github.com/klaytn/klaytn@v1.12.1/crypto/bn256/cloudflare/optate.go (about)

     1  // Copyright 2018 The klaytn Authors
     2  //
     3  // This file is derived from crypto/bn256/cloudflare/optate.go (2018/06/04).
     4  // See LICENSE in the top directory for the original copyright and license.
     5  
     6  package bn256
     7  
     8  func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2) (a, b, c *gfP2, rOut *twistPoint) {
     9  	// See the mixed addition algorithm from "Faster Computation of the
    10  	// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
    11  	B := (&gfP2{}).Mul(&p.x, &r.t)
    12  
    13  	D := (&gfP2{}).Add(&p.y, &r.z)
    14  	D.Square(D).Sub(D, r2).Sub(D, &r.t).Mul(D, &r.t)
    15  
    16  	H := (&gfP2{}).Sub(B, &r.x)
    17  	I := (&gfP2{}).Square(H)
    18  
    19  	E := (&gfP2{}).Add(I, I)
    20  	E.Add(E, E)
    21  
    22  	J := (&gfP2{}).Mul(H, E)
    23  
    24  	L1 := (&gfP2{}).Sub(D, &r.y)
    25  	L1.Sub(L1, &r.y)
    26  
    27  	V := (&gfP2{}).Mul(&r.x, E)
    28  
    29  	rOut = &twistPoint{}
    30  	rOut.x.Square(L1).Sub(&rOut.x, J).Sub(&rOut.x, V).Sub(&rOut.x, V)
    31  
    32  	rOut.z.Add(&r.z, H).Square(&rOut.z).Sub(&rOut.z, &r.t).Sub(&rOut.z, I)
    33  
    34  	t := (&gfP2{}).Sub(V, &rOut.x)
    35  	t.Mul(t, L1)
    36  	t2 := (&gfP2{}).Mul(&r.y, J)
    37  	t2.Add(t2, t2)
    38  	rOut.y.Sub(t, t2)
    39  
    40  	rOut.t.Square(&rOut.z)
    41  
    42  	t.Add(&p.y, &rOut.z).Square(t).Sub(t, r2).Sub(t, &rOut.t)
    43  
    44  	t2.Mul(L1, &p.x)
    45  	t2.Add(t2, t2)
    46  	a = (&gfP2{}).Sub(t2, t)
    47  
    48  	c = (&gfP2{}).MulScalar(&rOut.z, &q.y)
    49  	c.Add(c, c)
    50  
    51  	b = (&gfP2{}).Neg(L1)
    52  	b.MulScalar(b, &q.x).Add(b, b)
    53  
    54  	return
    55  }
    56  
    57  func lineFunctionDouble(r *twistPoint, q *curvePoint) (a, b, c *gfP2, rOut *twistPoint) {
    58  	// See the doubling algorithm for a=0 from "Faster Computation of the
    59  	// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
    60  	A := (&gfP2{}).Square(&r.x)
    61  	B := (&gfP2{}).Square(&r.y)
    62  	C := (&gfP2{}).Square(B)
    63  
    64  	D := (&gfP2{}).Add(&r.x, B)
    65  	D.Square(D).Sub(D, A).Sub(D, C).Add(D, D)
    66  
    67  	E := (&gfP2{}).Add(A, A)
    68  	E.Add(E, A)
    69  
    70  	G := (&gfP2{}).Square(E)
    71  
    72  	rOut = &twistPoint{}
    73  	rOut.x.Sub(G, D).Sub(&rOut.x, D)
    74  
    75  	rOut.z.Add(&r.y, &r.z).Square(&rOut.z).Sub(&rOut.z, B).Sub(&rOut.z, &r.t)
    76  
    77  	rOut.y.Sub(D, &rOut.x).Mul(&rOut.y, E)
    78  	t := (&gfP2{}).Add(C, C)
    79  	t.Add(t, t).Add(t, t)
    80  	rOut.y.Sub(&rOut.y, t)
    81  
    82  	rOut.t.Square(&rOut.z)
    83  
    84  	t.Mul(E, &r.t).Add(t, t)
    85  	b = (&gfP2{}).Neg(t)
    86  	b.MulScalar(b, &q.x)
    87  
    88  	a = (&gfP2{}).Add(&r.x, E)
    89  	a.Square(a).Sub(a, A).Sub(a, G)
    90  	t.Add(B, B).Add(t, t)
    91  	a.Sub(a, t)
    92  
    93  	c = (&gfP2{}).Mul(&rOut.z, &r.t)
    94  	c.Add(c, c).MulScalar(c, &q.y)
    95  
    96  	return
    97  }
    98  
    99  func mulLine(ret *gfP12, a, b, c *gfP2) {
   100  	a2 := &gfP6{}
   101  	a2.y.Set(a)
   102  	a2.z.Set(b)
   103  	a2.Mul(a2, &ret.x)
   104  	t3 := (&gfP6{}).MulScalar(&ret.y, c)
   105  
   106  	t := (&gfP2{}).Add(b, c)
   107  	t2 := &gfP6{}
   108  	t2.y.Set(a)
   109  	t2.z.Set(t)
   110  	ret.x.Add(&ret.x, &ret.y)
   111  
   112  	ret.y.Set(t3)
   113  
   114  	ret.x.Mul(&ret.x, t2).Sub(&ret.x, a2).Sub(&ret.x, &ret.y)
   115  	a2.MulTau(a2)
   116  	ret.y.Add(&ret.y, a2)
   117  }
   118  
   119  // sixuPlus2NAF is 6u+2 in non-adjacent form.
   120  var sixuPlus2NAF = []int8{
   121  	0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0,
   122  	0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1,
   123  	1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1,
   124  	1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 1,
   125  }
   126  
   127  // miller implements the Miller loop for calculating the Optimal Ate pairing.
   128  // See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf
   129  func miller(q *twistPoint, p *curvePoint) *gfP12 {
   130  	ret := (&gfP12{}).SetOne()
   131  
   132  	aAffine := &twistPoint{}
   133  	aAffine.Set(q)
   134  	aAffine.MakeAffine()
   135  
   136  	bAffine := &curvePoint{}
   137  	bAffine.Set(p)
   138  	bAffine.MakeAffine()
   139  
   140  	minusA := &twistPoint{}
   141  	minusA.Neg(aAffine)
   142  
   143  	r := &twistPoint{}
   144  	r.Set(aAffine)
   145  
   146  	r2 := (&gfP2{}).Square(&aAffine.y)
   147  
   148  	for i := len(sixuPlus2NAF) - 1; i > 0; i-- {
   149  		a, b, c, newR := lineFunctionDouble(r, bAffine)
   150  		if i != len(sixuPlus2NAF)-1 {
   151  			ret.Square(ret)
   152  		}
   153  
   154  		mulLine(ret, a, b, c)
   155  		r = newR
   156  
   157  		switch sixuPlus2NAF[i-1] {
   158  		case 1:
   159  			a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2)
   160  		case -1:
   161  			a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2)
   162  		default:
   163  			continue
   164  		}
   165  
   166  		mulLine(ret, a, b, c)
   167  		r = newR
   168  	}
   169  
   170  	// In order to calculate Q1 we have to convert q from the sextic twist
   171  	// to the full GF(p^12) group, apply the Frobenius there, and convert
   172  	// back.
   173  	//
   174  	// The twist isomorphism is (x', y') -> (xω², yω³). If we consider just
   175  	// x for a moment, then after applying the Frobenius, we have x̄ω^(2p)
   176  	// where x̄ is the conjugate of x. If we are going to apply the inverse
   177  	// isomorphism we need a value with a single coefficient of ω² so we
   178  	// rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of
   179  	// p, 2p-2 is a multiple of six. Therefore we can rewrite as
   180  	// x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the
   181  	// ω².
   182  	//
   183  	// A similar argument can be made for the y value.
   184  
   185  	q1 := &twistPoint{}
   186  	q1.x.Conjugate(&aAffine.x).Mul(&q1.x, xiToPMinus1Over3)
   187  	q1.y.Conjugate(&aAffine.y).Mul(&q1.y, xiToPMinus1Over2)
   188  	q1.z.SetOne()
   189  	q1.t.SetOne()
   190  
   191  	// For Q2 we are applying the p² Frobenius. The two conjugations cancel
   192  	// out and we are left only with the factors from the isomorphism. In
   193  	// the case of x, we end up with a pure number which is why
   194  	// xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We
   195  	// ignore this to end up with -Q2.
   196  
   197  	minusQ2 := &twistPoint{}
   198  	minusQ2.x.MulScalar(&aAffine.x, xiToPSquaredMinus1Over3)
   199  	minusQ2.y.Set(&aAffine.y)
   200  	minusQ2.z.SetOne()
   201  	minusQ2.t.SetOne()
   202  
   203  	r2.Square(&q1.y)
   204  	a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2)
   205  	mulLine(ret, a, b, c)
   206  	r = newR
   207  
   208  	r2.Square(&minusQ2.y)
   209  	a, b, c, _ = lineFunctionAdd(r, minusQ2, bAffine, r2)
   210  	mulLine(ret, a, b, c)
   211  
   212  	return ret
   213  }
   214  
   215  // finalExponentiation computes the (p¹²-1)/Order-th power of an element of
   216  // GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from
   217  // http://cryptojedi.org/papers/dclxvi-20100714.pdf)
   218  func finalExponentiation(in *gfP12) *gfP12 {
   219  	t1 := &gfP12{}
   220  
   221  	// This is the p^6-Frobenius
   222  	t1.x.Neg(&in.x)
   223  	t1.y.Set(&in.y)
   224  
   225  	inv := &gfP12{}
   226  	inv.Invert(in)
   227  	t1.Mul(t1, inv)
   228  
   229  	t2 := (&gfP12{}).FrobeniusP2(t1)
   230  	t1.Mul(t1, t2)
   231  
   232  	fp := (&gfP12{}).Frobenius(t1)
   233  	fp2 := (&gfP12{}).FrobeniusP2(t1)
   234  	fp3 := (&gfP12{}).Frobenius(fp2)
   235  
   236  	fu := (&gfP12{}).Exp(t1, u)
   237  	fu2 := (&gfP12{}).Exp(fu, u)
   238  	fu3 := (&gfP12{}).Exp(fu2, u)
   239  
   240  	y3 := (&gfP12{}).Frobenius(fu)
   241  	fu2p := (&gfP12{}).Frobenius(fu2)
   242  	fu3p := (&gfP12{}).Frobenius(fu3)
   243  	y2 := (&gfP12{}).FrobeniusP2(fu2)
   244  
   245  	y0 := &gfP12{}
   246  	y0.Mul(fp, fp2).Mul(y0, fp3)
   247  
   248  	y1 := (&gfP12{}).Conjugate(t1)
   249  	y5 := (&gfP12{}).Conjugate(fu2)
   250  	y3.Conjugate(y3)
   251  	y4 := (&gfP12{}).Mul(fu, fu2p)
   252  	y4.Conjugate(y4)
   253  
   254  	y6 := (&gfP12{}).Mul(fu3, fu3p)
   255  	y6.Conjugate(y6)
   256  
   257  	t0 := (&gfP12{}).Square(y6)
   258  	t0.Mul(t0, y4).Mul(t0, y5)
   259  	t1.Mul(y3, y5).Mul(t1, t0)
   260  	t0.Mul(t0, y2)
   261  	t1.Square(t1).Mul(t1, t0).Square(t1)
   262  	t0.Mul(t1, y1)
   263  	t1.Mul(t1, y0)
   264  	t0.Square(t0).Mul(t0, t1)
   265  
   266  	return t0
   267  }
   268  
   269  func optimalAte(a *twistPoint, b *curvePoint) *gfP12 {
   270  	e := miller(a, b)
   271  	ret := finalExponentiation(e)
   272  
   273  	if a.IsInfinity() || b.IsInfinity() {
   274  		ret.SetOne()
   275  	}
   276  	return ret
   277  }