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  // mul64 returns a * b.
    16  func mul64(a, b uint64) uint128 {
    17  	hi, lo := bits.Mul64(a, b)
    18  	return uint128{lo, hi}
    19  }
    20  
    21  // addMul64 returns v + a * b.
    22  func addMul64(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  // shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits.
    30  func shiftRightBy51(a uint128) uint64 {
    31  	return (a.hi << (64 - 51)) | (a.lo >> 51)
    32  }
    33  
    34  func feMulGeneric(v, a, b *Element) {
    35  	a0 := a.l0
    36  	a1 := a.l1
    37  	a2 := a.l2
    38  	a3 := a.l3
    39  	a4 := a.l4
    40  
    41  	b0 := b.l0
    42  	b1 := b.l1
    43  	b2 := b.l2
    44  	b3 := b.l3
    45  	b4 := b.l4
    46  
    47  	// Limb multiplication works like pen-and-paper columnar multiplication, but
    48  	// with 51-bit limbs instead of digits.
    49  	//
    50  	//                          a4   a3   a2   a1   a0  x
    51  	//                          b4   b3   b2   b1   b0  =
    52  	//                         ------------------------
    53  	//                        a4b0 a3b0 a2b0 a1b0 a0b0  +
    54  	//                   a4b1 a3b1 a2b1 a1b1 a0b1       +
    55  	//              a4b2 a3b2 a2b2 a1b2 a0b2            +
    56  	//         a4b3 a3b3 a2b3 a1b3 a0b3                 +
    57  	//    a4b4 a3b4 a2b4 a1b4 a0b4                      =
    58  	//   ----------------------------------------------
    59  	//      r8   r7   r6   r5   r4   r3   r2   r1   r0
    60  	//
    61  	// We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to
    62  	// reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5,
    63  	// r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc.
    64  	//
    65  	// Reduction can be carried out simultaneously to multiplication. For
    66  	// example, we do not compute r5: whenever the result of a multiplication
    67  	// belongs to r5, like a1b4, we multiply it by 19 and add the result to r0.
    68  	//
    69  	//            a4b0    a3b0    a2b0    a1b0    a0b0  +
    70  	//            a3b1    a2b1    a1b1    a0b1 19×a4b1  +
    71  	//            a2b2    a1b2    a0b2 19×a4b2 19×a3b2  +
    72  	//            a1b3    a0b3 19×a4b3 19×a3b3 19×a2b3  +
    73  	//            a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4  =
    74  	//           --------------------------------------
    75  	//              r4      r3      r2      r1      r0
    76  	//
    77  	// Finally we add up the columns into wide, overlapping limbs.
    78  
    79  	a1_19 := a1 * 19
    80  	a2_19 := a2 * 19
    81  	a3_19 := a3 * 19
    82  	a4_19 := a4 * 19
    83  
    84  	// r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
    85  	r0 := mul64(a0, b0)
    86  	r0 = addMul64(r0, a1_19, b4)
    87  	r0 = addMul64(r0, a2_19, b3)
    88  	r0 = addMul64(r0, a3_19, b2)
    89  	r0 = addMul64(r0, a4_19, b1)
    90  
    91  	// r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
    92  	r1 := mul64(a0, b1)
    93  	r1 = addMul64(r1, a1, b0)
    94  	r1 = addMul64(r1, a2_19, b4)
    95  	r1 = addMul64(r1, a3_19, b3)
    96  	r1 = addMul64(r1, a4_19, b2)
    97  
    98  	// r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
    99  	r2 := mul64(a0, b2)
   100  	r2 = addMul64(r2, a1, b1)
   101  	r2 = addMul64(r2, a2, b0)
   102  	r2 = addMul64(r2, a3_19, b4)
   103  	r2 = addMul64(r2, a4_19, b3)
   104  
   105  	// r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
   106  	r3 := mul64(a0, b3)
   107  	r3 = addMul64(r3, a1, b2)
   108  	r3 = addMul64(r3, a2, b1)
   109  	r3 = addMul64(r3, a3, b0)
   110  	r3 = addMul64(r3, a4_19, b4)
   111  
   112  	// r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
   113  	r4 := mul64(a0, b4)
   114  	r4 = addMul64(r4, a1, b3)
   115  	r4 = addMul64(r4, a2, b2)
   116  	r4 = addMul64(r4, a3, b1)
   117  	r4 = addMul64(r4, a4, b0)
   118  
   119  	// After the multiplication, we need to reduce (carry) the five coefficients
   120  	// to obtain a result with limbs that are at most slightly larger than 2⁵¹,
   121  	// to respect the Element invariant.
   122  	//
   123  	// Overall, the reduction works the same as carryPropagate, except with
   124  	// wider inputs: we take the carry for each coefficient by shifting it right
   125  	// by 51, and add it to the limb above it. The top carry is multiplied by 19
   126  	// according to the reduction identity and added to the lowest limb.
   127  	//
   128  	// The largest coefficient (r0) will be at most 111 bits, which guarantees
   129  	// that all carries are at most 111 - 51 = 60 bits, which fits in a uint64.
   130  	//
   131  	//     r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
   132  	//     r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²)
   133  	//     r0 < (1 + 19 × 4) × 2⁵² × 2⁵²
   134  	//     r0 < 2⁷ × 2⁵² × 2⁵²
   135  	//     r0 < 2¹¹¹
   136  	//
   137  	// Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most
   138  	// 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and
   139  	// allows us to easily apply the reduction identity.
   140  	//
   141  	//     r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
   142  	//     r4 < 5 × 2⁵² × 2⁵²
   143  	//     r4 < 2¹⁰⁷
   144  	//
   145  
   146  	c0 := shiftRightBy51(r0)
   147  	c1 := shiftRightBy51(r1)
   148  	c2 := shiftRightBy51(r2)
   149  	c3 := shiftRightBy51(r3)
   150  	c4 := shiftRightBy51(r4)
   151  
   152  	rr0 := r0.lo&maskLow51Bits + c4*19
   153  	rr1 := r1.lo&maskLow51Bits + c0
   154  	rr2 := r2.lo&maskLow51Bits + c1
   155  	rr3 := r3.lo&maskLow51Bits + c2
   156  	rr4 := r4.lo&maskLow51Bits + c3
   157  
   158  	// Now all coefficients fit into 64-bit registers but are still too large to
   159  	// be passed around as an Element. We therefore do one last carry chain,
   160  	// where the carries will be small enough to fit in the wiggle room above 2⁵¹.
   161  	*v = Element{rr0, rr1, rr2, rr3, rr4}
   162  	v.carryPropagate()
   163  }
   164  
   165  func feSquareGeneric(v, a *Element) {
   166  	l0 := a.l0
   167  	l1 := a.l1
   168  	l2 := a.l2
   169  	l3 := a.l3
   170  	l4 := a.l4
   171  
   172  	// Squaring works precisely like multiplication above, but thanks to its
   173  	// symmetry we get to group a few terms together.
   174  	//
   175  	//                          l4   l3   l2   l1   l0  x
   176  	//                          l4   l3   l2   l1   l0  =
   177  	//                         ------------------------
   178  	//                        l4l0 l3l0 l2l0 l1l0 l0l0  +
   179  	//                   l4l1 l3l1 l2l1 l1l1 l0l1       +
   180  	//              l4l2 l3l2 l2l2 l1l2 l0l2            +
   181  	//         l4l3 l3l3 l2l3 l1l3 l0l3                 +
   182  	//    l4l4 l3l4 l2l4 l1l4 l0l4                      =
   183  	//   ----------------------------------------------
   184  	//      r8   r7   r6   r5   r4   r3   r2   r1   r0
   185  	//
   186  	//            l4l0    l3l0    l2l0    l1l0    l0l0  +
   187  	//            l3l1    l2l1    l1l1    l0l1 19×l4l1  +
   188  	//            l2l2    l1l2    l0l2 19×l4l2 19×l3l2  +
   189  	//            l1l3    l0l3 19×l4l3 19×l3l3 19×l2l3  +
   190  	//            l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4  =
   191  	//           --------------------------------------
   192  	//              r4      r3      r2      r1      r0
   193  	//
   194  	// With precomputed 2×, 19×, and 2×19× terms, we can compute each limb with
   195  	// only three Mul64 and four Add64, instead of five and eight.
   196  
   197  	l0_2 := l0 * 2
   198  	l1_2 := l1 * 2
   199  
   200  	l1_38 := l1 * 38
   201  	l2_38 := l2 * 38
   202  	l3_38 := l3 * 38
   203  
   204  	l3_19 := l3 * 19
   205  	l4_19 := l4 * 19
   206  
   207  	// r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3)
   208  	r0 := mul64(l0, l0)
   209  	r0 = addMul64(r0, l1_38, l4)
   210  	r0 = addMul64(r0, l2_38, l3)
   211  
   212  	// r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
   213  	r1 := mul64(l0_2, l1)
   214  	r1 = addMul64(r1, l2_38, l4)
   215  	r1 = addMul64(r1, l3_19, l3)
   216  
   217  	// r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4
   218  	r2 := mul64(l0_2, l2)
   219  	r2 = addMul64(r2, l1, l1)
   220  	r2 = addMul64(r2, l3_38, l4)
   221  
   222  	// r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
   223  	r3 := mul64(l0_2, l3)
   224  	r3 = addMul64(r3, l1_2, l2)
   225  	r3 = addMul64(r3, l4_19, l4)
   226  
   227  	// r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2
   228  	r4 := mul64(l0_2, l4)
   229  	r4 = addMul64(r4, l1_2, l3)
   230  	r4 = addMul64(r4, l2, l2)
   231  
   232  	c0 := shiftRightBy51(r0)
   233  	c1 := shiftRightBy51(r1)
   234  	c2 := shiftRightBy51(r2)
   235  	c3 := shiftRightBy51(r3)
   236  	c4 := shiftRightBy51(r4)
   237  
   238  	rr0 := r0.lo&maskLow51Bits + c4*19
   239  	rr1 := r1.lo&maskLow51Bits + c0
   240  	rr2 := r2.lo&maskLow51Bits + c1
   241  	rr3 := r3.lo&maskLow51Bits + c2
   242  	rr4 := r4.lo&maskLow51Bits + c3
   243  
   244  	*v = Element{rr0, rr1, rr2, rr3, rr4}
   245  	v.carryPropagate()
   246  }
   247  
   248  // carryPropagateGeneric brings the limbs below 52 bits by applying the reduction
   249  // identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry.
   250  func (v *Element) carryPropagateGeneric() *Element {
   251  	c0 := v.l0 >> 51
   252  	c1 := v.l1 >> 51
   253  	c2 := v.l2 >> 51
   254  	c3 := v.l3 >> 51
   255  	c4 := v.l4 >> 51
   256  
   257  	// c4 is at most 64 - 51 = 13 bits, so c4*19 is at most 18 bits, and
   258  	// the final l0 will be at most 52 bits. Similarly for the rest.
   259  	v.l0 = v.l0&maskLow51Bits + c4*19
   260  	v.l1 = v.l1&maskLow51Bits + c0
   261  	v.l2 = v.l2&maskLow51Bits + c1
   262  	v.l3 = v.l3&maskLow51Bits + c2
   263  	v.l4 = v.l4&maskLow51Bits + c3
   264  
   265  	return v
   266  }
   267
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