github.com/ethw3/go-ethereuma@v0.0.0-20221013053120-c14602a4c23c/crypto/bn256/cloudflare/twist.go (about)

     1  package bn256
     2  
     3  import (
     4  	"math/big"
     5  )
     6  
     7  // twistPoint implements the elliptic curve y²=x³+3/ξ over GF(p²). Points are
     8  // kept in Jacobian form and t=z² when valid. The group G₂ is the set of
     9  // n-torsion points of this curve over GF(p²) (where n = Order)
    10  type twistPoint struct {
    11  	x, y, z, t gfP2
    12  }
    13  
    14  var twistB = &gfP2{
    15  	gfP{0x38e7ecccd1dcff67, 0x65f0b37d93ce0d3e, 0xd749d0dd22ac00aa, 0x0141b9ce4a688d4d},
    16  	gfP{0x3bf938e377b802a8, 0x020b1b273633535d, 0x26b7edf049755260, 0x2514c6324384a86d},
    17  }
    18  
    19  // twistGen is the generator of group G₂.
    20  var twistGen = &twistPoint{
    21  	gfP2{
    22  		gfP{0xafb4737da84c6140, 0x6043dd5a5802d8c4, 0x09e950fc52a02f86, 0x14fef0833aea7b6b},
    23  		gfP{0x8e83b5d102bc2026, 0xdceb1935497b0172, 0xfbb8264797811adf, 0x19573841af96503b},
    24  	},
    25  	gfP2{
    26  		gfP{0x64095b56c71856ee, 0xdc57f922327d3cbb, 0x55f935be33351076, 0x0da4a0e693fd6482},
    27  		gfP{0x619dfa9d886be9f6, 0xfe7fd297f59e9b78, 0xff9e1a62231b7dfe, 0x28fd7eebae9e4206},
    28  	},
    29  	gfP2{*newGFp(0), *newGFp(1)},
    30  	gfP2{*newGFp(0), *newGFp(1)},
    31  }
    32  
    33  func (c *twistPoint) String() string {
    34  	c.MakeAffine()
    35  	x, y := gfP2Decode(&c.x), gfP2Decode(&c.y)
    36  	return "(" + x.String() + ", " + y.String() + ")"
    37  }
    38  
    39  func (c *twistPoint) Set(a *twistPoint) {
    40  	c.x.Set(&a.x)
    41  	c.y.Set(&a.y)
    42  	c.z.Set(&a.z)
    43  	c.t.Set(&a.t)
    44  }
    45  
    46  // IsOnCurve returns true iff c is on the curve.
    47  func (c *twistPoint) IsOnCurve() bool {
    48  	c.MakeAffine()
    49  	if c.IsInfinity() {
    50  		return true
    51  	}
    52  
    53  	y2, x3 := &gfP2{}, &gfP2{}
    54  	y2.Square(&c.y)
    55  	x3.Square(&c.x).Mul(x3, &c.x).Add(x3, twistB)
    56  
    57  	if *y2 != *x3 {
    58  		return false
    59  	}
    60  	cneg := &twistPoint{}
    61  	cneg.Mul(c, Order)
    62  	return cneg.z.IsZero()
    63  }
    64  
    65  func (c *twistPoint) SetInfinity() {
    66  	c.x.SetZero()
    67  	c.y.SetOne()
    68  	c.z.SetZero()
    69  	c.t.SetZero()
    70  }
    71  
    72  func (c *twistPoint) IsInfinity() bool {
    73  	return c.z.IsZero()
    74  }
    75  
    76  func (c *twistPoint) Add(a, b *twistPoint) {
    77  	// For additional comments, see the same function in curve.go.
    78  
    79  	if a.IsInfinity() {
    80  		c.Set(b)
    81  		return
    82  	}
    83  	if b.IsInfinity() {
    84  		c.Set(a)
    85  		return
    86  	}
    87  
    88  	// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/addition/add-2007-bl.op3
    89  	z12 := (&gfP2{}).Square(&a.z)
    90  	z22 := (&gfP2{}).Square(&b.z)
    91  	u1 := (&gfP2{}).Mul(&a.x, z22)
    92  	u2 := (&gfP2{}).Mul(&b.x, z12)
    93  
    94  	t := (&gfP2{}).Mul(&b.z, z22)
    95  	s1 := (&gfP2{}).Mul(&a.y, t)
    96  
    97  	t.Mul(&a.z, z12)
    98  	s2 := (&gfP2{}).Mul(&b.y, t)
    99  
   100  	h := (&gfP2{}).Sub(u2, u1)
   101  	xEqual := h.IsZero()
   102  
   103  	t.Add(h, h)
   104  	i := (&gfP2{}).Square(t)
   105  	j := (&gfP2{}).Mul(h, i)
   106  
   107  	t.Sub(s2, s1)
   108  	yEqual := t.IsZero()
   109  	if xEqual && yEqual {
   110  		c.Double(a)
   111  		return
   112  	}
   113  	r := (&gfP2{}).Add(t, t)
   114  
   115  	v := (&gfP2{}).Mul(u1, i)
   116  
   117  	t4 := (&gfP2{}).Square(r)
   118  	t.Add(v, v)
   119  	t6 := (&gfP2{}).Sub(t4, j)
   120  	c.x.Sub(t6, t)
   121  
   122  	t.Sub(v, &c.x) // t7
   123  	t4.Mul(s1, j)  // t8
   124  	t6.Add(t4, t4) // t9
   125  	t4.Mul(r, t)   // t10
   126  	c.y.Sub(t4, t6)
   127  
   128  	t.Add(&a.z, &b.z) // t11
   129  	t4.Square(t)      // t12
   130  	t.Sub(t4, z12)    // t13
   131  	t4.Sub(t, z22)    // t14
   132  	c.z.Mul(t4, h)
   133  }
   134  
   135  func (c *twistPoint) Double(a *twistPoint) {
   136  	// See http://hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian-0/doubling/dbl-2009-l.op3
   137  	A := (&gfP2{}).Square(&a.x)
   138  	B := (&gfP2{}).Square(&a.y)
   139  	C := (&gfP2{}).Square(B)
   140  
   141  	t := (&gfP2{}).Add(&a.x, B)
   142  	t2 := (&gfP2{}).Square(t)
   143  	t.Sub(t2, A)
   144  	t2.Sub(t, C)
   145  	d := (&gfP2{}).Add(t2, t2)
   146  	t.Add(A, A)
   147  	e := (&gfP2{}).Add(t, A)
   148  	f := (&gfP2{}).Square(e)
   149  
   150  	t.Add(d, d)
   151  	c.x.Sub(f, t)
   152  
   153  	c.z.Mul(&a.y, &a.z)
   154  	c.z.Add(&c.z, &c.z)
   155  
   156  	t.Add(C, C)
   157  	t2.Add(t, t)
   158  	t.Add(t2, t2)
   159  	c.y.Sub(d, &c.x)
   160  	t2.Mul(e, &c.y)
   161  	c.y.Sub(t2, t)
   162  }
   163  
   164  func (c *twistPoint) Mul(a *twistPoint, scalar *big.Int) {
   165  	sum, t := &twistPoint{}, &twistPoint{}
   166  
   167  	for i := scalar.BitLen(); i >= 0; i-- {
   168  		t.Double(sum)
   169  		if scalar.Bit(i) != 0 {
   170  			sum.Add(t, a)
   171  		} else {
   172  			sum.Set(t)
   173  		}
   174  	}
   175  
   176  	c.Set(sum)
   177  }
   178  
   179  func (c *twistPoint) MakeAffine() {
   180  	if c.z.IsOne() {
   181  		return
   182  	} else if c.z.IsZero() {
   183  		c.x.SetZero()
   184  		c.y.SetOne()
   185  		c.t.SetZero()
   186  		return
   187  	}
   188  
   189  	zInv := (&gfP2{}).Invert(&c.z)
   190  	t := (&gfP2{}).Mul(&c.y, zInv)
   191  	zInv2 := (&gfP2{}).Square(zInv)
   192  	c.y.Mul(t, zInv2)
   193  	t.Mul(&c.x, zInv2)
   194  	c.x.Set(t)
   195  	c.z.SetOne()
   196  	c.t.SetOne()
   197  }
   198  
   199  func (c *twistPoint) Neg(a *twistPoint) {
   200  	c.x.Set(&a.x)
   201  	c.y.Neg(&a.y)
   202  	c.z.Set(&a.z)
   203  	c.t.SetZero()
   204  }