery elimination of terms:
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//
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// Since R is 2**257, we can divide by R with a bitwise shift if we can
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// ensure that the right-most 257 bits are all zero. We can make that true
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// by adding multiplies of p without affecting the value.
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//
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// So we eliminate limbs from right to left. Since the bottom 29 bits of p
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// are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
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// We can do that for 8 further limbs and then right shift to eliminate the
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// extra factor of R.
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for i := 0; ; i += 2 {
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tmp2[i+1] += tmp2[i] >> 29
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x = tmp2[i] & bottom29Bits
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xMask = nonZeroToAllOnes(x)
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tmp2[i] = 0
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// The bounds calculations for this loop are tricky. Each iteration of
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// the loop eliminates two words by adding values to words to their
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// right.
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//
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// The following table contains the amounts added to each word (as an
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// offset from the value of i at the top of the loop). The amounts are
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// accounted for from the first and second half of the loop separately
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// and are written as, for example, 28 to mean a value <2**28.
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//
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// Word: 3 4 5 6 7 8 9 10
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// Added in top half: 28 11 29 21 29 28
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// 28 29
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// 29
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// Added in bottom half: 29 10 28 21 28 28
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// 29
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//
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// The value that is currently offset 7 will be offset 5 for the next
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// iteration and then offset 3 for the iteration after that. Therefore
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// the total value added will be the values added at 7, 5 and 3.
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//
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// The following table accumulates these values. The sums at the bottom
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// are written as, for example, 29+28, to mean a value < 2**29+2**28.
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//
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// Word: 3 4 5 6 7 8 9 10 11 12 13
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// 28 11 10 29 21 29 28 28 28 28 28
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// 29 28 11 28 29 28 29 28 29 28
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// 29 28 21 21 29 21 29 21
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// 10 29 28 21 28 21 28
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// 28 29 28 29 28 29 28
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// 11 10 29 10 29 10
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// 29 28 11 28 11
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// 29 29
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// --------------------------------------------
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// 30+ 31+ 30+ 31+ 30+
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// 28+ 29+ 28+ 29+ 21+
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// 21+ 28+ 21+ 28+ 10
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// 10 21+ 10 21+
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// 11 11
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//
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// So the greatest amount is added to tmp2[10] and tmp2[12]. If
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// tmp2[10/12] has an initial value of <2**29, then the maximum value
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// will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
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// as required.
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tmp2[i+3] += (x << 10) & bottom28Bits
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tmp2[i+4] += (x >> 18)
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tmp2[i+6] += (x << 21) & bottom29Bits
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tmp2[i+7] += x >> 8
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// At position 200, which is the starting bit position for word 7, we
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// have a factor of 0xf000000 = 2**28 - 2**24.
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tmp2[i+7] += 0x10000000 & xMask
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tmp2[i+8] += (x - 1) & xMask
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tmp2[i+7] -= (x << 24) & bottom28Bits
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tmp2[i+8] -= x >> 4
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tmp2[i+8] += 0x20000000 & xMask
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tmp2[i+8] -= x
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tmp2[i+8] += (x << 28) & bottom29Bits
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tmp2[i+9] += ((x >> 1) - 1) & xMask
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if i+1 == p256Limbs {
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break
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}
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tmp2[i+2] += tmp2[i+1] >> 28
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x = tmp2[i+1] & bottom28Bits
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xMask = nonZeroToAllOnes(x)
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tmp2[i+1] = 0
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tmp2[i+4] += (x << 11) & bottom29Bits
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tmp2[i+5] += (x >> 18)
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tmp2[i+7] += (x << 21) & bottom28Bits
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tmp2[i+8] += x >> 7
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// At position 199, which is the starting bit of the 8th word when
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// dealing with a context starting on an odd word, we have a factor of
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// 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
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// word from i+1 is i+8.
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tmp2[i+8] += 0x20000000 & xMask
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tmp2[i+9] += (x - 1) & xMask
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tmp2[i+8] -= (x << 25) & bottom29Bits
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tmp2[i+9] -= x >> 4
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tmp2[i+9] += 0x10000000 & xMask
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tmp2[i+9] -= x
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tmp2[i+10] += (x - 1) & xMask
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}
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// We merge the right shift with a carry chain. The words above 2**257 have
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// widths of 28,29,... which we need to correct when copying them down.
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carry = 0
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for i := 0; i < 8; i++ {
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// The maximum value of tmp2[i + 9] occurs on the first iteration and
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// is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
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// therefore safe.
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out[i] = tmp2[i+9]
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out[i] += carry
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out[i] += (tmp2[i+10] << 28) & bottom29Bits
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carry = out[i] >> 29
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out[i] &= bottom29Bits
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i++
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out[i] = tmp2[i+9] >> 1
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out[i] += carry
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carry = out[i] >> 28
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out[i] &= bottom28Bits
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}
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out[8] = tmp2[17]
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out[8] += carry
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carry = out[8] >> 29
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out[8] &= bottom29Bits
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p256ReduceCarry(out, carry)
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}
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// p256Square sets out=in*in.
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//
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// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
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// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
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func p256Square(out, in *[p256Limbs]uint32) {
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var tmp [17]uint64
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tmp[0] = uint64(in[0]) * uint64(in[0])
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tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
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tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
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uint64(in[1])*(uint64(in[1])<<1)
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tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
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uint64(in[1])*(uint64(in[2])<<1)
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tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
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uint64(in[1])*(uint64(in[3])<<2) +
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uint64(in[2])*uint64(in[2])
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tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
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uint64(in[1])*(uint64(in[4])<<1) +
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uint64(in[2])*(uint64(in[3])<<1)
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tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
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uint64(in[1])*(uint64(in[5])<<2) +
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uint64(in[2])*(uint64(in[4])<<1) +
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uint64(in[3])*(uint64(in[3])<<1)
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tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
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uint64(in[1])*(uint64(in[6])<<1) +
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uint64(in[2])*(uint64(in[5])<<1) +
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uint64(in[3])*(uint64(in[4])<<1)
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// tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
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// which is < 2**64 as required.
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tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
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uint64(in[1])*(uint64(in[7])<<2) +
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uint64(in[2])*(uint64(in[6])<<1) +
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uint64(in[3])*(uint64(in[5])<<2) +
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uint64(in[4])*uint64(in[4])
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tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
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uint64(in[2])*(uint64(in[7])<<1) +
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uint64(in[3])*(uint64(in[6])<<1) +
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uint64(in[4])*(uint64(in[5])<<1)
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tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
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uint64(in[3])*(uint64(in[7])<<2) +
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uint64(in[4])*(uint64(in[6])<<1) +
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uint64(in[5])*(uint64(in[5])<<1)
|
||||
tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
|
||||
uint64(in[4])*(uint64(in[7])<<1) +
|
||||
uint64(in[5])*(uint64(in[6])<<1)
|
||||
tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
|
||||
uint64(in[5])*(uint64(in[7])<<2) +
|
||||
uint64(in[6])*uint64(in[6])
|
||||
tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
|
||||
uint64(in[6])*(uint64(in[7])<<1)
|
||||
tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
|
||||
uint64(in[7])*(uint64(in[7])<<1)
|
||||
tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
|
||||
tmp[16] = uint64(in[8]) * uint64(in[8])
|
||||
|
||||
p256ReduceDegree(out, tmp)
|
||||
}
|
||||
|
||||
// p256Mul sets out=in*in2.
|
||||
//
|
||||
// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
|
||||
// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Mul(out, in, in2 *[p256Limbs]uint32) {
|
||||
var tmp [17]uint64
|
||||
|
||||
tmp[0] = uint64(in[0]) * uint64(in2[0])
|
||||
tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
|
||||
uint64(in[1])*(uint64(in2[0])<<0)
|
||||
tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
|
||||
uint64(in[1])*(uint64(in2[1])<<1) +
|
||||
uint64(in[2])*(uint64(in2[0])<<0)
|
||||
tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
|
||||
uint64(in[1])*(uint64(in2[2])<<0) +
|
||||
uint64(in[2])*(uint64(in2[1])<<0) +
|
||||
uint64(in[3])*(uint64(in2[0])<<0)
|
||||
tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
|
||||
uint64(in[1])*(uint64(in2[3])<<1) +
|
||||
uint64(in[2])*(uint64(in2[2])<<0) +
|
||||
uint64(in[3])*(uint64(in2[1])<<1) +
|
||||
uint64(in[4])*(uint64(in2[0])<<0)
|
||||
tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
|
||||
uint64(in[1])*(uint64(in2[4])<<0) +
|
||||
uint64(in[2])*(uint64(in2[3])<<0) +
|
||||
uint64(in[3])*(uint64(in2[2])<<0) +
|
||||
uint64(in[4])*(uint64(in2[1])<<0) +
|
||||
uint64(in[5])*(uint64(in2[0])<<0)
|
||||
tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
|
||||
uint64(in[1])*(uint64(in2[5])<<1) +
|
||||
uint64(in[2])*(uint64(in2[4])<<0) +
|
||||
uint64(in[3])*(uint64(in2[3])<<1) +
|
||||
uint64(in[4])*(uint64(in2[2])<<0) +
|
||||
uint64(in[5])*(uint64(in2[1])<<1) +
|
||||
uint64(in[6])*(uint64(in2[0])<<0)
|
||||
tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
|
||||
uint64(in[1])*(uint64(in2[6])<<0) +
|
||||
uint64(in[2])*(uint64(in2[5])<<0) +
|
||||
uint64(in[3])*(uint64(in2[4])<<0) +
|
||||
uint64(in[4])*(uint64(in2[3])<<0) +
|
||||
uint64(in[5])*(uint64(in2[2])<<0) +
|
||||
uint64(in[6])*(uint64(in2[1])<<0) +
|
||||
uint64(in[7])*(uint64(in2[0])<<0)
|
||||
// tmp[8] has the greatest value but doesn't overflow. See logic in
|
||||
// p256Square.
|
||||
tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
|
||||
uint64(in[1])*(uint64(in2[7])<<1) +
|
||||
uint64(in[2])*(uint64(in2[6])<<0) +
|
||||
uint64(in[3])*(uint64(in2[5])<<1) +
|
||||
uint64(in[4])*(uint64(in2[4])<<0) +
|
||||
uint64(in[5])*(uint64(in2[3])<<1) +
|
||||
uint64(in[6])*(uint64(in2[2])<<0) +
|
||||
uint64(in[7])*(uint64(in2[1])<<1) +
|
||||
uint64(in[8])*(uint64(in2[0])<<0)
|
||||
tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
|
||||
uint64(in[2])*(uint64(in2[7])<<0) +
|
||||
uint64(in[3])*(uint64(in2[6])<<0) +
|
||||
uint64(in[4])*(uint64(in2[5])<<0) +
|
||||
uint64(in[5])*(uint64(in2[4])<<0) +
|
||||
uint64(in[6])*(uint64(in2[3])<<0) +
|
||||
uint64(in[7])*(uint64(in2[2])<<0) +
|
||||
uint64(in[8])*(uint64(in2[1])<<0)
|
||||
tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
|
||||
uint64(in[3])*(uint64(in2[7])<<1) +
|
||||
uint64(in[4])*(uint64(in2[6])<<0) +
|
||||
uint64(in[5])*(uint64(in2[5])<<1) +
|
||||
uint64(in[6])*(uint64(in2[4])<<0) +
|
||||
uint64(in[7])*(uint64(in2[3])<<1) +
|
||||
uint64(in[8])*(uint64(in2[2])<<0)
|
||||
tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
|
||||
uint64(in[4])*(uint64(in2[7])<<0) +
|
||||
uint64(in[5])*(uint64(in2[6])<<0) +
|
||||
uint64(in[6])*(uint64(in2[5])<<0) +
|
||||
uint64(in[7])*(uint64(in2[4])<<0) +
|
||||
uint64(in[8])*(uint64(in2[3])<<0)
|
||||
tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
|
||||
uint64(in[5])*(uint64(in2[7])<<1) +
|
||||
uint64(in[6])*(uint64(in2[6])<<0) +
|
||||
uint64(in[7])*(uint64(in2[5])<<1) +
|
||||
uint64(in[8])*(uint64(in2[4])<<0)
|
||||
tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
|
||||
uint64(in[6])*(uint64(in2[7])<<0) +
|
||||
uint64(in[7])*(uint64(in2[6])<<0) +
|
||||
uint64(in[8])*(uint64(in2[5])<<0)
|
||||
tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
|
||||
uint64(in[7])*(uint64(in2[7])<<1) +
|
||||
uint64(in[8])*(uint64(in2[6])<<0)
|
||||
tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
|
||||
uint64(in[8])*(uint64(in2[7])<<0)
|
||||
tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
|
||||
|
||||
p256ReduceDegree(out, tmp)
|
||||
}
|
||||
|
||||
func p256Assign(out, in *[p256Limbs]uint32) {
|
||||
*out = *in
|
||||
}
|
||||
|
||||
// p256Invert calculates |out| = |in|^{-1}
|
||||
//
|
||||
// Based on Fermat's Little Theorem:
|
||||
//
|
||||
// a^p = a (mod p)
|
||||
// a^{p-1} = 1 (mod p)
|
||||
// a^{p-2} = a^{-1} (mod p)
|
||||
func p256Invert(out, in *[p256Limbs]uint32) {
|
||||
var ftmp, ftmp2 [p256Limbs]uint32
|
||||
|
||||
// each e_I will hold |in|^{2^I - 1}
|
||||
var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
|
||||
|
||||
p256Square(&ftmp, in) // 2^1
|
||||
p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
|
||||
p256Assign(&e2, &ftmp)
|
||||
p256Square(&ftmp, &ftmp) // 2^3 - 2^1
|
||||
p256Square(&ftmp, &ftmp) // 2^4 - 2^2
|
||||
p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
|
||||
p256Assign(&e4, &ftmp)
|
||||
p256Square(&ftmp, &ftmp) // 2^5 - 2^1
|
||||
p256Square(&ftmp, &ftmp) // 2^6 - 2^2
|
||||
p256Square(&ftmp, &ftmp) // 2^7 - 2^3
|
||||
p256Square(&ftmp, &ftmp) // 2^8 - 2^4
|
||||
p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
|
||||
p256Assign(&e8, &ftmp)
|
||||
for i := 0; i < 8; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^16 - 2^8
|
||||
p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
|
||||
p256Assign(&e16, &ftmp)
|
||||
for i := 0; i < 16; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^32 - 2^16
|
||||
p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
|
||||
p256Assign(&e32, &ftmp)
|
||||
for i := 0; i < 32; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^64 - 2^32
|
||||
p256Assign(&e64, &ftmp)
|
||||
p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
|
||||
for i := 0; i < 192; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^256 - 2^224 + 2^192
|
||||
|
||||
p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
|
||||
for i := 0; i < 16; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^80 - 2^16
|
||||
p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
|
||||
for i := 0; i < 8; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^88 - 2^8
|
||||
p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
|
||||
for i := 0; i < 4; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^92 - 2^4
|
||||
p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
|
||||
p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
|
||||
p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
|
||||
p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
|
||||
p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
|
||||
p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
|
||||
p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
|
||||
|
||||
p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
|
||||
}
|
||||
|
||||
// p256Scalar3 sets out=3*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar3(out *[p256Limbs]uint32) {
|
||||
var carry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
out[i] *= 3
|
||||
out[i] += carry
|
||||
carry = out[i] >> 29
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
out[i] *= 3
|
||||
out[i] += carry
|
||||
carry = out[i] >> 28
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256Scalar4 sets out=4*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar4(out *[p256Limbs]uint32) {
|
||||
var carry, nextCarry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
nextCarry = out[i] >> 27
|
||||
out[i] <<= 2
|
||||
out[i] &= bottom29Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 29)
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
nextCarry = out[i] >> 26
|
||||
out[i] <<= 2
|
||||
out[i] &= bottom28Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 28)
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256Scalar8 sets out=8*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar8(out *[p256Limbs]uint32) {
|
||||
var carry, nextCarry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
nextCarry = out[i] >> 26
|
||||
out[i] <<= 3
|
||||
out[i] &= bottom29Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 29)
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
nextCarry = out[i] >> 25
|
||||
out[i] <<= 3
|
||||
out[i] &= bottom28Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 28)
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// Group operations:
|
||||
//
|
||||
// Elements of the elliptic curve group are represented in Jacobian
|
||||
@ -908,16 +269,6 @@ func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) {
|
||||
p256Diff(yOut, yOut, &tmp)
|
||||
}
|
||||
|
||||
// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
|
||||
//
|
||||
// On entry: mask is either 0 or 0xffffffff.
|
||||
func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
|
||||
for i := 0; i < p256Limbs; i++ {
|
||||
tmp := mask & (in[i] ^ out[i])
|
||||
out[i] ^= tmp
|
||||
}
|
||||
}
|
||||
|
||||
// p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
|
||||
//
|
||||
// On entry: index < 16, table[0] must be zero.
|
||||
@ -1124,50 +475,3 @@ func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8
|
||||
nIsInfinityMask &^= pIsNoninfiniteMask
|
||||
}
|
||||
}
|
||||
|
||||
// p256FromBig sets out = R*in.
|
||||
func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
|
||||
tmp := new(big.Int).Lsh(in, 257)
|
||||
tmp.Mod(tmp, p256Params.P)
|
||||
|
||||
for i := 0; i < p256Limbs; i++ {
|
||||
if bits := tmp.Bits(); len(bits) > 0 {
|
||||
out[i] = uint32(bits[0]) & bottom29Bits
|
||||
} else {
|
||||
out[i] = 0
|
||||
}
|
||||
tmp.Rsh(tmp, 29)
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
if bits := tmp.Bits(); len(bits) > 0 {
|
||||
out[i] = uint32(bits[0]) & bottom28Bits
|
||||
} else {
|
||||
out[i] = 0
|
||||
}
|
||||
tmp.Rsh(tmp, 28)
|
||||
}
|
||||
}
|
||||
|
||||
// p256ToBig returns a *big.Int containing the value of in.
|
||||
func p256ToBig(in *[p256Limbs]uint32) *big.Int {
|
||||
result, tmp := new(big.Int), new(big.Int)
|
||||
|
||||
result.SetInt64(int64(in[p256Limbs-1]))
|
||||
for i := p256Limbs - 2; i >= 0; i-- {
|
||||
if (i & 1) == 0 {
|
||||
result.Lsh(result, 29)
|
||||
} else {
|
||||
result.Lsh(result, 28)
|
||||
}
|
||||
tmp.SetInt64(int64(in[i]))
|
||||
result.Add(result, tmp)
|
||||
}
|
||||
|
||||
result.Mul(result, p256RInverse)
|
||||
result.Mod(result, p256Params.P)
|
||||
return result
|
||||
}
|
||||
|
||||
705
src/crypto/elliptic/p256_generic_field.go
Normal file
705
src/crypto/elliptic/p256_generic_field.go
Normal file
@ -0,0 +1,705 @@
|
||||
// Copyright 2013 The Go Authors. All rights reserved.
|
||||
// Use of this source code is governed by a BSD-style
|
||||
// license that can be found in the LICENSE file.
|
||||
|
||||
//go:build !amd64 && !arm64
|
||||
|
||||
package elliptic
|
||||
|
||||
import "math/big"
|
||||
|
||||
// Field elements are represented as nine, unsigned 32-bit words.
|
||||
//
|
||||
// The value of a field element is:
|
||||
// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
|
||||
//
|
||||
// That is, each limb is alternately 29 or 28-bits wide in little-endian
|
||||
// order.
|
||||
//
|
||||
// This means that a field element hits 2**257, rather than 2**256 as we would
|
||||
// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
|
||||
// problems when multiplying as terms end up one bit short of a limb which
|
||||
// would require much bit-shifting to correct.
|
||||
//
|
||||
// Finally, the values stored in a field element are in Montgomery form. So the
|
||||
// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
|
||||
// 2**257.
|
||||
|
||||
const (
|
||||
p256Limbs = 9
|
||||
bottom29Bits = 0x1fffffff
|
||||
)
|
||||
|
||||
var (
|
||||
// p256One is the number 1 as a field element.
|
||||
p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
|
||||
p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
|
||||
// p256P is the prime modulus as a field element.
|
||||
p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
|
||||
// p2562P is the twice prime modulus as a field element.
|
||||
p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
|
||||
)
|
||||
|
||||
// Field element operations:
|
||||
|
||||
const bottom28Bits = 0xfffffff
|
||||
|
||||
// nonZeroToAllOnes returns:
|
||||
//
|
||||
// 0xffffffff for 0 < x <= 2**31
|
||||
// 0 for x == 0 or x > 2**31.
|
||||
func nonZeroToAllOnes(x uint32) uint32 {
|
||||
return ((x - 1) >> 31) - 1
|
||||
}
|
||||
|
||||
// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
|
||||
// which is a term at 2**257.
|
||||
//
|
||||
// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
|
||||
// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
|
||||
func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
|
||||
carry_mask := nonZeroToAllOnes(carry)
|
||||
|
||||
inout[0] += carry << 1
|
||||
inout[3] += 0x10000000 & carry_mask
|
||||
// carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
|
||||
// previous line therefore this doesn't underflow.
|
||||
inout[3] -= carry << 11
|
||||
inout[4] += (0x20000000 - 1) & carry_mask
|
||||
inout[5] += (0x10000000 - 1) & carry_mask
|
||||
inout[6] += (0x20000000 - 1) & carry_mask
|
||||
inout[6] -= carry << 22
|
||||
// This may underflow if carry is non-zero but, if so, we'll fix it in the
|
||||
// next line.
|
||||
inout[7] -= 1 & carry_mask
|
||||
inout[7] += carry << 25
|
||||
}
|
||||
|
||||
// p256Sum sets out = in+in2.
|
||||
//
|
||||
// On entry: in[i]+in2[i] must not overflow a 32-bit word.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Sum(out, in, in2 *[p256Limbs]uint32) {
|
||||
carry := uint32(0)
|
||||
for i := 0; ; i++ {
|
||||
out[i] = in[i] + in2[i]
|
||||
out[i] += carry
|
||||
carry = out[i] >> 29
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
out[i] = in[i] + in2[i]
|
||||
out[i] += carry
|
||||
carry = out[i] >> 28
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
const (
|
||||
two30m2 = 1<<30 - 1<<2
|
||||
two30p13m2 = 1<<30 + 1<<13 - 1<<2
|
||||
two31m2 = 1<<31 - 1<<2
|
||||
two31m3 = 1<<31 - 1<<3
|
||||
two31p24m2 = 1<<31 + 1<<24 - 1<<2
|
||||
two30m27m2 = 1<<30 - 1<<27 - 1<<2
|
||||
)
|
||||
|
||||
// p256Zero31 is 0 mod p.
|
||||
var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
|
||||
|
||||
// p256Diff sets out = in-in2.
|
||||
//
|
||||
// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
|
||||
// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Diff(out, in, in2 *[p256Limbs]uint32) {
|
||||
var carry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
out[i] = in[i] - in2[i]
|
||||
out[i] += p256Zero31[i]
|
||||
out[i] += carry
|
||||
carry = out[i] >> 29
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
out[i] = in[i] - in2[i]
|
||||
out[i] += p256Zero31[i]
|
||||
out[i] += carry
|
||||
carry = out[i] >> 28
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
|
||||
// the same 29,28,... bit positions as a field element.
|
||||
//
|
||||
// The values in field elements are in Montgomery form: x*R mod p where R =
|
||||
// 2**257. Since we just multiplied two Montgomery values together, the result
|
||||
// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
|
||||
// in Montgomery form.
|
||||
//
|
||||
// On entry: tmp[i] < 2**64.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
|
||||
// The following table may be helpful when reading this code:
|
||||
//
|
||||
// Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
|
||||
// Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
|
||||
// Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
|
||||
// (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
|
||||
var tmp2 [18]uint32
|
||||
var carry, x, xMask uint32
|
||||
|
||||
// tmp contains 64-bit words with the same 29,28,29-bit positions as a
|
||||
// field element. So the top of an element of tmp might overlap with
|
||||
// another element two positions down. The following loop eliminates
|
||||
// this overlap.
|
||||
tmp2[0] = uint32(tmp[0]) & bottom29Bits
|
||||
|
||||
tmp2[1] = uint32(tmp[0]) >> 29
|
||||
tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
|
||||
tmp2[1] += uint32(tmp[1]) & bottom28Bits
|
||||
carry = tmp2[1] >> 28
|
||||
tmp2[1] &= bottom28Bits
|
||||
|
||||
for i := 2; i < 17; i++ {
|
||||
tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
|
||||
tmp2[i] += (uint32(tmp[i-1])) >> 28
|
||||
tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
|
||||
tmp2[i] += uint32(tmp[i]) & bottom29Bits
|
||||
tmp2[i] += carry
|
||||
carry = tmp2[i] >> 29
|
||||
tmp2[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == 17 {
|
||||
break
|
||||
}
|
||||
tmp2[i] = uint32(tmp[i-2]>>32) >> 25
|
||||
tmp2[i] += uint32(tmp[i-1]) >> 29
|
||||
tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
|
||||
tmp2[i] += uint32(tmp[i]) & bottom28Bits
|
||||
tmp2[i] += carry
|
||||
carry = tmp2[i] >> 28
|
||||
tmp2[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
tmp2[17] = uint32(tmp[15]>>32) >> 25
|
||||
tmp2[17] += uint32(tmp[16]) >> 29
|
||||
tmp2[17] += uint32(tmp[16]>>32) << 3
|
||||
tmp2[17] += carry
|
||||
|
||||
// Montgomery elimination of terms:
|
||||
//
|
||||
// Since R is 2**257, we can divide by R with a bitwise shift if we can
|
||||
// ensure that the right-most 257 bits are all zero. We can make that true
|
||||
// by adding multiplies of p without affecting the value.
|
||||
//
|
||||
// So we eliminate limbs from right to left. Since the bottom 29 bits of p
|
||||
// are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
|
||||
// We can do that for 8 further limbs and then right shift to eliminate the
|
||||
// extra factor of R.
|
||||
for i := 0; ; i += 2 {
|
||||
tmp2[i+1] += tmp2[i] >> 29
|
||||
x = tmp2[i] & bottom29Bits
|
||||
xMask = nonZeroToAllOnes(x)
|
||||
tmp2[i] = 0
|
||||
|
||||
// The bounds calculations for this loop are tricky. Each iteration of
|
||||
// the loop eliminates two words by adding values to words to their
|
||||
// right.
|
||||
//
|
||||
// The following table contains the amounts added to each word (as an
|
||||
// offset from the value of i at the top of the loop). The amounts are
|
||||
// accounted for from the first and second half of the loop separately
|
||||
// and are written as, for example, 28 to mean a value <2**28.
|
||||
//
|
||||
// Word: 3 4 5 6 7 8 9 10
|
||||
// Added in top half: 28 11 29 21 29 28
|
||||
// 28 29
|
||||
// 29
|
||||
// Added in bottom half: 29 10 28 21 28 28
|
||||
// 29
|
||||
//
|
||||
// The value that is currently offset 7 will be offset 5 for the next
|
||||
// iteration and then offset 3 for the iteration after that. Therefore
|
||||
// the total value added will be the values added at 7, 5 and 3.
|
||||
//
|
||||
// The following table accumulates these values. The sums at the bottom
|
||||
// are written as, for example, 29+28, to mean a value < 2**29+2**28.
|
||||
//
|
||||
// Word: 3 4 5 6 7 8 9 10 11 12 13
|
||||
// 28 11 10 29 21 29 28 28 28 28 28
|
||||
// 29 28 11 28 29 28 29 28 29 28
|
||||
// 29 28 21 21 29 21 29 21
|
||||
// 10 29 28 21 28 21 28
|
||||
// 28 29 28 29 28 29 28
|
||||
// 11 10 29 10 29 10
|
||||
// 29 28 11 28 11
|
||||
// 29 29
|
||||
// --------------------------------------------
|
||||
// 30+ 31+ 30+ 31+ 30+
|
||||
// 28+ 29+ 28+ 29+ 21+
|
||||
// 21+ 28+ 21+ 28+ 10
|
||||
// 10 21+ 10 21+
|
||||
// 11 11
|
||||
//
|
||||
// So the greatest amount is added to tmp2[10] and tmp2[12]. If
|
||||
// tmp2[10/12] has an initial value of <2**29, then the maximum value
|
||||
// will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
|
||||
// as required.
|
||||
tmp2[i+3] += (x << 10) & bottom28Bits
|
||||
tmp2[i+4] += (x >> 18)
|
||||
|
||||
tmp2[i+6] += (x << 21) & bottom29Bits
|
||||
tmp2[i+7] += x >> 8
|
||||
|
||||
// At position 200, which is the starting bit position for word 7, we
|
||||
// have a factor of 0xf000000 = 2**28 - 2**24.
|
||||
tmp2[i+7] += 0x10000000 & xMask
|
||||
tmp2[i+8] += (x - 1) & xMask
|
||||
tmp2[i+7] -= (x << 24) & bottom28Bits
|
||||
tmp2[i+8] -= x >> 4
|
||||
|
||||
tmp2[i+8] += 0x20000000 & xMask
|
||||
tmp2[i+8] -= x
|
||||
tmp2[i+8] += (x << 28) & bottom29Bits
|
||||
tmp2[i+9] += ((x >> 1) - 1) & xMask
|
||||
|
||||
if i+1 == p256Limbs {
|
||||
break
|
||||
}
|
||||
tmp2[i+2] += tmp2[i+1] >> 28
|
||||
x = tmp2[i+1] & bottom28Bits
|
||||
xMask = nonZeroToAllOnes(x)
|
||||
tmp2[i+1] = 0
|
||||
|
||||
tmp2[i+4] += (x << 11) & bottom29Bits
|
||||
tmp2[i+5] += (x >> 18)
|
||||
|
||||
tmp2[i+7] += (x << 21) & bottom28Bits
|
||||
tmp2[i+8] += x >> 7
|
||||
|
||||
// At position 199, which is the starting bit of the 8th word when
|
||||
// dealing with a context starting on an odd word, we have a factor of
|
||||
// 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
|
||||
// word from i+1 is i+8.
|
||||
tmp2[i+8] += 0x20000000 & xMask
|
||||
tmp2[i+9] += (x - 1) & xMask
|
||||
tmp2[i+8] -= (x << 25) & bottom29Bits
|
||||
tmp2[i+9] -= x >> 4
|
||||
|
||||
tmp2[i+9] += 0x10000000 & xMask
|
||||
tmp2[i+9] -= x
|
||||
tmp2[i+10] += (x - 1) & xMask
|
||||
}
|
||||
|
||||
// We merge the right shift with a carry chain. The words above 2**257 have
|
||||
// widths of 28,29,... which we need to correct when copying them down.
|
||||
carry = 0
|
||||
for i := 0; i < 8; i++ {
|
||||
// The maximum value of tmp2[i + 9] occurs on the first iteration and
|
||||
// is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
|
||||
// therefore safe.
|
||||
out[i] = tmp2[i+9]
|
||||
out[i] += carry
|
||||
out[i] += (tmp2[i+10] << 28) & bottom29Bits
|
||||
carry = out[i] >> 29
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
out[i] = tmp2[i+9] >> 1
|
||||
out[i] += carry
|
||||
carry = out[i] >> 28
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
out[8] = tmp2[17]
|
||||
out[8] += carry
|
||||
carry = out[8] >> 29
|
||||
out[8] &= bottom29Bits
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256Square sets out=in*in.
|
||||
//
|
||||
// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Square(out, in *[p256Limbs]uint32) {
|
||||
var tmp [17]uint64
|
||||
|
||||
tmp[0] = uint64(in[0]) * uint64(in[0])
|
||||
tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
|
||||
tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
|
||||
uint64(in[1])*(uint64(in[1])<<1)
|
||||
tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
|
||||
uint64(in[1])*(uint64(in[2])<<1)
|
||||
tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
|
||||
uint64(in[1])*(uint64(in[3])<<2) +
|
||||
uint64(in[2])*uint64(in[2])
|
||||
tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
|
||||
uint64(in[1])*(uint64(in[4])<<1) +
|
||||
uint64(in[2])*(uint64(in[3])<<1)
|
||||
tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
|
||||
uint64(in[1])*(uint64(in[5])<<2) +
|
||||
uint64(in[2])*(uint64(in[4])<<1) +
|
||||
uint64(in[3])*(uint64(in[3])<<1)
|
||||
tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
|
||||
uint64(in[1])*(uint64(in[6])<<1) +
|
||||
uint64(in[2])*(uint64(in[5])<<1) +
|
||||
uint64(in[3])*(uint64(in[4])<<1)
|
||||
// tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
|
||||
// which is < 2**64 as required.
|
||||
tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
|
||||
uint64(in[1])*(uint64(in[7])<<2) +
|
||||
uint64(in[2])*(uint64(in[6])<<1) +
|
||||
uint64(in[3])*(uint64(in[5])<<2) +
|
||||
uint64(in[4])*uint64(in[4])
|
||||
tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
|
||||
uint64(in[2])*(uint64(in[7])<<1) +
|
||||
uint64(in[3])*(uint64(in[6])<<1) +
|
||||
uint64(in[4])*(uint64(in[5])<<1)
|
||||
tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
|
||||
uint64(in[3])*(uint64(in[7])<<2) +
|
||||
uint64(in[4])*(uint64(in[6])<<1) +
|
||||
uint64(in[5])*(uint64(in[5])<<1)
|
||||
tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
|
||||
uint64(in[4])*(uint64(in[7])<<1) +
|
||||
uint64(in[5])*(uint64(in[6])<<1)
|
||||
tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
|
||||
uint64(in[5])*(uint64(in[7])<<2) +
|
||||
uint64(in[6])*uint64(in[6])
|
||||
tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
|
||||
uint64(in[6])*(uint64(in[7])<<1)
|
||||
tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
|
||||
uint64(in[7])*(uint64(in[7])<<1)
|
||||
tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
|
||||
tmp[16] = uint64(in[8]) * uint64(in[8])
|
||||
|
||||
p256ReduceDegree(out, tmp)
|
||||
}
|
||||
|
||||
// p256Mul sets out=in*in2.
|
||||
//
|
||||
// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
|
||||
// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Mul(out, in, in2 *[p256Limbs]uint32) {
|
||||
var tmp [17]uint64
|
||||
|
||||
tmp[0] = uint64(in[0]) * uint64(in2[0])
|
||||
tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
|
||||
uint64(in[1])*(uint64(in2[0])<<0)
|
||||
tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
|
||||
uint64(in[1])*(uint64(in2[1])<<1) +
|
||||
uint64(in[2])*(uint64(in2[0])<<0)
|
||||
tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
|
||||
uint64(in[1])*(uint64(in2[2])<<0) +
|
||||
uint64(in[2])*(uint64(in2[1])<<0) +
|
||||
uint64(in[3])*(uint64(in2[0])<<0)
|
||||
tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
|
||||
uint64(in[1])*(uint64(in2[3])<<1) +
|
||||
uint64(in[2])*(uint64(in2[2])<<0) +
|
||||
uint64(in[3])*(uint64(in2[1])<<1) +
|
||||
uint64(in[4])*(uint64(in2[0])<<0)
|
||||
tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
|
||||
uint64(in[1])*(uint64(in2[4])<<0) +
|
||||
uint64(in[2])*(uint64(in2[3])<<0) +
|
||||
uint64(in[3])*(uint64(in2[2])<<0) +
|
||||
uint64(in[4])*(uint64(in2[1])<<0) +
|
||||
uint64(in[5])*(uint64(in2[0])<<0)
|
||||
tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
|
||||
uint64(in[1])*(uint64(in2[5])<<1) +
|
||||
uint64(in[2])*(uint64(in2[4])<<0) +
|
||||
uint64(in[3])*(uint64(in2[3])<<1) +
|
||||
uint64(in[4])*(uint64(in2[2])<<0) +
|
||||
uint64(in[5])*(uint64(in2[1])<<1) +
|
||||
uint64(in[6])*(uint64(in2[0])<<0)
|
||||
tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
|
||||
uint64(in[1])*(uint64(in2[6])<<0) +
|
||||
uint64(in[2])*(uint64(in2[5])<<0) +
|
||||
uint64(in[3])*(uint64(in2[4])<<0) +
|
||||
uint64(in[4])*(uint64(in2[3])<<0) +
|
||||
uint64(in[5])*(uint64(in2[2])<<0) +
|
||||
uint64(in[6])*(uint64(in2[1])<<0) +
|
||||
uint64(in[7])*(uint64(in2[0])<<0)
|
||||
// tmp[8] has the greatest value but doesn't overflow. See logic in
|
||||
// p256Square.
|
||||
tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
|
||||
uint64(in[1])*(uint64(in2[7])<<1) +
|
||||
uint64(in[2])*(uint64(in2[6])<<0) +
|
||||
uint64(in[3])*(uint64(in2[5])<<1) +
|
||||
uint64(in[4])*(uint64(in2[4])<<0) +
|
||||
uint64(in[5])*(uint64(in2[3])<<1) +
|
||||
uint64(in[6])*(uint64(in2[2])<<0) +
|
||||
uint64(in[7])*(uint64(in2[1])<<1) +
|
||||
uint64(in[8])*(uint64(in2[0])<<0)
|
||||
tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
|
||||
uint64(in[2])*(uint64(in2[7])<<0) +
|
||||
uint64(in[3])*(uint64(in2[6])<<0) +
|
||||
uint64(in[4])*(uint64(in2[5])<<0) +
|
||||
uint64(in[5])*(uint64(in2[4])<<0) +
|
||||
uint64(in[6])*(uint64(in2[3])<<0) +
|
||||
uint64(in[7])*(uint64(in2[2])<<0) +
|
||||
uint64(in[8])*(uint64(in2[1])<<0)
|
||||
tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
|
||||
uint64(in[3])*(uint64(in2[7])<<1) +
|
||||
uint64(in[4])*(uint64(in2[6])<<0) +
|
||||
uint64(in[5])*(uint64(in2[5])<<1) +
|
||||
uint64(in[6])*(uint64(in2[4])<<0) +
|
||||
uint64(in[7])*(uint64(in2[3])<<1) +
|
||||
uint64(in[8])*(uint64(in2[2])<<0)
|
||||
tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
|
||||
uint64(in[4])*(uint64(in2[7])<<0) +
|
||||
uint64(in[5])*(uint64(in2[6])<<0) +
|
||||
uint64(in[6])*(uint64(in2[5])<<0) +
|
||||
uint64(in[7])*(uint64(in2[4])<<0) +
|
||||
uint64(in[8])*(uint64(in2[3])<<0)
|
||||
tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
|
||||
uint64(in[5])*(uint64(in2[7])<<1) +
|
||||
uint64(in[6])*(uint64(in2[6])<<0) +
|
||||
uint64(in[7])*(uint64(in2[5])<<1) +
|
||||
uint64(in[8])*(uint64(in2[4])<<0)
|
||||
tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
|
||||
uint64(in[6])*(uint64(in2[7])<<0) +
|
||||
uint64(in[7])*(uint64(in2[6])<<0) +
|
||||
uint64(in[8])*(uint64(in2[5])<<0)
|
||||
tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
|
||||
uint64(in[7])*(uint64(in2[7])<<1) +
|
||||
uint64(in[8])*(uint64(in2[6])<<0)
|
||||
tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
|
||||
uint64(in[8])*(uint64(in2[7])<<0)
|
||||
tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
|
||||
|
||||
p256ReduceDegree(out, tmp)
|
||||
}
|
||||
|
||||
func p256Assign(out, in *[p256Limbs]uint32) {
|
||||
*out = *in
|
||||
}
|
||||
|
||||
// p256Invert calculates |out| = |in|^{-1}
|
||||
//
|
||||
// Based on Fermat's Little Theorem:
|
||||
//
|
||||
// a^p = a (mod p)
|
||||
// a^{p-1} = 1 (mod p)
|
||||
// a^{p-2} = a^{-1} (mod p)
|
||||
func p256Invert(out, in *[p256Limbs]uint32) {
|
||||
var ftmp, ftmp2 [p256Limbs]uint32
|
||||
|
||||
// each e_I will hold |in|^{2^I - 1}
|
||||
var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
|
||||
|
||||
p256Square(&ftmp, in) // 2^1
|
||||
p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
|
||||
p256Assign(&e2, &ftmp)
|
||||
p256Square(&ftmp, &ftmp) // 2^3 - 2^1
|
||||
p256Square(&ftmp, &ftmp) // 2^4 - 2^2
|
||||
p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
|
||||
p256Assign(&e4, &ftmp)
|
||||
p256Square(&ftmp, &ftmp) // 2^5 - 2^1
|
||||
p256Square(&ftmp, &ftmp) // 2^6 - 2^2
|
||||
p256Square(&ftmp, &ftmp) // 2^7 - 2^3
|
||||
p256Square(&ftmp, &ftmp) // 2^8 - 2^4
|
||||
p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
|
||||
p256Assign(&e8, &ftmp)
|
||||
for i := 0; i < 8; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^16 - 2^8
|
||||
p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
|
||||
p256Assign(&e16, &ftmp)
|
||||
for i := 0; i < 16; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^32 - 2^16
|
||||
p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
|
||||
p256Assign(&e32, &ftmp)
|
||||
for i := 0; i < 32; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^64 - 2^32
|
||||
p256Assign(&e64, &ftmp)
|
||||
p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
|
||||
for i := 0; i < 192; i++ {
|
||||
p256Square(&ftmp, &ftmp)
|
||||
} // 2^256 - 2^224 + 2^192
|
||||
|
||||
p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
|
||||
for i := 0; i < 16; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^80 - 2^16
|
||||
p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
|
||||
for i := 0; i < 8; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^88 - 2^8
|
||||
p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
|
||||
for i := 0; i < 4; i++ {
|
||||
p256Square(&ftmp2, &ftmp2)
|
||||
} // 2^92 - 2^4
|
||||
p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
|
||||
p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
|
||||
p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
|
||||
p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
|
||||
p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
|
||||
p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
|
||||
p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
|
||||
|
||||
p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
|
||||
}
|
||||
|
||||
// p256Scalar3 sets out=3*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar3(out *[p256Limbs]uint32) {
|
||||
var carry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
out[i] *= 3
|
||||
out[i] += carry
|
||||
carry = out[i] >> 29
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
out[i] *= 3
|
||||
out[i] += carry
|
||||
carry = out[i] >> 28
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256Scalar4 sets out=4*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar4(out *[p256Limbs]uint32) {
|
||||
var carry, nextCarry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
nextCarry = out[i] >> 27
|
||||
out[i] <<= 2
|
||||
out[i] &= bottom29Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 29)
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
nextCarry = out[i] >> 26
|
||||
out[i] <<= 2
|
||||
out[i] &= bottom28Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 28)
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256Scalar8 sets out=8*out.
|
||||
//
|
||||
// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
|
||||
func p256Scalar8(out *[p256Limbs]uint32) {
|
||||
var carry, nextCarry uint32
|
||||
|
||||
for i := 0; ; i++ {
|
||||
nextCarry = out[i] >> 26
|
||||
out[i] <<= 3
|
||||
out[i] &= bottom29Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 29)
|
||||
out[i] &= bottom29Bits
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
nextCarry = out[i] >> 25
|
||||
out[i] <<= 3
|
||||
out[i] &= bottom28Bits
|
||||
out[i] += carry
|
||||
carry = nextCarry + (out[i] >> 28)
|
||||
out[i] &= bottom28Bits
|
||||
}
|
||||
|
||||
p256ReduceCarry(out, carry)
|
||||
}
|
||||
|
||||
// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
|
||||
//
|
||||
// On entry: mask is either 0 or 0xffffffff.
|
||||
func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
|
||||
for i := 0; i < p256Limbs; i++ {
|
||||
tmp := mask & (in[i] ^ out[i])
|
||||
out[i] ^= tmp
|
||||
}
|
||||
}
|
||||
|
||||
// p256FromBig sets out = R*in.
|
||||
func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
|
||||
tmp := new(big.Int).Lsh(in, 257)
|
||||
tmp.Mod(tmp, p256Params.P)
|
||||
|
||||
for i := 0; i < p256Limbs; i++ {
|
||||
if bits := tmp.Bits(); len(bits) > 0 {
|
||||
out[i] = uint32(bits[0]) & bottom29Bits
|
||||
} else {
|
||||
out[i] = 0
|
||||
}
|
||||
tmp.Rsh(tmp, 29)
|
||||
|
||||
i++
|
||||
if i == p256Limbs {
|
||||
break
|
||||
}
|
||||
|
||||
if bits := tmp.Bits(); len(bits) > 0 {
|
||||
out[i] = uint32(bits[0]) & bottom28Bits
|
||||
} else {
|
||||
out[i] = 0
|
||||
}
|
||||
tmp.Rsh(tmp, 28)
|
||||
}
|
||||
}
|
||||
|
||||
// p256ToBig returns a *big.Int containing the value of in.
|
||||
func p256ToBig(in *[p256Limbs]uint32) *big.Int {
|
||||
result, tmp := new(big.Int), new(big.Int)
|
||||
|
||||
result.SetInt64(int64(in[p256Limbs-1]))
|
||||
for i := p256Limbs - 2; i >= 0; i-- {
|
||||
if (i & 1) == 0 {
|
||||
result.Lsh(result, 29)
|
||||
} else {
|
||||
result.Lsh(result, 28)
|
||||
}
|
||||
tmp.SetInt64(int64(in[i]))
|
||||
result.Add(result, tmp)
|
||||
}
|
||||
|
||||
result.Mul(result, p256RInverse)
|
||||
result.Mod(result, p256Params.P)
|
||||
return result
|
||||
}
|
||||
Loading…
Reference in New Issue
Block a user