curve.go raw

   1  package x448
   2  
   3  import (
   4  	fp "github.com/cloudflare/circl/math/fp448"
   5  )
   6  
   7  // ladderJoye calculates a fixed-point multiplication with the generator point.
   8  // The algorithm is the right-to-left Joye's ladder as described
   9  // in "How to precompute a ladder" in SAC'2017.
  10  func ladderJoye(k *Key) {
  11  	w := [5]fp.Elt{} // [mu,x1,z1,x2,z2] order must be preserved.
  12  	w[1] = fp.Elt{   // x1 = S
  13  		0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  14  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  15  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  16  		0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff,
  17  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  18  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  19  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  20  	}
  21  	fp.SetOne(&w[2]) // z1 = 1
  22  	w[3] = fp.Elt{   // x2 = G-S
  23  		0x20, 0x27, 0x9d, 0xc9, 0x7d, 0x19, 0xb1, 0xac,
  24  		0xf8, 0xba, 0x69, 0x1c, 0xff, 0x33, 0xac, 0x23,
  25  		0x51, 0x1b, 0xce, 0x3a, 0x64, 0x65, 0xbd, 0xf1,
  26  		0x23, 0xf8, 0xc1, 0x84, 0x9d, 0x45, 0x54, 0x29,
  27  		0x67, 0xb9, 0x81, 0x1c, 0x03, 0xd1, 0xcd, 0xda,
  28  		0x7b, 0xeb, 0xff, 0x1a, 0x88, 0x03, 0xcf, 0x3a,
  29  		0x42, 0x44, 0x32, 0x01, 0x25, 0xb7, 0xfa, 0xf0,
  30  	}
  31  	fp.SetOne(&w[4]) // z2 = 1
  32  
  33  	const n = 448
  34  	const h = 2
  35  	swap := uint(1)
  36  	for s := 0; s < n-h; s++ {
  37  		i := (s + h) / 8
  38  		j := (s + h) % 8
  39  		bit := uint((k[i] >> uint(j)) & 1)
  40  		copy(w[0][:], tableGenerator[s*Size:(s+1)*Size])
  41  		diffAdd(&w, swap^bit)
  42  		swap = bit
  43  	}
  44  	for s := 0; s < h; s++ {
  45  		double(&w[1], &w[2])
  46  	}
  47  	toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
  48  }
  49  
  50  // ladderMontgomery calculates a generic scalar point multiplication
  51  // The algorithm implemented is the left-to-right Montgomery's ladder.
  52  func ladderMontgomery(k, xP *Key) {
  53  	w := [5]fp.Elt{}      // [x1, x2, z2, x3, z3] order must be preserved.
  54  	w[0] = *(*fp.Elt)(xP) // x1 = xP
  55  	fp.SetOne(&w[1])      // x2 = 1
  56  	w[3] = *(*fp.Elt)(xP) // x3 = xP
  57  	fp.SetOne(&w[4])      // z3 = 1
  58  
  59  	move := uint(0)
  60  	for s := 448 - 1; s >= 0; s-- {
  61  		i := s / 8
  62  		j := s % 8
  63  		bit := uint((k[i] >> uint(j)) & 1)
  64  		ladderStep(&w, move^bit)
  65  		move = bit
  66  	}
  67  	toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
  68  }
  69  
  70  func toAffine(k *[fp.Size]byte, x, z *fp.Elt) {
  71  	fp.Inv(z, z)
  72  	fp.Mul(x, x, z)
  73  	_ = fp.ToBytes(k[:], x)
  74  }
  75  
  76  var lowOrderPoints = [3]fp.Elt{
  77  	{ /* (0,_,1) point of order 2 on Curve448 */
  78  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  79  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  80  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  81  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  82  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  83  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  84  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  85  	},
  86  	{ /* (1,_,1) a point of order 4 on the twist of Curve448 */
  87  		0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  88  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  89  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  90  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  91  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  92  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  93  		0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  94  	},
  95  	{ /* (-1,_,1) point of order 4 on Curve448 */
  96  		0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  97  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  98  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
  99  		0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff,
 100  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
 101  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
 102  		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
 103  	},
 104  }
 105