github.com/linapex/ethereum-dpos-chinese@v0.0.0-20190316121959-b78b3a4a1ece/crypto/bn256/cloudflare/twist.go (about)

     1  
     2  //<developer>
     3  //    <name>linapex 曹一峰</name>
     4  //    <email>linapex@163.com</email>
     5  //    <wx>superexc</wx>
     6  //    <qqgroup>128148617</qqgroup>
     7  //    <url>https://jsq.ink</url>
     8  //    <role>pku engineer</role>
     9  //    <date>2019-03-16 12:09:36</date>
    10  //</624342625641566208>
    11  
    12  package bn256
    13  
    14  import (
    15  	"math/big"
    16  )
    17  
    18  //Twistpoint在gf（p²）上实现椭圆曲线y²=x³+3/ξ。点是
    19  //以雅可比形式保存，有效时t=z²。G组是一组
    20  //n——该曲线在gf（p²）上的扭转点（其中n=阶数）
    21  type twistPoint struct {
    22  	x, y, z, t gfP2
    23  }
    24  
    25  var twistB = &gfP2{
    26  	gfP{0x38e7ecccd1dcff67, 0x65f0b37d93ce0d3e, 0xd749d0dd22ac00aa, 0x0141b9ce4a688d4d},
    27  	gfP{0x3bf938e377b802a8, 0x020b1b273633535d, 0x26b7edf049755260, 0x2514c6324384a86d},
    28  }
    29  
    30  //TwistGen是G组的发生器。
    31  var twistGen = &twistPoint{
    32  	gfP2{
    33  		gfP{0xafb4737da84c6140, 0x6043dd5a5802d8c4, 0x09e950fc52a02f86, 0x14fef0833aea7b6b},
    34  		gfP{0x8e83b5d102bc2026, 0xdceb1935497b0172, 0xfbb8264797811adf, 0x19573841af96503b},
    35  	},
    36  	gfP2{
    37  		gfP{0x64095b56c71856ee, 0xdc57f922327d3cbb, 0x55f935be33351076, 0x0da4a0e693fd6482},
    38  		gfP{0x619dfa9d886be9f6, 0xfe7fd297f59e9b78, 0xff9e1a62231b7dfe, 0x28fd7eebae9e4206},
    39  	},
    40  	gfP2{*newGFp(0), *newGFp(1)},
    41  	gfP2{*newGFp(0), *newGFp(1)},
    42  }
    43  
    44  func (c *twistPoint) String() string {
    45  	c.MakeAffine()
    46  	x, y := gfP2Decode(&c.x), gfP2Decode(&c.y)
    47  	return "(" + x.String() + ", " + y.String() + ")"
    48  }
    49  
    50  func (c *twistPoint) Set(a *twistPoint) {
    51  	c.x.Set(&a.x)
    52  	c.y.Set(&a.y)
    53  	c.z.Set(&a.z)
    54  	c.t.Set(&a.t)
    55  }
    56  
    57  //is on curve返回真的iff c在曲线上。
    58  func (c *twistPoint) IsOnCurve() bool {
    59  	c.MakeAffine()
    60  	if c.IsInfinity() {
    61  		return true
    62  	}
    63  
    64  	y2, x3 := &gfP2{}, &gfP2{}
    65  	y2.Square(&c.y)
    66  	x3.Square(&c.x).Mul(x3, &c.x).Add(x3, twistB)
    67  
    68  	if *y2 != *x3 {
    69  		return false
    70  	}
    71  	cneg := &twistPoint{}
    72  	cneg.Mul(c, Order)
    73  	return cneg.z.IsZero()
    74  }
    75  
    76  func (c *twistPoint) SetInfinity() {
    77  	c.x.SetZero()
    78  	c.y.SetOne()
    79  	c.z.SetZero()
    80  	c.t.SetZero()
    81  }
    82  
    83  func (c *twistPoint) IsInfinity() bool {
    84  	return c.z.IsZero()
    85  }
    86  
    87  func (c *twistPoint) Add(a, b *twistPoint) {
    88  //有关其他注释，请参见curve.go中的相同函数。
    89  
    90  	if a.IsInfinity() {
    91  		c.Set(b)
    92  		return
    93  	}
    94  	if b.IsInfinity() {
    95  		c.Set(a)
    96  		return
    97  	}
    98  
    99  //见http://hyper椭圆形.org/efd/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
   100  	z12 := (&gfP2{}).Square(&a.z)
   101  	z22 := (&gfP2{}).Square(&b.z)
   102  	u1 := (&gfP2{}).Mul(&a.x, z22)
   103  	u2 := (&gfP2{}).Mul(&b.x, z12)
   104  
   105  	t := (&gfP2{}).Mul(&b.z, z22)
   106  	s1 := (&gfP2{}).Mul(&a.y, t)
   107  
   108  	t.Mul(&a.z, z12)
   109  	s2 := (&gfP2{}).Mul(&b.y, t)
   110  
   111  	h := (&gfP2{}).Sub(u2, u1)
   112  	xEqual := h.IsZero()
   113  
   114  	t.Add(h, h)
   115  	i := (&gfP2{}).Square(t)
   116  	j := (&gfP2{}).Mul(h, i)
   117  
   118  	t.Sub(s2, s1)
   119  	yEqual := t.IsZero()
   120  	if xEqual && yEqual {
   121  		c.Double(a)
   122  		return
   123  	}
   124  	r := (&gfP2{}).Add(t, t)
   125  
   126  	v := (&gfP2{}).Mul(u1, i)
   127  
   128  	t4 := (&gfP2{}).Square(r)
   129  	t.Add(v, v)
   130  	t6 := (&gfP2{}).Sub(t4, j)
   131  	c.x.Sub(t6, t)
   132  
   133  t.Sub(v, &c.x) //T7
   134  t4.Mul(s1, j)  //T8
   135  t6.Add(t4, t4) //T9
   136  t4.Mul(r, t)   //T10
   137  	c.y.Sub(t4, t6)
   138  
   139  t.Add(&a.z, &b.z) //T11
   140  t4.Square(t)      //T12
   141  t.Sub(t4, z12)    //T13
   142  t4.Sub(t, z22)    //T14
   143  	c.z.Mul(t4, h)
   144  }
   145  
   146  func (c *twistPoint) Double(a *twistPoint) {
   147  //请参阅http://hyper椭圆形.org/efd/g1p/auto-code/shortw/jacobian-0/double/dbl-2009-l.op3
   148  	A := (&gfP2{}).Square(&a.x)
   149  	B := (&gfP2{}).Square(&a.y)
   150  	C := (&gfP2{}).Square(B)
   151  
   152  	t := (&gfP2{}).Add(&a.x, B)
   153  	t2 := (&gfP2{}).Square(t)
   154  	t.Sub(t2, A)
   155  	t2.Sub(t, C)
   156  	d := (&gfP2{}).Add(t2, t2)
   157  	t.Add(A, A)
   158  	e := (&gfP2{}).Add(t, A)
   159  	f := (&gfP2{}).Square(e)
   160  
   161  	t.Add(d, d)
   162  	c.x.Sub(f, t)
   163  
   164  	t.Add(C, C)
   165  	t2.Add(t, t)
   166  	t.Add(t2, t2)
   167  	c.y.Sub(d, &c.x)
   168  	t2.Mul(e, &c.y)
   169  	c.y.Sub(t2, t)
   170  
   171  	t.Mul(&a.y, &a.z)
   172  	c.z.Add(t, t)
   173  }
   174  
   175  func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int) {
   176  	sum, t := &twistPoint{}, &twistPoint{}
   177  
   178  	for i := scalar.BitLen(); i >= 0; i-- {
   179  		t.Double(sum)
   180  		if scalar.Bit(i) != 0 {
   181  			sum.Add(t, a)
   182  		} else {
   183  			sum.Set(t)
   184  		}
   185  	}
   186  
   187  	c.Set(sum)
   188  }
   189  
   190  func (c *twistPoint) MakeAffine() {
   191  	if c.z.IsOne() {
   192  		return
   193  	} else if c.z.IsZero() {
   194  		c.x.SetZero()
   195  		c.y.SetOne()
   196  		c.t.SetZero()
   197  		return
   198  	}
   199  
   200  	zInv := (&gfP2{}).Invert(&c.z)
   201  	t := (&gfP2{}).Mul(&c.y, zInv)
   202  	zInv2 := (&gfP2{}).Square(zInv)
   203  	c.y.Mul(t, zInv2)
   204  	t.Mul(&c.x, zInv2)
   205  	c.x.Set(t)
   206  	c.z.SetOne()
   207  	c.t.SetOne()
   208  }
   209  
   210  func (c *twistPoint) Neg(a *twistPoint) {
   211  	c.x.Set(&a.x)
   212  	c.y.Neg(&a.y)
   213  	c.z.Set(&a.z)
   214  	c.t.SetZero()
   215  }
   216