github.com/ConsenSys/Quorum@v20.10.0+incompatible/crypto/bn256/cloudflare/gfp12.go (about)

     1  package bn256
     2  
     3  // For details of the algorithms used, see "Multiplication and Squaring on
     4  // Pairing-Friendly Fields, Devegili et al.
     5  // http://eprint.iacr.org/2006/471.pdf.
     6  
     7  import (
     8  	"math/big"
     9  )
    10  
    11  // gfP12 implements the field of size p¹² as a quadratic extension of gfP6
    12  // where ω²=τ.
    13  type gfP12 struct {
    14  	x, y gfP6 // value is xω + y
    15  }
    16  
    17  func (e *gfP12) String() string {
    18  	return "(" + e.x.String() + "," + e.y.String() + ")"
    19  }
    20  
    21  func (e *gfP12) Set(a *gfP12) *gfP12 {
    22  	e.x.Set(&a.x)
    23  	e.y.Set(&a.y)
    24  	return e
    25  }
    26  
    27  func (e *gfP12) SetZero() *gfP12 {
    28  	e.x.SetZero()
    29  	e.y.SetZero()
    30  	return e
    31  }
    32  
    33  func (e *gfP12) SetOne() *gfP12 {
    34  	e.x.SetZero()
    35  	e.y.SetOne()
    36  	return e
    37  }
    38  
    39  func (e *gfP12) IsZero() bool {
    40  	return e.x.IsZero() && e.y.IsZero()
    41  }
    42  
    43  func (e *gfP12) IsOne() bool {
    44  	return e.x.IsZero() && e.y.IsOne()
    45  }
    46  
    47  func (e *gfP12) Conjugate(a *gfP12) *gfP12 {
    48  	e.x.Neg(&a.x)
    49  	e.y.Set(&a.y)
    50  	return e
    51  }
    52  
    53  func (e *gfP12) Neg(a *gfP12) *gfP12 {
    54  	e.x.Neg(&a.x)
    55  	e.y.Neg(&a.y)
    56  	return e
    57  }
    58  
    59  // Frobenius computes (xω+y)^p = x^p ω·ξ^((p-1)/6) + y^p
    60  func (e *gfP12) Frobenius(a *gfP12) *gfP12 {
    61  	e.x.Frobenius(&a.x)
    62  	e.y.Frobenius(&a.y)
    63  	e.x.MulScalar(&e.x, xiToPMinus1Over6)
    64  	return e
    65  }
    66  
    67  // FrobeniusP2 computes (xω+y)^p² = x^p² ω·ξ^((p²-1)/6) + y^p²
    68  func (e *gfP12) FrobeniusP2(a *gfP12) *gfP12 {
    69  	e.x.FrobeniusP2(&a.x)
    70  	e.x.MulGFP(&e.x, xiToPSquaredMinus1Over6)
    71  	e.y.FrobeniusP2(&a.y)
    72  	return e
    73  }
    74  
    75  func (e *gfP12) FrobeniusP4(a *gfP12) *gfP12 {
    76  	e.x.FrobeniusP4(&a.x)
    77  	e.x.MulGFP(&e.x, xiToPSquaredMinus1Over3)
    78  	e.y.FrobeniusP4(&a.y)
    79  	return e
    80  }
    81  
    82  func (e *gfP12) Add(a, b *gfP12) *gfP12 {
    83  	e.x.Add(&a.x, &b.x)
    84  	e.y.Add(&a.y, &b.y)
    85  	return e
    86  }
    87  
    88  func (e *gfP12) Sub(a, b *gfP12) *gfP12 {
    89  	e.x.Sub(&a.x, &b.x)
    90  	e.y.Sub(&a.y, &b.y)
    91  	return e
    92  }
    93  
    94  func (e *gfP12) Mul(a, b *gfP12) *gfP12 {
    95  	tx := (&gfP6{}).Mul(&a.x, &b.y)
    96  	t := (&gfP6{}).Mul(&b.x, &a.y)
    97  	tx.Add(tx, t)
    98  
    99  	ty := (&gfP6{}).Mul(&a.y, &b.y)
   100  	t.Mul(&a.x, &b.x).MulTau(t)
   101  
   102  	e.x.Set(tx)
   103  	e.y.Add(ty, t)
   104  	return e
   105  }
   106  
   107  func (e *gfP12) MulScalar(a *gfP12, b *gfP6) *gfP12 {
   108  	e.x.Mul(&e.x, b)
   109  	e.y.Mul(&e.y, b)
   110  	return e
   111  }
   112  
   113  func (c *gfP12) Exp(a *gfP12, power *big.Int) *gfP12 {
   114  	sum := (&gfP12{}).SetOne()
   115  	t := &gfP12{}
   116  
   117  	for i := power.BitLen() - 1; i >= 0; i-- {
   118  		t.Square(sum)
   119  		if power.Bit(i) != 0 {
   120  			sum.Mul(t, a)
   121  		} else {
   122  			sum.Set(t)
   123  		}
   124  	}
   125  
   126  	c.Set(sum)
   127  	return c
   128  }
   129  
   130  func (e *gfP12) Square(a *gfP12) *gfP12 {
   131  	// Complex squaring algorithm
   132  	v0 := (&gfP6{}).Mul(&a.x, &a.y)
   133  
   134  	t := (&gfP6{}).MulTau(&a.x)
   135  	t.Add(&a.y, t)
   136  	ty := (&gfP6{}).Add(&a.x, &a.y)
   137  	ty.Mul(ty, t).Sub(ty, v0)
   138  	t.MulTau(v0)
   139  	ty.Sub(ty, t)
   140  
   141  	e.x.Add(v0, v0)
   142  	e.y.Set(ty)
   143  	return e
   144  }
   145  
   146  func (e *gfP12) Invert(a *gfP12) *gfP12 {
   147  	// See "Implementing cryptographic pairings", M. Scott, section 3.2.
   148  	// ftp://136.206.11.249/pub/crypto/pairings.pdf
   149  	t1, t2 := &gfP6{}, &gfP6{}
   150  
   151  	t1.Square(&a.x)
   152  	t2.Square(&a.y)
   153  	t1.MulTau(t1).Sub(t2, t1)
   154  	t2.Invert(t1)
   155  
   156  	e.x.Neg(&a.x)
   157  	e.y.Set(&a.y)
   158  	e.MulScalar(e, t2)
   159  	return e
   160  }