| // 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 |
| } |
