version3/cpp/fp12.cpp

/*
Licensed to the Apache Software Foundation (ASF) under one
or more contributor license agreements.  See the NOTICE file
distributed with this work for additional information
regarding copyright ownership.  The ASF licenses this file
to you under the Apache License, Version 2.0 (the
"License"); you may not use this file except in compliance
with the License.  You may obtain a copy of the License at

  http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing,
software distributed under the License is distributed on an
"AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
KIND, either express or implied.  See the License for the
specific language governing permissions and limitations
under the License.
*/

/* AMCL Fp^12 functions */
/* SU=m, m is Stack Usage (no lazy )*/
/* FP12 elements are of the form a+i.b+i^2.c */

#include "fp12_YYY.h"
#include "config_curve_ZZZ.h"

using namespace XXX;

/* return 1 if b==c, no branching */
static int teq(sign32 b,sign32 c)
{
    sign32 x=b^c;
    x-=1;  // if x=0, x now -1
    return (int)((x>>31)&1);
}


/* Constant time select from pre-computed table */
static void FP12_select(YYY::FP12 *f,YYY::FP12 g[],sign32 b)
{
    YYY::FP12 invf;
    sign32 m=b>>31;
    sign32 babs=(b^m)-m;

    babs=(babs-1)/2;

    FP12_cmove(f,&g[0],teq(babs,0));  // conditional move
    FP12_cmove(f,&g[1],teq(babs,1));
    FP12_cmove(f,&g[2],teq(babs,2));
    FP12_cmove(f,&g[3],teq(babs,3));
    FP12_cmove(f,&g[4],teq(babs,4));
    FP12_cmove(f,&g[5],teq(babs,5));
    FP12_cmove(f,&g[6],teq(babs,6));
    FP12_cmove(f,&g[7],teq(babs,7));

    FP12_copy(&invf,f);
    FP12_conj(&invf,&invf);  // 1/f
    FP12_cmove(f,&invf,(int)(m&1));
}

/* test x==0 ? */
/* SU= 8 */
int YYY::FP12_iszilch(FP12 *x)
{
    if (FP4_iszilch(&(x->a)) && FP4_iszilch(&(x->b)) && FP4_iszilch(&(x->c))) return 1;
    return 0;
}

/* test x==1 ? */
/* SU= 8 */
int YYY::FP12_isunity(FP12 *x)
{
    if (FP4_isunity(&(x->a)) && FP4_iszilch(&(x->b)) && FP4_iszilch(&(x->c))) return 1;
    return 0;
}

/* FP12 copy w=x */
/* SU= 16 */
void YYY::FP12_copy(FP12 *w,FP12 *x)
{
    if (x==w) return;
    FP4_copy(&(w->a),&(x->a));
    FP4_copy(&(w->b),&(x->b));
    FP4_copy(&(w->c),&(x->c));
	w->type=x->type;
}

/* FP12 w=1 */
/* SU= 8 */
void YYY::FP12_one(FP12 *w)
{
    FP4_one(&(w->a));
    FP4_zero(&(w->b));
    FP4_zero(&(w->c));
	w->type=FP_UNITY;
}

/* return 1 if x==y, else 0 */
/* SU= 16 */
int YYY::FP12_equals(FP12 *x,FP12 *y)
{
    if (FP4_equals(&(x->a),&(y->a)) && FP4_equals(&(x->b),&(y->b)) && FP4_equals(&(x->b),&(y->b)))
        return 1;
    return 0;
}

/* Set w=conj(x) */
/* SU= 8 */
void YYY::FP12_conj(FP12 *w,FP12 *x)
{
    FP12_copy(w,x);
    FP4_conj(&(w->a),&(w->a));
    FP4_nconj(&(w->b),&(w->b));
    FP4_conj(&(w->c),&(w->c));
}

/* Create FP12 from FP4 */
/* SU= 8 */
void YYY::FP12_from_FP4(FP12 *w,FP4 *a)
{
    FP4_copy(&(w->a),a);
    FP4_zero(&(w->b));
    FP4_zero(&(w->c));
	w->type=FP_SPARSER;
}

/* Create FP12 from 3 FP4's */
/* SU= 16 */
void YYY::FP12_from_FP4s(FP12 *w,FP4 *a,FP4 *b,FP4 *c)
{
    FP4_copy(&(w->a),a);
    FP4_copy(&(w->b),b);
    FP4_copy(&(w->c),c);
	w->type=FP_DENSE;
}

/* Granger-Scott Unitary Squaring. This does not benefit from lazy reduction */
/* SU= 600 */
void YYY::FP12_usqr(FP12 *w,FP12 *x)
{
    FP4 A,B,C,D;

    FP4_copy(&A,&(x->a));   

    FP4_sqr(&(w->a),&(x->a));  // Wa XES=2
    FP4_add(&D,&(w->a),&(w->a)); // Wa XES=4
    FP4_add(&(w->a),&D,&(w->a)); // Wa XES=6

    FP4_norm(&(w->a));
    FP4_nconj(&A,&A);

    FP4_add(&A,&A,&A);
    FP4_add(&(w->a),&(w->a),&A); // Wa XES=8
    FP4_sqr(&B,&(x->c));
    FP4_times_i(&B);

    FP4_add(&D,&B,&B);
    FP4_add(&B,&B,&D);
    FP4_norm(&B);

    FP4_sqr(&C,&(x->b));

    FP4_add(&D,&C,&C);
    FP4_add(&C,&C,&D);

    FP4_norm(&C);
    FP4_conj(&(w->b),&(x->b));
    FP4_add(&(w->b),&(w->b),&(w->b));
    FP4_nconj(&(w->c),&(x->c));

    FP4_add(&(w->c),&(w->c),&(w->c));
    FP4_add(&(w->b),&B,&(w->b));
    FP4_add(&(w->c),&C,&(w->c));

	w->type=FP_DENSE;
    FP12_reduce(w);	    /* reduce here as in pow function repeated squarings would trigger multiple reductions */
}

/* FP12 squaring w=x^2 */
/* SU= 600 */
void YYY::FP12_sqr(FP12 *w,FP12 *x)
{
    /* Use Chung-Hasan SQR2 method from http://cacr.uwaterloo.ca/techreports/2006/cacr2006-24.pdf */

    FP4 A,B,C,D;

	if (x->type<=FP_UNITY)
	{
		FP12_copy(w,x);
		return;
	}

    FP4_sqr(&A,&(x->a));
    FP4_mul(&B,&(x->b),&(x->c));
    FP4_add(&B,&B,&B);
	FP4_norm(&B);
    FP4_sqr(&C,&(x->c));

    FP4_mul(&D,&(x->a),&(x->b));
    FP4_add(&D,&D,&D);
    FP4_add(&(w->c),&(x->a),&(x->c));
    FP4_add(&(w->c),&(x->b),&(w->c));
	FP4_norm(&(w->c));	

    FP4_sqr(&(w->c),&(w->c));

    FP4_copy(&(w->a),&A);
    FP4_add(&A,&A,&B);

    FP4_norm(&A);

    FP4_add(&A,&A,&C);
    FP4_add(&A,&A,&D);

    FP4_norm(&A);
    FP4_neg(&A,&A);
    FP4_times_i(&B);
    FP4_times_i(&C);

    FP4_add(&(w->a),&(w->a),&B);
    FP4_add(&(w->b),&C,&D);
    FP4_add(&(w->c),&(w->c),&A);

	if (x->type==FP_SPARSER)
		w->type=FP_SPARSE;
	else
		w->type=FP_DENSE;
    FP12_norm(w);
}

// Use FP12_mul when both multiplicands are dense
// Use FP12smul when it is known that both multiplicands are line functions
// Use FP12ssmul when it is suspected that one or both multiplicands could have some sparsity


/* FP12 full multiplication w=w*y */
void YYY::FP12_mul(FP12 *w,FP12 *y)
{
	FP4 z0,z1,z2,z3,t0,t1;

	FP4_mul(&z0,&(w->a),&(y->a));  // xa.ya   always 11x11

	FP4_mul(&z2,&(w->b),&(y->b));  // xb.yb  could be 00x00 or 01x01 or or 10x10 or 11x00 or 11x10 or 11x01 or 11x11 

	FP4_add(&t0,&(w->a),&(w->b));  // (xa+xb)
	FP4_add(&t1,&(y->a),&(y->b));  // (ya+yb)

	FP4_norm(&t0);
	FP4_norm(&t1);

	FP4_mul(&z1,&t0,&t1); // (xa+xb)(ya+yb)  always 11x11
	FP4_add(&t0,&(w->b),&(w->c));  // (xb+xc)
	FP4_add(&t1,&(y->b),&(y->c));  // (yb+yc)

	FP4_norm(&t0);
	FP4_norm(&t1);

	FP4_mul(&z3,&t0,&t1);	// (xb+xc)(yb+yc)   could be anything...
	FP4_neg(&t0,&z0);		// -(xa.ya)
	FP4_neg(&t1,&z2);		// -(xb.yb)

	FP4_add(&z1,&z1,&t0);  
	FP4_add(&(w->b),&z1,&t1); // /wb = (xa+xb)(ya+yb) -(xa.ya) -(xb.yb)						= xa.yb + xb.ya

	FP4_add(&z3,&z3,&t1);        // (xb+xc)(yb+yc) -(xb.yb)
	FP4_add(&z2,&z2,&t0);        // (xb.yb) - (xa.ya)

	FP4_add(&t0,&(w->a),&(w->c));  // (xa+xc)
	FP4_add(&t1,&(y->a),&(y->c));  // (ya+yc)

	FP4_norm(&t0);
	FP4_norm(&t1);

	FP4_mul(&t0,&t1,&t0);	// (xa+xc)(ya+yc)    always 11x11
	FP4_add(&z2,&z2,&t0);	// (xb.yb) - (xa.ya) + (xa+xc)(ya+yc)

	FP4_mul(&t0,&(w->c),&(y->c)); // (xc.yc)  could be anything	
	FP4_neg(&t1,&t0);			  // -(xc.yc) 

	FP4_add(&(w->c),&z2,&t1);		// wc = (xb.yb) - (xa.ya) + (xa+xc)(ya+yc) - (xc.yc)	=  xb.yb + xc.ya + xa.yc
	FP4_add(&z3,&z3,&t1);			// (xb+xc)(yb+yc) -(xb.yb) - (xc.yc)					=  xb.yc + xc.yb
	FP4_times_i(&t0);				// i.(xc.yc)
	FP4_add(&(w->b),&(w->b),&t0);   // wb = (xa+xb)(ya+yb) -(xa.ya) -(xb.yb) +i(xc.yc)
	FP4_norm(&z3);
	FP4_times_i(&z3);				// i[(xb+xc)(yb+yc) -(xb.yb) - (xc.yc)]					= i(xb.yc + xc.yb)
	FP4_add(&(w->a),&z0,&z3);		// wa = xa.ya + i(xb.yc + xc.yb)

    FP12_norm(w);
	w->type=FP_DENSE;
}

/* FP12 full multiplication w=w*y */
/* Supports sparse multiplicands */
/* Usually w is denser than y */
void YYY::FP12_ssmul(FP12 *w,FP12 *y)
{
	FP4 z0,z1,z2,z3,t0,t1;
	if (w->type==FP_UNITY)
	{
		FP12_copy(w,y);
		return;
	}
	if (y->type==FP_UNITY)
		return;

	if (y->type >= FP_SPARSE)
	{
		FP4_mul(&z0,&(w->a),&(y->a));  // xa.ya   always 11x11

#if SEXTIC_TWIST_ZZZ == M_TYPE
		if (y->type==FP_SPARSE || w->type==FP_SPARSE)
		{
			FP2_mul(&z2.b,&(w->b).b,&(y->b).b);
			FP2_zero(&z2.a);
			if (y->type!=FP_SPARSE)
				FP2_mul(&z2.a,&(w->b).b,&(y->b).a);
			if (w->type!=FP_SPARSE)
				FP2_mul(&z2.a,&(w->b).a,&(y->b).b);
			FP4_times_i(&z2);
		}
		else
#endif
			FP4_mul(&z2,&(w->b),&(y->b));  // xb.yb  could be 00x00 or 01x01 or or 10x10 or 11x00 or 11x10 or 11x01 or 11x11 

		FP4_add(&t0,&(w->a),&(w->b));  // (xa+xb)
		FP4_add(&t1,&(y->a),&(y->b));  // (ya+yb)

		FP4_norm(&t0);
		FP4_norm(&t1);

		FP4_mul(&z1,&t0,&t1); // (xa+xb)(ya+yb)  always 11x11
		FP4_add(&t0,&(w->b),&(w->c));  // (xb+xc)
		FP4_add(&t1,&(y->b),&(y->c));  // (yb+yc)

		FP4_norm(&t0);
		FP4_norm(&t1);

		FP4_mul(&z3,&t0,&t1);	// (xb+xc)(yb+yc)   could be anything...
		FP4_neg(&t0,&z0);		// -(xa.ya)
		FP4_neg(&t1,&z2);		// -(xb.yb)

		FP4_add(&z1,&z1,&t0);  
		FP4_add(&(w->b),&z1,&t1); // /wb = (xa+xb)(ya+yb) -(xa.ya) -(xb.yb)						= xa.yb + xb.ya

		FP4_add(&z3,&z3,&t1);        // (xb+xc)(yb+yc) -(xb.yb)
		FP4_add(&z2,&z2,&t0);        // (xb.yb) - (xa.ya)

		FP4_add(&t0,&(w->a),&(w->c));  // (xa+xc)
		FP4_add(&t1,&(y->a),&(y->c));  // (ya+yc)

		FP4_norm(&t0);
		FP4_norm(&t1);

		FP4_mul(&t0,&t1,&t0);	// (xa+xc)(ya+yc)    always 11x11
		FP4_add(&z2,&z2,&t0);	// (xb.yb) - (xa.ya) + (xa+xc)(ya+yc)

#if SEXTIC_TWIST_ZZZ == D_TYPE
		if (y->type==FP_SPARSE || w->type==FP_SPARSE)
		{
			FP2_mul(&t0.a,&(w->c).a,&(y->c).a);
			FP2_zero(&t0.b);
			if (y->type!=FP_SPARSE)
				FP2_mul(&t0.b,&(w->c).a,&(y->c).b);
			if (w->type!=FP_SPARSE)
				FP2_mul(&t0.b,&(w->c).b,&(y->c).a);
		}
		else
#endif
			FP4_mul(&t0,&(w->c),&(y->c)); // (xc.yc)  could be anything
			
		FP4_neg(&t1,&t0);			  // -(xc.yc) 

		FP4_add(&(w->c),&z2,&t1);		// wc = (xb.yb) - (xa.ya) + (xa+xc)(ya+yc) - (xc.yc)	=  xb.yb + xc.ya + xa.yc
		FP4_add(&z3,&z3,&t1);			// (xb+xc)(yb+yc) -(xb.yb) - (xc.yc)					=  xb.yc + xc.yb
		FP4_times_i(&t0);				// i.(xc.yc)
		FP4_add(&(w->b),&(w->b),&t0);   // wb = (xa+xb)(ya+yb) -(xa.ya) -(xb.yb) +i(xc.yc)
		FP4_norm(&z3);
		FP4_times_i(&z3);				// i[(xb+xc)(yb+yc) -(xb.yb) - (xc.yc)]					= i(xb.yc + xc.yb)
		FP4_add(&(w->a),&z0,&z3);		// wa = xa.ya + i(xb.yc + xc.yb)
	} else {
		if (w->type==FP_SPARSER)
		{
			FP12_smul(w,y);
			return;
		}
 // dense by sparser - 13m 
#if SEXTIC_TWIST_ZZZ == D_TYPE
		FP4_copy(&z3,&(w->b));
		FP4_mul(&z0,&(w->a),&(y->a));

		FP4_pmul(&z2,&(w->b),&(y->b).a);
		FP4_add(&(w->b),&(w->a),&(w->b));
		FP4_copy(&t1,&(y->a));
		FP2_add(&t1.a,&t1.a,&(y->b).a);

		FP4_norm(&t1);
		FP4_norm(&(w->b));

		FP4_mul(&(w->b),&(w->b),&t1);
		FP4_add(&z3,&z3,&(w->c));
		FP4_norm(&z3);
		FP4_pmul(&z3,&z3,&(y->b).a);
		FP4_neg(&t0,&z0);
		FP4_neg(&t1,&z2);

		FP4_add(&(w->b),&(w->b),&t0);   // z1=z1-z0
		FP4_add(&(w->b),&(w->b),&t1);   // z1=z1-z2

		FP4_add(&z3,&z3,&t1);        // z3=z3-z2
		FP4_add(&z2,&z2,&t0);        // z2=z2-z0

		FP4_add(&t0,&(w->a),&(w->c));
		FP4_norm(&t0);
		FP4_norm(&z3);

		FP4_mul(&t0,&(y->a),&t0);
		FP4_add(&(w->c),&z2,&t0);

		FP4_times_i(&z3);
		FP4_add(&(w->a),&z0,&z3);
#endif
#if SEXTIC_TWIST_ZZZ == M_TYPE
		FP4_mul(&z0,&(w->a),&(y->a));
		FP4_add(&t0,&(w->a),&(w->b));
		FP4_norm(&t0);

		FP4_mul(&z1,&t0,&(y->a));
		FP4_add(&t0,&(w->b),&(w->c));
		FP4_norm(&t0);

		FP4_pmul(&z3,&t0,&(y->c).b);
		FP4_times_i(&z3);

		FP4_neg(&t0,&z0);
		FP4_add(&z1,&z1,&t0);   // z1=z1-z0

		FP4_copy(&(w->b),&z1);
		FP4_copy(&z2,&t0);

		FP4_add(&t0,&(w->a),&(w->c));
		FP4_add(&t1,&(y->a),&(y->c));

		FP4_norm(&t0);
		FP4_norm(&t1);

		FP4_mul(&t0,&t1,&t0);
		FP4_add(&z2,&z2,&t0);

		FP4_pmul(&t0,&(w->c),&(y->c).b);
		FP4_times_i(&t0);
		FP4_neg(&t1,&t0);
		FP4_times_i(&t0);

		FP4_add(&(w->c),&z2,&t1);
		FP4_add(&z3,&z3,&t1);

		FP4_add(&(w->b),&(w->b),&t0);
		FP4_norm(&z3);
		FP4_times_i(&z3);
		FP4_add(&(w->a),&z0,&z3);

#endif
	}
	w->type=FP_DENSE;
    FP12_norm(w);
}

/* FP12 multiplication w=w*y */
/* catering for special case that arises from special form of ATE pairing line function */
/* w and y are both sparser line functions - cost = 6m */ 
void YYY::FP12_smul(FP12 *w,FP12 *y)
{
	FP2 w1,w2,w3,ta,tb,tc,td,te,t;

//	if (type==D_TYPE)
//	{ 
#if SEXTIC_TWIST_ZZZ == D_TYPE
	FP2_mul(&w1,&(w->a).a,&(y->a).a); // A1.A2
	FP2_mul(&w2,&(w->a).b,&(y->a).b); // B1.B2
	FP2_mul(&w3,&(w->b).a,&(y->b).a); // C1.C2

	FP2_add(&ta,&(w->a).a,&(w->a).b); // A1+B1
	FP2_add(&tb,&(y->a).a,&(y->a).b); // A2+B2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&tc,&ta,&tb);			// (A1+B1)(A2+B2)
	FP2_add(&t,&w1,&w2);
	FP2_neg(&t,&t);
	FP2_add(&tc,&tc,&t);			// (A1+B1)(A2+B2)-A1.A2-B1*B2 =  (A1.B2+A2.B1)		
				
	FP2_add(&ta,&(w->a).a,&(w->b).a); // A1+C1
	FP2_add(&tb,&(y->a).a,&(y->b).a); // A2+C2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&td,&ta,&tb);			// (A1+C1)(A2+C2)
	FP2_add(&t,&w1,&w3);
	FP2_neg(&t,&t);
	FP2_add(&td,&td,&t);			// (A1+C1)(A2+C2)-A1.A2-C1*C2 =  (A1.C2+A2.C1)		

	FP2_add(&ta,&(w->a).b,&(w->b).a); // B1+C1
	FP2_add(&tb,&(y->a).b,&(y->b).a); // B2+C2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&te,&ta,&tb);			// (B1+C1)(B2+C2)
	FP2_add(&t,&w2,&w3);
	FP2_neg(&t,&t);
	FP2_add(&te,&te,&t);			// (B1+C1)(B2+C2)-B1.B2-C1*C2 =  (B1.C2+B2.C1)		

	FP2_mul_ip(&w2);
	FP2_add(&w1,&w1,&w2);
	FP4_from_FP2s(&(w->a),&w1,&tc);
	FP4_from_FP2s(&(w->b),&td,&te); // only norm these 2
	FP4_from_FP2(&(w->c),&w3);

	FP4_norm(&(w->a));
	FP4_norm(&(w->b));
#endif
//	} else { 
#if SEXTIC_TWIST_ZZZ == M_TYPE
	FP2_mul(&w1,&(w->a).a,&(y->a).a); // A1.A2
	FP2_mul(&w2,&(w->a).b,&(y->a).b); // B1.B2
	FP2_mul(&w3,&(w->c).b,&(y->c).b); // F1.F2

	FP2_add(&ta,&(w->a).a,&(w->a).b); // A1+B1
	FP2_add(&tb,&(y->a).a,&(y->a).b); // A2+B2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&tc,&ta,&tb);			// (A1+B1)(A2+B2)
	FP2_add(&t,&w1,&w2);
	FP2_neg(&t,&t);
	FP2_add(&tc,&tc,&t);			// (A1+B1)(A2+B2)-A1.A2-B1*B2 =  (A1.B2+A2.B1)		
				
	FP2_add(&ta,&(w->a).a,&(w->c).b); // A1+F1
	FP2_add(&tb,&(y->a).a,&(y->c).b); // A2+F2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&td,&ta,&tb);			// (A1+F1)(A2+F2)
	FP2_add(&t,&w1,&w3);
	FP2_neg(&t,&t);
	FP2_add(&td,&td,&t);			// (A1+F1)(A2+F2)-A1.A2-F1*F2 =  (A1.F2+A2.F1)		

	FP2_add(&ta,&(w->a).b,&(w->c).b); // B1+F1
	FP2_add(&tb,&(y->a).b,&(y->c).b); // B2+F2
	FP2_norm(&ta);
	FP2_norm(&tb);
	FP2_mul(&te,&ta,&tb);			// (B1+F1)(B2+F2)
	FP2_add(&t,&w2,&w3);
	FP2_neg(&t,&t);
	FP2_add(&te,&te,&t);			// (B1+F1)(B2+F2)-B1.B2-F1*F2 =  (B1.F2+B2.F1)	

	FP2_mul_ip(&w2);
	FP2_add(&w1,&w1,&w2);
	FP4_from_FP2s(&(w->a),&w1,&tc);

	FP2_mul_ip(&w3);
	FP2_norm(&w3);
	FP4_from_FP2H(&(w->b),&w3);

	FP2_norm(&te);
	FP2_mul_ip(&te);
	FP4_from_FP2s(&(w->c),&te,&td);

	FP4_norm(&(w->a));
	FP4_norm(&(w->c));
#endif

//	}
	w->type=FP_SPARSE;
}

/* Set w=1/x */
/* SU= 600 */
void YYY::FP12_inv(FP12 *w,FP12 *x)
{
    FP4 f0,f1,f2,f3;

    FP4_sqr(&f0,&(x->a));
    FP4_mul(&f1,&(x->b),&(x->c));
    FP4_times_i(&f1);
    FP4_sub(&f0,&f0,&f1);  /* y.a */
	FP4_norm(&f0); 		

    FP4_sqr(&f1,&(x->c));
    FP4_times_i(&f1);
    FP4_mul(&f2,&(x->a),&(x->b));
    FP4_sub(&f1,&f1,&f2);  /* y.b */
	FP4_norm(&f1); 

    FP4_sqr(&f2,&(x->b));
    FP4_mul(&f3,&(x->a),&(x->c));
    FP4_sub(&f2,&f2,&f3);  /* y.c */
	FP4_norm(&f2); 

    FP4_mul(&f3,&(x->b),&f2);
    FP4_times_i(&f3);
    FP4_mul(&(w->a),&f0,&(x->a));
    FP4_add(&f3,&(w->a),&f3);
    FP4_mul(&(w->c),&f1,&(x->c));
    FP4_times_i(&(w->c));

    FP4_add(&f3,&(w->c),&f3);
	FP4_norm(&f3);
	
    FP4_inv(&f3,&f3);

    FP4_mul(&(w->a),&f0,&f3);
    FP4_mul(&(w->b),&f1,&f3);
    FP4_mul(&(w->c),&f2,&f3);
	w->type=FP_DENSE;
}

/* constant time powering by small integer of max length bts */

void YYY::FP12_pinpow(FP12 *r,int e,int bts)
{
    int i,b;
    FP12 R[2];

    FP12_one(&R[0]);
    FP12_copy(&R[1],r);

    for (i=bts-1; i>=0; i--)
    {
        b=(e>>i)&1;
        FP12_mul(&R[1-b],&R[b]);
        FP12_usqr(&R[b],&R[b]);
    }
    FP12_copy(r,&R[0]);
}

/* Compressed powering of unitary elements y=x^(e mod r) */

void YYY::FP12_compow(FP4 *c,FP12 *x,BIG e,BIG r)
{
    FP12 g1,g2;
	FP4 cp,cpm1,cpm2;
    FP2 f;
	BIG q,a,b,m;

    BIG_rcopy(a,Fra);
    BIG_rcopy(b,Frb);
    FP2_from_BIGs(&f,a,b);

    BIG_rcopy(q,Modulus);

    FP12_copy(&g1,x);
	FP12_copy(&g2,x);

    BIG_copy(m,q);
    BIG_mod(m,r);

    BIG_copy(a,e);
    BIG_mod(a,m);

    BIG_copy(b,e);
    BIG_sdiv(b,m);

    FP12_trace(c,&g1);

	if (BIG_iszilch(b))
	{
		FP4_xtr_pow(c,c,e);
		return;
	}

    FP12_frob(&g2,&f);
    FP12_trace(&cp,&g2);

    FP12_conj(&g1,&g1);
    FP12_mul(&g2,&g1);
    FP12_trace(&cpm1,&g2);
    FP12_mul(&g2,&g1);
    FP12_trace(&cpm2,&g2);

    FP4_xtr_pow2(c,&cp,c,&cpm1,&cpm2,a,b);

}

/* Note this is simple square and multiply, so not side-channel safe */
/* But fast for final exponentiation where exponent is not a secret */

void YYY::FP12_pow(FP12 *r,FP12 *a,BIG b)
{
    FP12 w,sf;
    BIG b1,b3;
    int i,nb,bt;
	BIG_copy(b1,b);
	BIG_norm(b1);
	BIG_pmul(b3,b1,3);
	BIG_norm(b3);
	FP12_copy(&sf,a);
	FP12_norm(&sf);
    FP12_copy(&w,&sf);

	nb=BIG_nbits(b3);
	for (i=nb-2;i>=1;i--)
	{
		FP12_usqr(&w,&w);
		bt=BIG_bit(b3,i)-BIG_bit(b1,i);
		if (bt==1)
			FP12_mul(&w,&sf);
		if (bt==-1)
		{
			FP12_conj(&sf,&sf);
			FP12_mul(&w,&sf);
			FP12_conj(&sf,&sf);
		}
	}

	FP12_copy(r,&w);
	FP12_reduce(r);


}


/* p=q0^u0.q1^u1.q2^u2.q3^u3 */
/* Side channel attack secure */
// Bos & Costello https://eprint.iacr.org/2013/458.pdf
// Faz-Hernandez & Longa & Sanchez  https://eprint.iacr.org/2013/158.pdf

void YYY::FP12_pow4(FP12 *p,FP12 *q,BIG u[4])
{
    int i,j,k,nb,pb,bt;
	FP12 g[8],r;
	BIG t[4],mt;
    sign8 w[NLEN_XXX*BASEBITS_XXX+1];
    sign8 s[NLEN_XXX*BASEBITS_XXX+1];

    for (i=0; i<4; i++)
        BIG_copy(t[i],u[i]);


// Precomputed table
    FP12_copy(&g[0],&q[0]); // q[0]
    FP12_copy(&g[1],&g[0]);
	FP12_mul(&g[1],&q[1]);	// q[0].q[1]
    FP12_copy(&g[2],&g[0]);
	FP12_mul(&g[2],&q[2]);	// q[0].q[2]
	FP12_copy(&g[3],&g[1]);
	FP12_mul(&g[3],&q[2]);	// q[0].q[1].q[2]
	FP12_copy(&g[4],&g[0]);
	FP12_mul(&g[4],&q[3]);  // q[0].q[3]
	FP12_copy(&g[5],&g[1]);
	FP12_mul(&g[5],&q[3]);	// q[0].q[1].q[3]
	FP12_copy(&g[6],&g[2]);
	FP12_mul(&g[6],&q[3]);	// q[0].q[2].q[3]
	FP12_copy(&g[7],&g[3]);
	FP12_mul(&g[7],&q[3]);	// q[0].q[1].q[2].q[3]

// Make it odd
	pb=1-BIG_parity(t[0]);
	BIG_inc(t[0],pb);
	BIG_norm(t[0]);

// Number of bits
    BIG_zero(mt);
    for (i=0; i<4; i++)
    {
        BIG_or(mt,mt,t[i]);
    }
    nb=1+BIG_nbits(mt);

// Sign pivot 
	s[nb-1]=1;
	for (i=0;i<nb-1;i++)
	{
        BIG_fshr(t[0],1);
		s[i]=2*BIG_parity(t[0])-1;
	}

// Recoded exponent
    for (i=0; i<nb; i++)
    {
		w[i]=0;
		k=1;
		for (j=1; j<4; j++)
		{
			bt=s[i]*BIG_parity(t[j]);
			BIG_fshr(t[j],1);

			BIG_dec(t[j],(bt>>1));
			BIG_norm(t[j]);
			w[i]+=bt*k;
			k*=2;
        }
    }		

// Main loop
	FP12_select(p,g,2*w[nb-1]+1);
    for (i=nb-2; i>=0; i--)
    {
        FP12_select(&r,g,2*w[i]+s[i]);
		FP12_usqr(p,p);
        FP12_mul(p,&r);
    }
// apply correction
	FP12_conj(&r,&q[0]);   
	FP12_mul(&r,p);
	FP12_cmove(p,&r,pb);

	FP12_reduce(p);
}

/* Set w=w^p using Frobenius */
/* SU= 160 */
void YYY::FP12_frob(FP12 *w,FP2 *f)
{
    FP2 f2,f3;
    FP2_sqr(&f2,f);     /* f2=f^2 */
    FP2_mul(&f3,&f2,f); /* f3=f^3 */

    FP4_frob(&(w->a),&f3);
    FP4_frob(&(w->b),&f3);
    FP4_frob(&(w->c),&f3);

    FP4_pmul(&(w->b),&(w->b),f);
    FP4_pmul(&(w->c),&(w->c),&f2);
	w->type=FP_DENSE;
}

/* SU= 8 */
/* normalise all components of w */
void YYY::FP12_norm(FP12 *w)
{
    FP4_norm(&(w->a));
    FP4_norm(&(w->b));
    FP4_norm(&(w->c));
}

/* SU= 8 */
/* reduce all components of w */
void YYY::FP12_reduce(FP12 *w)
{
    FP4_reduce(&(w->a));
    FP4_reduce(&(w->b));
    FP4_reduce(&(w->c));
}

/* trace function w=trace(x) */
/* SU= 8 */
void YYY::FP12_trace(FP4 *w,FP12 *x)
{
    FP4_imul(w,&(x->a),3);
    FP4_reduce(w);
}

/* SU= 8 */
/* Output w in hex */
void YYY::FP12_output(FP12 *w)
{
    printf("[");
    FP4_output(&(w->a));
    printf(",");
    FP4_output(&(w->b));
    printf(",");
    FP4_output(&(w->c));
    printf("]");
}

/* SU= 64 */
/* Convert g to octet string w */
void YYY::FP12_toOctet(octet *W,FP12 *g)
{
    BIG a;
    W->len=12*MODBYTES_XXX;

    FP_redc(a,&(g->a.a.a));
    BIG_toBytes(&(W->val[0]),a);
    FP_redc(a,&(g->a.a.b));
    BIG_toBytes(&(W->val[MODBYTES_XXX]),a);
    FP_redc(a,&(g->a.b.a));
    BIG_toBytes(&(W->val[2*MODBYTES_XXX]),a);
    FP_redc(a,&(g->a.b.b));
    BIG_toBytes(&(W->val[3*MODBYTES_XXX]),a);
    FP_redc(a,&(g->b.a.a));
    BIG_toBytes(&(W->val[4*MODBYTES_XXX]),a);
    FP_redc(a,&(g->b.a.b));
    BIG_toBytes(&(W->val[5*MODBYTES_XXX]),a);
    FP_redc(a,&(g->b.b.a));
    BIG_toBytes(&(W->val[6*MODBYTES_XXX]),a);
    FP_redc(a,&(g->b.b.b));
    BIG_toBytes(&(W->val[7*MODBYTES_XXX]),a);
    FP_redc(a,&(g->c.a.a));
    BIG_toBytes(&(W->val[8*MODBYTES_XXX]),a);
    FP_redc(a,&(g->c.a.b));
    BIG_toBytes(&(W->val[9*MODBYTES_XXX]),a);
    FP_redc(a,&(g->c.b.a));
    BIG_toBytes(&(W->val[10*MODBYTES_XXX]),a);
    FP_redc(a,&(g->c.b.b));
    BIG_toBytes(&(W->val[11*MODBYTES_XXX]),a);
}

/* SU= 24 */
/* Restore g from octet string w */
void YYY::FP12_fromOctet(FP12 *g,octet *W)
{
	BIG b;
    BIG_fromBytes(b,&W->val[0]);
    FP_nres(&(g->a.a.a),b);
    BIG_fromBytes(b,&W->val[MODBYTES_XXX]);
    FP_nres(&(g->a.a.b),b);
    BIG_fromBytes(b,&W->val[2*MODBYTES_XXX]);
    FP_nres(&(g->a.b.a),b);
    BIG_fromBytes(b,&W->val[3*MODBYTES_XXX]);
    FP_nres(&(g->a.b.b),b);
    BIG_fromBytes(b,&W->val[4*MODBYTES_XXX]);
    FP_nres(&(g->b.a.a),b);
    BIG_fromBytes(b,&W->val[5*MODBYTES_XXX]);
    FP_nres(&(g->b.a.b),b);
    BIG_fromBytes(b,&W->val[6*MODBYTES_XXX]);
    FP_nres(&(g->b.b.a),b);
    BIG_fromBytes(b,&W->val[7*MODBYTES_XXX]);
    FP_nres(&(g->b.b.b),b);
    BIG_fromBytes(b,&W->val[8*MODBYTES_XXX]);
    FP_nres(&(g->c.a.a),b);
    BIG_fromBytes(b,&W->val[9*MODBYTES_XXX]);
    FP_nres(&(g->c.a.b),b);
    BIG_fromBytes(b,&W->val[10*MODBYTES_XXX]);
    FP_nres(&(g->c.b.a),b);
    BIG_fromBytes(b,&W->val[11*MODBYTES_XXX]);
    FP_nres(&(g->c.b.b),b);
}

/* Move g to f
if d=1 */
void YYY::FP12_cmove(FP12 *f,FP12 *g,int d)
{
    FP4_cmove(&(f->a),&(g->a),d);
    FP4_cmove(&(f->b),&(g->b),d);
    FP4_cmove(&(f->c),&(g->c),d);
	d=~(d-1);
	f->type^=(f->type^g->type)&d;
}