fe_generic.go

     1  // Copyright (c) 2017 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package field
     6  
     7  import "math/bits"
     8  
     9  // uint128 holds a 128-bit number as two 64-bit limbs, for use with the
    10  // bits.Mul64 and bits.Add64 intrinsics.
    11  type uint128 struct {
    12  	lo, hi uint64
    13  }
    14  
    15  // mul returns a * b.
    16  func mul(a, b uint64) uint128 {
    17  	hi, lo := bits.Mul64(a, b)
    18  	return uint128{lo, hi}
    19  }
    20  
    21  // addMul returns v + a * b.
    22  func addMul(v uint128, a, b uint64) uint128 {
    23  	hi, lo := bits.Mul64(a, b)
    24  	lo, c := bits.Add64(lo, v.lo, 0)
    25  	hi, _ = bits.Add64(hi, v.hi, c)
    26  	return uint128{lo, hi}
    27  }
    28  
    29  // mul19 returns v * 19.
    30  func mul19(v uint64) uint64 {
    31  	// Using this approach seems to yield better optimizations than *19.
    32  	return v + (v+v<<3)<<1
    33  }
    34  
    35  // addMul19 returns v + 19 * a * b, where a and b are at most 52 bits.
    36  func addMul19(v uint128, a, b uint64) uint128 {
    37  	hi, lo := bits.Mul64(mul19(a), b)
    38  	lo, c := bits.Add64(lo, v.lo, 0)
    39  	hi, _ = bits.Add64(hi, v.hi, c)
    40  	return uint128{lo, hi}
    41  }
    42  
    43  // addMul38 returns v + 38 * a * b, where a and b are at most 52 bits.
    44  func addMul38(v uint128, a, b uint64) uint128 {
    45  	hi, lo := bits.Mul64(mul19(a), b*2)
    46  	lo, c := bits.Add64(lo, v.lo, 0)
    47  	hi, _ = bits.Add64(hi, v.hi, c)
    48  	return uint128{lo, hi}
    49  }
    50  
    51  // shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits.
    52  func shiftRightBy51(a uint128) uint64 {
    53  	return (a.hi << (64 - 51)) | (a.lo >> 51)
    54  }
    55  
    56  func feMulGeneric(v, a, b *Element) {
    57  	a0 := a.l0
    58  	a1 := a.l1
    59  	a2 := a.l2
    60  	a3 := a.l3
    61  	a4 := a.l4
    62  
    63  	b0 := b.l0
    64  	b1 := b.l1
    65  	b2 := b.l2
    66  	b3 := b.l3
    67  	b4 := b.l4
    68  
    69  	// Limb multiplication works like pen-and-paper columnar multiplication, but
    70  	// with 51-bit limbs instead of digits.
    71  	//
    72  	//                          a4   a3   a2   a1   a0  x
    73  	//                          b4   b3   b2   b1   b0  =
    74  	//                         ------------------------
    75  	//                        a4b0 a3b0 a2b0 a1b0 a0b0  +
    76  	//                   a4b1 a3b1 a2b1 a1b1 a0b1       +
    77  	//              a4b2 a3b2 a2b2 a1b2 a0b2            +
    78  	//         a4b3 a3b3 a2b3 a1b3 a0b3                 +
    79  	//    a4b4 a3b4 a2b4 a1b4 a0b4                      =
    80  	//   ----------------------------------------------
    81  	//      r8   r7   r6   r5   r4   r3   r2   r1   r0
    82  	//
    83  	// We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to
    84  	// reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5,
    85  	// r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc.
    86  	//
    87  	// Reduction can be carried out simultaneously to multiplication. For
    88  	// example, we do not compute r5: whenever the result of a multiplication
    89  	// belongs to r5, like a1b4, we multiply it by 19 and add the result to r0.
    90  	//
    91  	//            a4b0    a3b0    a2b0    a1b0    a0b0  +
    92  	//            a3b1    a2b1    a1b1    a0b1 19×a4b1  +
    93  	//            a2b2    a1b2    a0b2 19×a4b2 19×a3b2  +
    94  	//            a1b3    a0b3 19×a4b3 19×a3b3 19×a2b3  +
    95  	//            a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4  =
    96  	//           --------------------------------------
    97  	//              r4      r3      r2      r1      r0
    98  	//
    99  	// Finally we add up the columns into wide, overlapping limbs.
   100  
   101  	// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
   102  	r0 := mul(a0, b0)
   103  	r0 = addMul19(r0, a1, b4)
   104  	r0 = addMul19(r0, a2, b3)
   105  	r0 = addMul19(r0, a3, b2)
   106  	r0 = addMul19(r0, a4, b1)
   107  
   108  	// r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
   109  	r1 := mul(a0, b1)
   110  	r1 = addMul(r1, a1, b0)
   111  	r1 = addMul19(r1, a2, b4)
   112  	r1 = addMul19(r1, a3, b3)
   113  	r1 = addMul19(r1, a4, b2)
   114  
   115  	// r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
   116  	r2 := mul(a0, b2)
   117  	r2 = addMul(r2, a1, b1)
   118  	r2 = addMul(r2, a2, b0)
   119  	r2 = addMul19(r2, a3, b4)
   120  	r2 = addMul19(r2, a4, b3)
   121  
   122  	// r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
   123  	r3 := mul(a0, b3)
   124  	r3 = addMul(r3, a1, b2)
   125  	r3 = addMul(r3, a2, b1)
   126  	r3 = addMul(r3, a3, b0)
   127  	r3 = addMul19(r3, a4, b4)
   128  
   129  	// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
   130  	r4 := mul(a0, b4)
   131  	r4 = addMul(r4, a1, b3)
   132  	r4 = addMul(r4, a2, b2)
   133  	r4 = addMul(r4, a3, b1)
   134  	r4 = addMul(r4, a4, b0)
   135  
   136  	// After the multiplication, we need to reduce (carry) the five coefficients
   137  	// to obtain a result with limbs that are at most slightly larger than 2⁵¹,
   138  	// to respect the Element invariant.
   139  	//
   140  	// Overall, the reduction works the same as carryPropagate, except with
   141  	// wider inputs: we take the carry for each coefficient by shifting it right
   142  	// by 51, and add it to the limb above it. The top carry is multiplied by 19
   143  	// according to the reduction identity and added to the lowest limb.
   144  	//
   145  	// The largest coefficient (r0) will be at most 111 bits, which guarantees
   146  	// that all carries are at most 111 - 51 = 60 bits, which fits in a uint64.
   147  	//
   148  	//     r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
   149  	//     r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²)
   150  	//     r0 < (1 + 19 × 4) × 2⁵² × 2⁵²
   151  	//     r0 < 2⁷ × 2⁵² × 2⁵²
   152  	//     r0 < 2¹¹¹
   153  	//
   154  	// Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most
   155  	// 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and
   156  	// allows us to easily apply the reduction identity.
   157  	//
   158  	//     r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
   159  	//     r4 < 5 × 2⁵² × 2⁵²
   160  	//     r4 < 2¹⁰⁷
   161  	//
   162  
   163  	c0 := shiftRightBy51(r0)
   164  	c1 := shiftRightBy51(r1)
   165  	c2 := shiftRightBy51(r2)
   166  	c3 := shiftRightBy51(r3)
   167  	c4 := shiftRightBy51(r4)
   168  
   169  	rr0 := r0.lo&maskLow51Bits + mul19(c4)
   170  	rr1 := r1.lo&maskLow51Bits + c0
   171  	rr2 := r2.lo&maskLow51Bits + c1
   172  	rr3 := r3.lo&maskLow51Bits + c2
   173  	rr4 := r4.lo&maskLow51Bits + c3
   174  
   175  	// Now all coefficients fit into 64-bit registers but are still too large to
   176  	// be passed around as an Element. We therefore do one last carry chain,
   177  	// where the carries will be small enough to fit in the wiggle room above 2⁵¹.
   178  
   179  	v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
   180  	v.l1 = rr1&maskLow51Bits + rr0>>51
   181  	v.l2 = rr2&maskLow51Bits + rr1>>51
   182  	v.l3 = rr3&maskLow51Bits + rr2>>51
   183  	v.l4 = rr4&maskLow51Bits + rr3>>51
   184  }
   185  
   186  func feSquareGeneric(v, a *Element) {
   187  	l0 := a.l0
   188  	l1 := a.l1
   189  	l2 := a.l2
   190  	l3 := a.l3
   191  	l4 := a.l4
   192  
   193  	// Squaring works precisely like multiplication above, but thanks to its
   194  	// symmetry we get to group a few terms together.
   195  	//
   196  	//                          l4   l3   l2   l1   l0  x
   197  	//                          l4   l3   l2   l1   l0  =
   198  	//                         ------------------------
   199  	//                        l4l0 l3l0 l2l0 l1l0 l0l0  +
   200  	//                   l4l1 l3l1 l2l1 l1l1 l0l1       +
   201  	//              l4l2 l3l2 l2l2 l1l2 l0l2            +
   202  	//         l4l3 l3l3 l2l3 l1l3 l0l3                 +
   203  	//    l4l4 l3l4 l2l4 l1l4 l0l4                      =
   204  	//   ----------------------------------------------
   205  	//      r8   r7   r6   r5   r4   r3   r2   r1   r0
   206  	//
   207  	//            l4l0    l3l0    l2l0    l1l0    l0l0  +
   208  	//            l3l1    l2l1    l1l1    l0l1 19×l4l1  +
   209  	//            l2l2    l1l2    l0l2 19×l4l2 19×l3l2  +
   210  	//            l1l3    l0l3 19×l4l3 19×l3l3 19×l2l3  +
   211  	//            l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4  =
   212  	//           --------------------------------------
   213  	//              r4      r3      r2      r1      r0
   214  
   215  	// r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3)
   216  	r0 := mul(l0, l0)
   217  	r0 = addMul38(r0, l1, l4)
   218  	r0 = addMul38(r0, l2, l3)
   219  
   220  	// r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
   221  	r1 := mul(l0*2, l1)
   222  	r1 = addMul38(r1, l2, l4)
   223  	r1 = addMul19(r1, l3, l3)
   224  
   225  	// r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4
   226  	r2 := mul(l0*2, l2)
   227  	r2 = addMul(r2, l1, l1)
   228  	r2 = addMul38(r2, l3, l4)
   229  
   230  	// r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
   231  	r3 := mul(l0*2, l3)
   232  	r3 = addMul(r3, l1*2, l2)
   233  	r3 = addMul19(r3, l4, l4)
   234  
   235  	// r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2
   236  	r4 := mul(l0*2, l4)
   237  	r4 = addMul(r4, l1*2, l3)
   238  	r4 = addMul(r4, l2, l2)
   239  
   240  	c0 := shiftRightBy51(r0)
   241  	c1 := shiftRightBy51(r1)
   242  	c2 := shiftRightBy51(r2)
   243  	c3 := shiftRightBy51(r3)
   244  	c4 := shiftRightBy51(r4)
   245  
   246  	rr0 := r0.lo&maskLow51Bits + mul19(c4)
   247  	rr1 := r1.lo&maskLow51Bits + c0
   248  	rr2 := r2.lo&maskLow51Bits + c1
   249  	rr3 := r3.lo&maskLow51Bits + c2
   250  	rr4 := r4.lo&maskLow51Bits + c3
   251  
   252  	v.l0 = rr0&maskLow51Bits + mul19(rr4>>51)
   253  	v.l1 = rr1&maskLow51Bits + rr0>>51
   254  	v.l2 = rr2&maskLow51Bits + rr1>>51
   255  	v.l3 = rr3&maskLow51Bits + rr2>>51
   256  	v.l4 = rr4&maskLow51Bits + rr3>>51
   257  }
   258  
   259  // carryPropagate brings the limbs below 52 bits by applying the reduction
   260  // identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry.
   261  func (v *Element) carryPropagate() *Element {
   262  	// (l4>>51) is at most 64 - 51 = 13 bits, so (l4>>51)*19 is at most 18 bits, and
   263  	// the final l0 will be at most 52 bits. Similarly for the rest.
   264  	l0 := v.l0
   265  	v.l0 = v.l0&maskLow51Bits + mul19(v.l4>>51)
   266  	v.l4 = v.l4&maskLow51Bits + v.l3>>51
   267  	v.l3 = v.l3&maskLow51Bits + v.l2>>51
   268  	v.l2 = v.l2&maskLow51Bits + v.l1>>51
   269  	v.l1 = v.l1&maskLow51Bits + l0>>51
   270  
   271  	return v
   272  }
   273
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