github.com/ebceco/ebc@v1.8.19-0.20190309150932-8cb0b9e06484/crypto/bn256/cloudflare/optate.go (about)

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