version3/cpp/fp.cpp - incubator-milagro-crypto - Git at Google

 /*
 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 mod p functions */
 /* Small Finite Field arithmetic */
 /* SU=m, SU is Stack Usage (NOT_SPECIAL Modulus) */

 #include "fp_YYY.h"

 using namespace XXX;

 /* Fast Modular Reduction Methods */

 /* r=d mod m */
 /* d MUST be normalised */
 /* Products must be less than pR in all cases !!! */
 /* So when multiplying two numbers, their product *must* be less than MODBITS_YYY+BASEBITS_XXX*NLEN_XXX */
 /* Results *may* be one bit bigger than MODBITS_YYY */

 #if MODTYPE_YYY == PSEUDO_MERSENNE
 /* r=d mod m */

 /* Converts from BIG integer to residue form mod Modulus */
 void YYY::FP_nres(FP *y,BIG x)
 {
 	BIG_copy(y->g,x);
 	y->XES=1;
 }

 /* Converts from residue form back to BIG integer form */
 void YYY::FP_redc(BIG x,FP *y)
 {
 	BIG_copy(x,y->g);
 }

 /* reduce a DBIG to a BIG exploiting the special form of the modulus */
 void YYY::FP_mod(BIG r,DBIG d)
 {
     BIG t,b;
     chunk v,tw;
     BIG_split(t,b,d,MODBITS_YYY);

     /* Note that all of the excess gets pushed into t. So if squaring a value with a 4-bit excess, this results in
        t getting all 8 bits of the excess product! So products must be less than pR which is Montgomery compatible */

     if (MConst < NEXCESS_XXX)
     {
         BIG_imul(t,t,MConst);
         BIG_norm(t);
 		BIG_add(r,t,b);
 		BIG_norm(r);
         tw=r[NLEN_XXX-1];
         r[NLEN_XXX-1]&=TMASK_YYY;
         r[0]+=MConst*((tw>>TBITS_YYY));
     }
     else
     {
         v=BIG_pmul(t,t,MConst);
 		BIG_add(r,t,b);
 		BIG_norm(r);
         tw=r[NLEN_XXX-1];
         r[NLEN_XXX-1]&=TMASK_YYY;
 #if CHUNK == 16
         r[1]+=muladd(MConst,((tw>>TBITS_YYY)+(v<<(BASEBITS_XXX-TBITS_YYY))),0,&r[0]);
 #else
         r[0]+=MConst*((tw>>TBITS_YYY)+(v<<(BASEBITS_XXX-TBITS_YYY)));
 #endif
     }
     BIG_norm(r);
 }
 #endif

 /* This only applies to Curve C448, so specialised (for now) */
 #if MODTYPE_YYY == GENERALISED_MERSENNE

 void YYY::FP_nres(FP *y,BIG x)
 {
 	BIG_copy(y->g,x);
 	y->XES=1;
 }

 /* Converts from residue form back to BIG integer form */
 void YYY::FP_redc(BIG x,FP *y)
 {
 	BIG_copy(x,y->g);
 }

 /* reduce a DBIG to a BIG exploiting the special form of a modulus 2^m - 2^n -c */
 void YYY::FP_mod(BIG r,DBIG d)
 {
     BIG t,b,t2,b2;
 	int BTset=MBITS_YYY/2;
     chunk carry;
     BIG_split(t,b,d,MBITS_YYY);

     BIG_dscopy(d,t);
     BIG_dshl(d,BTset);

     BIG_split(t2,b2,d,MBITS_YYY);

 	BIG_add(b,b,b2);  // 2
 	BIG_add(t,t,t2);  // 2

     BIG_shl(t2,BTset);

 	BIG_add(b,b,t2); // 3
     BIG_norm(t);

 //	carry=0;
 // Now multiply t by MConst..(?) and extract carry
 //	if (MConst!=1)
 //		carry=BIG_pmul(t,t,MConst);

 	BIG_add(r,t,b);
 	BIG_norm(r);

     carry=r[NLEN_XXX-1]>>TBITS_YYY;// + (carry<<(BASEBITS_XXX-TBITS_YYY));
     r[NLEN_XXX-1]&=TMASK_YYY;

     r[BTset/BASEBITS_XXX]+=carry<<(BTset%BASEBITS_XXX); /* need to check that this falls mid-word */
 //	if (MConst!=1) carry*=MConst;
 	r[0]+=carry;

     BIG_norm(r);
 }

 #endif

 #if MODTYPE_YYY == MONTGOMERY_FRIENDLY

 /* convert to Montgomery n-residue form */
 void YYY::FP_nres(FP *y,BIG x)
 {
     DBIG d;
     BIG r;
     BIG_rcopy(r,R2modp);
 	BIG_mul(d,x,r);
 	FP_mod(y->g,d);
 	y->XES=2;
 }

 /* convert back to regular form */
 void YYY::FP_redc(BIG x,FP *y)
 {
     DBIG d;
     BIG_dzero(d);
     BIG_dscopy(d,y->g);
     FP_mod(x,d);
 }

 /* fast modular reduction from DBIG to BIG exploiting special form of the modulus */
 void YYY::FP_mod(BIG a,DBIG d)
 {
     int i;

     for (i=0; i<NLEN_XXX; i++)
         d[NLEN_XXX+i]+=muladd(d[i],MConst-1,d[i],&d[NLEN_XXX+i-1]);

     BIG_sducopy(a,d);
     BIG_norm(a);
 }

 #endif

 #if MODTYPE_YYY == NOT_SPECIAL

 /* convert to Montgomery n-residue form */
 void YYY::FP_nres(FP *y,BIG x)
 {
     DBIG d;
     BIG r;
     BIG_rcopy(r,R2modp);
 	BIG_mul(d,x,r);
 	FP_mod(y->g,d);
 	y->XES=2;
 }

 /* convert back to regular form */
 void YYY::FP_redc(BIG x,FP *y)
 {
     DBIG d;
     BIG_dzero(d);
     BIG_dscopy(d,y->g);
     FP_mod(x,d);
 }


 /* reduce a DBIG to a BIG using Montgomery's no trial division method */
 /* d is expected to be dnormed before entry */
 /* SU= 112 */
 void YYY::FP_mod(BIG a,DBIG d)
 {
 	BIG mdls;
 	BIG_rcopy(mdls,Modulus);
 	BIG_monty(a,mdls,MConst,d);
 }

 #endif

 /* test x==0 ? */
 /* SU= 48 */
 int YYY::FP_iszilch(FP *x)
 {
     BIG m,t;
     BIG_rcopy(m,Modulus);
 	BIG_copy(t,x->g);
     BIG_mod(t,m);
     return BIG_iszilch(t);
 }

 void YYY::FP_copy(FP *y,FP *x)
 {
 	BIG_copy(y->g,x->g);
 	y->XES=x->XES;
 }

 void YYY::FP_rcopy(FP *y, const BIG c)
 {
 	BIG b;
 	BIG_rcopy(b,c);
 	FP_nres(y,b);
 }

 /* Swap a and b if d=1 */
 void YYY::FP_cswap(FP *a,FP *b,int d)
 {
 	sign32 t,c=d;
 	BIG_cswap(a->g,b->g,d);

     c=~(c-1);
 	t=c&((a->XES)^(b->XES));
 	a->XES^=t;
 	b->XES^=t;

 }

 /* Move b to a if d=1 */
 void YYY::FP_cmove(FP *a,FP *b,int d)
 {
 	sign32 c=-d;

 	BIG_cmove(a->g,b->g,d);
 	a->XES^=(a->XES^b->XES)&c;
 }

 void YYY::FP_zero(FP *x)
 {
 	BIG_zero(x->g);
 	x->XES=1;
 }

 int YYY::FP_equals(FP *x,FP *y)
 {
 	FP xg,yg;
 	FP_copy(&xg,x);
 	FP_copy(&yg,y);
 	FP_reduce(&xg); FP_reduce(&yg);

 	if (BIG_comp(xg.g,yg.g)==0) return 1;
 	return 0;
 }

 /* output FP */
 /* SU= 48 */
 void YYY::FP_output(FP *r)
 {
     BIG c;
     FP_redc(c,r);
     BIG_output(c);
 }

 void YYY::FP_rawoutput(FP *r)
 {
     BIG_rawoutput(r->g);
 }

 #ifdef GET_STATS
 int tsqr=0,rsqr=0,tmul=0,rmul=0;
 int tadd=0,radd=0,tneg=0,rneg=0;
 int tdadd=0,rdadd=0,tdneg=0,rdneg=0;
 #endif

 #ifdef FUSED_MODMUL

 /* Insert fastest code here */

 #endif

 /* r=a*b mod Modulus */
 /* product must be less that p.R - and we need to know this in advance! */
 /* SU= 88 */
 void YYY::FP_mul(FP *r,FP *a,FP *b)
 {
     DBIG d;
     chunk ea,eb;

 	if ((sign64)a->XES*b->XES>(sign64)FEXCESS_YYY)
 	{
 #ifdef DEBUG_REDUCE
         printf("Product too large - reducing it\n");
 #endif
         FP_reduce(a);  /* it is sufficient to fully reduce just one of them < p */
 	}

 #ifdef FUSED_MODMUL
 	FP_modmul(r->g,a->g,b->g);
 #else
     BIG_mul(d,a->g,b->g);
     FP_mod(r->g,d);
 #endif
 	r->XES=2;
 }


 /* multiplication by an integer, r=a*c */
 /* SU= 136 */
 void YYY::FP_imul(FP *r,FP *a,int c)
 {
     DBIG d;
 	BIG k;
 	FP f;

     int s=0;

     if (c<0)
     {
         c=-c;
         s=1;
     }

 #if MODTYPE_YYY==PSEUDO_MERSENNE || MODTYPE_YYY==GENERALISED_MERSENNE

 	BIG_pxmul(d,a->g,c);
 	FP_mod(r->g,d);
 	r->XES=2;

 #else
 	//Montgomery
 	if (a->XES*c<=FEXCESS_YYY)
 	{
 		BIG_pmul(r->g,a->g,c);
 		r->XES=a->XES*c;    // careful here - XES jumps!
 	}
 	else
 	{// don't want to do this - only a problem for Montgomery modulus and larger constants
 		BIG_zero(k);
 		BIG_inc(k,c);
 		BIG_norm(k);
 		FP_nres(&f,k);
 		FP_mul(r,a,&f);
 	}
 #endif

     if (s)
 	{
 		FP_neg(r,r);
 		FP_norm(r);
 	}
 }

 /* Set r=a^2 mod m */
 /* SU= 88 */
 void YYY::FP_sqr(FP *r,FP *a)
 {
     DBIG d;

 	if ((sign64)a->XES*a->XES>(sign64)FEXCESS_YYY)
 	{
 #ifdef DEBUG_REDUCE
         printf("Product too large - reducing it\n");
 #endif
         FP_reduce(a);
     }

     BIG_sqr(d,a->g);
     FP_mod(r->g,d);
 	r->XES=2;
 }

 /* SU= 16 */
 /* Set r=a+b */
 void YYY::FP_add(FP *r,FP *a,FP *b)
 {
     BIG_add(r->g,a->g,b->g);
 	r->XES=a->XES+b->XES;
 	if (r->XES>FEXCESS_YYY)
 	{
 #ifdef DEBUG_REDUCE
         printf("Sum too large - reducing it \n");
 #endif
         FP_reduce(r);
     }
 }

 /* Set r=a-b mod m */
 /* SU= 56 */
 void YYY::FP_sub(FP *r,FP *a,FP *b)
 {
     FP n;
     FP_neg(&n,b);
     FP_add(r,a,&n);
 }

 // https://graphics.stanford.edu/~seander/bithacks.html
 // constant time log to base 2 (or number of bits in)

 static int logb2(unsign32 v)
 {
     int r;
     v |= v >> 1;
     v |= v >> 2;
     v |= v >> 4;
     v |= v >> 8;
     v |= v >> 16;

     v = v - ((v >> 1) & 0x55555555);
     v = (v & 0x33333333) + ((v >> 2) & 0x33333333);
     r = (((v + (v >> 4)) & 0xF0F0F0F) * 0x1010101) >> 24;
     return r;
 }

 // find appoximation to quotient of a/m
 // Out by at most 2.
 // Note that MAXXES is bounded to be 2-bits less than half a word
 static int quo(BIG n,BIG m)
 {
 	int sh;
 	chunk num,den;
 	int hb=CHUNK/2;
 	if (TBITS_YYY<hb)
 	{
 		sh=hb-TBITS_YYY;
 		num=(n[NLEN_XXX-1]<<sh)|(n[NLEN_XXX-2]>>(BASEBITS_XXX-sh));
 		den=(m[NLEN_XXX-1]<<sh)|(m[NLEN_XXX-2]>>(BASEBITS_XXX-sh));
 	}
 	else
 	{
 		num=n[NLEN_XXX-1];
 		den=m[NLEN_XXX-1];
 	}
 	return (int)(num/(den+1));
 }

 /* SU= 48 */
 /* Fully reduce a mod Modulus */
 void YYY::FP_reduce(FP *a)
 {
     BIG m,r;
 	int sr,sb,q;
 	chunk carry;

     BIG_rcopy(m,Modulus);
 	BIG_norm(a->g);

 	if (a->XES>16)
 	{
 		q=quo(a->g,m);
 		carry=BIG_pmul(r,m,q);
 		r[NLEN_XXX-1]+=(carry<<BASEBITS_XXX); // correction - put any carry out back in again
 		BIG_sub(a->g,a->g,r);
 		BIG_norm(a->g);
 		sb=2;
 	}
 	else sb=logb2(a->XES-1);  // sb does not depend on the actual data

 	BIG_fshl(m,sb);
 	while (sb>0)
 	{
 // constant time...
 		sr=BIG_ssn(r,a->g,m);  // optimized combined shift, subtract and norm
 		BIG_cmove(a->g,r,1-sr);
 		sb--;
 	}

 	a->XES=1;
 }

 void YYY::FP_norm(FP *x)
 {
 	BIG_norm(x->g);
 }

 /* Set r=-a mod Modulus */
 /* SU= 64 */
 void YYY::FP_neg(FP *r,FP *a)
 {
     int sb;
     BIG m;

     BIG_rcopy(m,Modulus);

     sb=logb2(a->XES-1);
     BIG_fshl(m,sb);
     BIG_sub(r->g,m,a->g);
 	r->XES=((sign32)1<<sb)+1;  // +1 to cover case where a is zero ?

     if (r->XES>FEXCESS_YYY)
     {
 #ifdef DEBUG_REDUCE
         printf("Negation too large -  reducing it \n");
 #endif
         FP_reduce(r);
 	}

 }

 /* Set r=a/2. */
 /* SU= 56 */
 void YYY::FP_div2(FP *r,FP *a)
 {
     BIG m;
     BIG_rcopy(m,Modulus);
     FP_copy(r,a);
     if (BIG_parity(a->g)==0)
     {

         BIG_fshr(r->g,1);
     }
     else
     {
         BIG_add(r->g,r->g,m);
         BIG_norm(r->g);
         BIG_fshr(r->g,1);
     }
 }

 #if MODTYPE_YYY==PSEUDO_MERSENNE  || MODTYPE_YYY==GENERALISED_MERSENNE

 // See eprint paper https://eprint.iacr.org/2018/1038
 // If p=3 mod 4 r= x^{(p-3)/4}, if p=5 mod 8 r=x^{(p-5)/8}

 void YYY::FP_fpow(FP *r,FP *x)
 {
 	int i,j,k,bw,w,nw,lo,m,n,c;
 	FP xp[11],t,key;
 	const int ac[]={1,2,3,6,12,15,30,60,120,240,255};
 // phase 1
 	FP_copy(&xp[0],x);	// 1
 	FP_sqr(&xp[1],x); // 2
 	FP_mul(&xp[2],&xp[1],x);  //3
 	FP_sqr(&xp[3],&xp[2]);  // 6
 	FP_sqr(&xp[4],&xp[3]); // 12
 	FP_mul(&xp[5],&xp[4],&xp[2]); // 15
 	FP_sqr(&xp[6],&xp[5]); // 30
 	FP_sqr(&xp[7],&xp[6]); // 60
 	FP_sqr(&xp[8],&xp[7]); // 120
 	FP_sqr(&xp[9],&xp[8]); // 240
 	FP_mul(&xp[10],&xp[9],&xp[5]); // 255

 #if MODTYPE_YYY==PSEUDO_MERSENNE
 	n=MODBITS_YYY;
 #endif
 #if MODTYPE_YYY==GENERALISED_MERSENNE  // Goldilocks ONLY
 	n=MODBITS_YYY/2;
 #endif

     if (MOD8_YYY==5)
     {
 		n-=3;
 		c=(MConst+5)/8;
 	} else {
 		n-=2;
 		c=(MConst+3)/4;
 	}

 	bw=0; w=1; while (w<c) {w*=2; bw+=1;}
 	k=w-c;

 	if (k!=0)
 	{
 		i=10; while (ac[i]>k) i--;
 		FP_copy(&key,&xp[i]);
 		k-=ac[i];
 	}
 	while (k!=0)
 	{
 		i--;
 		if (ac[i]>k) continue;
 		FP_mul(&key,&key,&xp[i]);
 		k-=ac[i];
 	}

 // phase 2
 	FP_copy(&xp[1],&xp[2]);
 	FP_copy(&xp[2],&xp[5]);
 	FP_copy(&xp[3],&xp[10]);

 	j=3; m=8;
 	nw=n-bw;
 	while (2*m<nw)
 	{
 		FP_copy(&t,&xp[j++]);
 		for (i=0;i<m;i++)
 			FP_sqr(&t,&t);
 		FP_mul(&xp[j],&xp[j-1],&t);
 		m*=2;
 	}

 	lo=nw-m;
 	FP_copy(r,&xp[j]);

 	while (lo!=0)
 	{
 		m/=2; j--;
 		if (lo<m) continue;
 		lo-=m;
 		FP_copy(&t,r);
 		for (i=0;i<m;i++)
 			FP_sqr(&t,&t);
 		FP_mul(r,&t,&xp[j]);
 	}
 // phase 3

 	if (bw!=0)
 	{
 		for (i=0;i<bw;i++ )
 			FP_sqr(r,r);
 		FP_mul(r,r,&key);
 	}

 #if MODTYPE_YYY==GENERALISED_MERSENNE  // Goldilocks ONLY
 	FP_copy(&key,r);
 	FP_sqr(&t,&key);
 	FP_mul(r,&t,x);
 	for (i=0;i<n+1;i++)
 		FP_sqr(r,r);
 	FP_mul(r,r,&key);
 #endif

 }

 void YYY::FP_inv(FP *r,FP *x)
 {
 	FP y,t;
 	FP_fpow(&y,x);
     if (MOD8_YYY==5)
     { // r=x^3.y^8
 		FP_sqr(&t,x);
 		FP_mul(&t,&t,x);
 		FP_sqr(&y,&y);
 		FP_sqr(&y,&y);
 		FP_sqr(&y,&y);
 		FP_mul(r,&t,&y);
 	} else {
 		FP_sqr(&y,&y);
 		FP_sqr(&y,&y);
 		FP_mul(r,&y,x);
 	}
 }

 #else

 void YYY::FP_pow(FP *r,FP *a,BIG b)
 {
 	sign8 w[1+(NLEN_XXX*BASEBITS_XXX+3)/4];
 	FP tb[16];
 	BIG t;
 	int i,nb;

 	FP_norm(a);
     BIG_norm(b);
 	BIG_copy(t,b);
 	nb=1+(BIG_nbits(t)+3)/4;
     // convert exponent to 4-bit window
     for (i=0; i<nb; i++)
     {
         w[i]=BIG_lastbits(t,4);
         BIG_dec(t,w[i]);
         BIG_norm(t);
         BIG_fshr(t,4);
     }

 	FP_one(&tb[0]);
 	FP_copy(&tb[1],a);
 	for (i=2;i<16;i++)
 		FP_mul(&tb[i],&tb[i-1],a);

 	FP_copy(r,&tb[w[nb-1]]);
     for (i=nb-2; i>=0; i--)
     {
 		FP_sqr(r,r);
 		FP_sqr(r,r);
 		FP_sqr(r,r);
 		FP_sqr(r,r);
 		FP_mul(r,r,&tb[w[i]]);
 	}
     FP_reduce(r);
 }

 /* set w=1/x */
 void YYY::FP_inv(FP *w,FP *x)
 {
  	BIG m2;
 	BIG_rcopy(m2,Modulus);
 	BIG_dec(m2,2);
 	BIG_norm(m2);
 	FP_pow(w,x,m2);
 }

 #endif

 /* SU=8 */
 /* set n=1 */
 void YYY::FP_one(FP *n)
 {
 	BIG b;
     BIG_one(b);
     FP_nres(n,b);
 }


 /* is r a QR? */
 int YYY::FP_qr(FP *r)
 {
     int j;
     BIG m;
 	BIG b;
     BIG_rcopy(m,Modulus);
     FP_redc(b,r);
     j=BIG_jacobi(b,m);
     FP_nres(r,b);
     if (j==1) return 1;
     return 0;

 }

 /* Set a=sqrt(b) mod Modulus */
 /* SU= 160 */
 void YYY::FP_sqrt(FP *r,FP *a)
 {
 	FP v,i;
     BIG b;
     BIG m;
     BIG_rcopy(m,Modulus);
     BIG_mod(a->g,m);
     BIG_copy(b,m);
     if (MOD8_YYY==5)
     {
         FP_copy(&i,a);
         BIG_fshl(i.g,1);
 #if MODTYPE_YYY == PSEUDO_MERSENNE   || MODTYPE_YYY==GENERALISED_MERSENNE
 		FP_fpow(&v,&i);
 #else
         BIG_dec(b,5);
         BIG_norm(b);
         BIG_fshr(b,3); /* (p-5)/8 */
         FP_pow(&v,&i,b);
 #endif
         FP_mul(&i,&i,&v);
         FP_mul(&i,&i,&v);
         BIG_dec(i.g,1);
         FP_mul(r,a,&v);
         FP_mul(r,r,&i);
         FP_reduce(r);
     }
     if (MOD8_YYY==3 || MOD8_YYY==7)
     {
 #if MODTYPE_YYY == PSEUDO_MERSENNE   || MODTYPE_YYY==GENERALISED_MERSENNE
 		FP_fpow(r,a);
 		FP_mul(r,r,a);
 #else
         BIG_inc(b,1);
         BIG_norm(b);
         BIG_fshr(b,2); /* (p+1)/4 */
         FP_pow(r,a,b);
 #endif
     }
 }
	/*
	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 mod p functions */
	/* Small Finite Field arithmetic */
	/* SU=m, SU is Stack Usage (NOT_SPECIAL Modulus) */

	#include "fp_YYY.h"

	using namespace XXX;

	/* Fast Modular Reduction Methods */

	/* r=d mod m */
	/* d MUST be normalised */
	/* Products must be less than pR in all cases !!! */
	/* So when multiplying two numbers, their product must be less than MODBITS_YYY+BASEBITS_XXXNLEN_XXX /
	/* Results may be one bit bigger than MODBITS_YYY */

	#if MODTYPE_YYY == PSEUDO_MERSENNE
	/* r=d mod m */

	/* Converts from BIG integer to residue form mod Modulus */
	void YYY::FP_nres(FP *y,BIG x)
	{
	BIG_copy(y->g,x);
	y->XES=1;
	}

	/* Converts from residue form back to BIG integer form */
	void YYY::FP_redc(BIG x,FP *y)
	{
	BIG_copy(x,y->g);
	}

	/* reduce a DBIG to a BIG exploiting the special form of the modulus */
	void YYY::FP_mod(BIG r,DBIG d)
	{
	BIG t,b;
	chunk v,tw;
	BIG_split(t,b,d,MODBITS_YYY);

	/* Note that all of the excess gets pushed into t. So if squaring a value with a 4-bit excess, this results in
	t getting all 8 bits of the excess product! So products must be less than pR which is Montgomery compatible */

	if (MConst < NEXCESS_XXX)
	{
	BIG_imul(t,t,MConst);
	BIG_norm(t);
	BIG_add(r,t,b);
	BIG_norm(r);
	tw=r[NLEN_XXX-1];
	r[NLEN_XXX-1]&=TMASK_YYY;
	r[0]+=MConst*((tw>>TBITS_YYY));
	}
	else
	{
	v=BIG_pmul(t,t,MConst);
	BIG_add(r,t,b);
	BIG_norm(r);
	tw=r[NLEN_XXX-1];
	r[NLEN_XXX-1]&=TMASK_YYY;
	#if CHUNK == 16
	r[1]+=muladd(MConst,((tw>>TBITS_YYY)+(v<<(BASEBITS_XXX-TBITS_YYY))),0,&r[0]);
	#else
	r[0]+=MConst*((tw>>TBITS_YYY)+(v<<(BASEBITS_XXX-TBITS_YYY)));
	#endif
	}
	BIG_norm(r);
	}
	#endif

	/* This only applies to Curve C448, so specialised (for now) */
	#if MODTYPE_YYY == GENERALISED_MERSENNE

	void YYY::FP_nres(FP *y,BIG x)
	{
	BIG_copy(y->g,x);
	y->XES=1;
	}

	/* Converts from residue form back to BIG integer form */
	void YYY::FP_redc(BIG x,FP *y)
	{
	BIG_copy(x,y->g);
	}

	/* reduce a DBIG to a BIG exploiting the special form of a modulus 2^m - 2^n -c */
	void YYY::FP_mod(BIG r,DBIG d)
	{
	BIG t,b,t2,b2;
	int BTset=MBITS_YYY/2;
	chunk carry;
	BIG_split(t,b,d,MBITS_YYY);

	BIG_dscopy(d,t);
	BIG_dshl(d,BTset);

	BIG_split(t2,b2,d,MBITS_YYY);

	BIG_add(b,b,b2); // 2
	BIG_add(t,t,t2); // 2

	BIG_shl(t2,BTset);

	BIG_add(b,b,t2); // 3
	BIG_norm(t);

	// carry=0;
	// Now multiply t by MConst..(?) and extract carry
	// if (MConst!=1)
	// carry=BIG_pmul(t,t,MConst);

	BIG_add(r,t,b);
	BIG_norm(r);

	carry=r[NLEN_XXX-1]>>TBITS_YYY;// + (carry<<(BASEBITS_XXX-TBITS_YYY));
	r[NLEN_XXX-1]&=TMASK_YYY;

	r[BTset/BASEBITS_XXX]+=carry<<(BTset%BASEBITS_XXX); /* need to check that this falls mid-word */
	// if (MConst!=1) carry*=MConst;
	r[0]+=carry;

	BIG_norm(r);
	}

	#endif

	#if MODTYPE_YYY == MONTGOMERY_FRIENDLY

	/* convert to Montgomery n-residue form */
	void YYY::FP_nres(FP *y,BIG x)
	{
	DBIG d;
	BIG r;
	BIG_rcopy(r,R2modp);
	BIG_mul(d,x,r);
	FP_mod(y->g,d);
	y->XES=2;
	}

	/* convert back to regular form */
	void YYY::FP_redc(BIG x,FP *y)
	{
	DBIG d;
	BIG_dzero(d);
	BIG_dscopy(d,y->g);
	FP_mod(x,d);
	}

	/* fast modular reduction from DBIG to BIG exploiting special form of the modulus */
	void YYY::FP_mod(BIG a,DBIG d)
	{
	int i;

	for (i=0; i<NLEN_XXX; i++)
	d[NLEN_XXX+i]+=muladd(d[i],MConst-1,d[i],&d[NLEN_XXX+i-1]);

	BIG_sducopy(a,d);
	BIG_norm(a);
	}

	#endif

	#if MODTYPE_YYY == NOT_SPECIAL

	/* convert to Montgomery n-residue form */
	void YYY::FP_nres(FP *y,BIG x)
	{
	DBIG d;
	BIG r;
	BIG_rcopy(r,R2modp);
	BIG_mul(d,x,r);
	FP_mod(y->g,d);
	y->XES=2;
	}

	/* convert back to regular form */
	void YYY::FP_redc(BIG x,FP *y)
	{
	DBIG d;
	BIG_dzero(d);
	BIG_dscopy(d,y->g);
	FP_mod(x,d);
	}


	/* reduce a DBIG to a BIG using Montgomery's no trial division method */
	/* d is expected to be dnormed before entry */
	/* SU= 112 */
	void YYY::FP_mod(BIG a,DBIG d)
	{
	BIG mdls;
	BIG_rcopy(mdls,Modulus);
	BIG_monty(a,mdls,MConst,d);
	}

	#endif

	/* test x==0 ? */
	/* SU= 48 */
	int YYY::FP_iszilch(FP *x)
	{
	BIG m,t;
	BIG_rcopy(m,Modulus);
	BIG_copy(t,x->g);
	BIG_mod(t,m);
	return BIG_iszilch(t);
	}

	void YYY::FP_copy(FP y,FP x)
	{
	BIG_copy(y->g,x->g);
	y->XES=x->XES;
	}

	void YYY::FP_rcopy(FP *y, const BIG c)
	{
	BIG b;
	BIG_rcopy(b,c);
	FP_nres(y,b);
	}

	/* Swap a and b if d=1 */
	void YYY::FP_cswap(FP a,FP b,int d)
	{
	sign32 t,c=d;
	BIG_cswap(a->g,b->g,d);

	c=~(c-1);
	t=c&((a->XES)^(b->XES));
	a->XES^=t;
	b->XES^=t;

	}

	/* Move b to a if d=1 */
	void YYY::FP_cmove(FP a,FP b,int d)
	{
	sign32 c=-d;

	BIG_cmove(a->g,b->g,d);
	a->XES^=(a->XES^b->XES)&c;
	}

	void YYY::FP_zero(FP *x)
	{
	BIG_zero(x->g);
	x->XES=1;
	}

	int YYY::FP_equals(FP x,FP y)
	{
	FP xg,yg;
	FP_copy(&xg,x);
	FP_copy(&yg,y);
	FP_reduce(&xg); FP_reduce(&yg);

	if (BIG_comp(xg.g,yg.g)==0) return 1;
	return 0;
	}

	/* output FP */
	/* SU= 48 */
	void YYY::FP_output(FP *r)
	{
	BIG c;
	FP_redc(c,r);
	BIG_output(c);
	}

	void YYY::FP_rawoutput(FP *r)
	{
	BIG_rawoutput(r->g);
	}

	#ifdef GET_STATS
	int tsqr=0,rsqr=0,tmul=0,rmul=0;
	int tadd=0,radd=0,tneg=0,rneg=0;
	int tdadd=0,rdadd=0,tdneg=0,rdneg=0;
	#endif

	#ifdef FUSED_MODMUL

	/* Insert fastest code here */

	#endif

	/* r=ab mod Modulus /
	/* product must be less that p.R - and we need to know this in advance! */
	/* SU= 88 */
	void YYY::FP_mul(FP r,FP a,FP *b)
	{
	DBIG d;
	chunk ea,eb;

	if ((sign64)a->XES*b->XES>(sign64)FEXCESS_YYY)
	{
	#ifdef DEBUG_REDUCE
	printf("Product too large - reducing it\n");
	#endif
	FP_reduce(a); /* it is sufficient to fully reduce just one of them < p */
	}

	#ifdef FUSED_MODMUL
	FP_modmul(r->g,a->g,b->g);
	#else
	BIG_mul(d,a->g,b->g);
	FP_mod(r->g,d);
	#endif
	r->XES=2;
	}


	/* multiplication by an integer, r=ac /
	/* SU= 136 */
	void YYY::FP_imul(FP r,FP a,int c)
	{
	DBIG d;
	BIG k;
	FP f;

	int s=0;

	if (c<0)
	{
	c=-c;
	s=1;
	}

	#if MODTYPE_YYY==PSEUDO_MERSENNE \|\| MODTYPE_YYY==GENERALISED_MERSENNE

	BIG_pxmul(d,a->g,c);
	FP_mod(r->g,d);
	r->XES=2;

	#else
	//Montgomery
	if (a->XES*c<=FEXCESS_YYY)
	{
	BIG_pmul(r->g,a->g,c);
	r->XES=a->XES*c; // careful here - XES jumps!
	}
	else
	{// don't want to do this - only a problem for Montgomery modulus and larger constants
	BIG_zero(k);
	BIG_inc(k,c);
	BIG_norm(k);
	FP_nres(&f,k);
	FP_mul(r,a,&f);
	}
	#endif

	if (s)
	{
	FP_neg(r,r);
	FP_norm(r);
	}
	}

	/* Set r=a^2 mod m */
	/* SU= 88 */
	void YYY::FP_sqr(FP r,FP a)
	{
	DBIG d;

	if ((sign64)a->XES*a->XES>(sign64)FEXCESS_YYY)
	{
	#ifdef DEBUG_REDUCE
	printf("Product too large - reducing it\n");
	#endif
	FP_reduce(a);
	}

	BIG_sqr(d,a->g);
	FP_mod(r->g,d);
	r->XES=2;
	}

	/* SU= 16 */
	/* Set r=a+b */
	void YYY::FP_add(FP r,FP a,FP *b)
	{
	BIG_add(r->g,a->g,b->g);
	r->XES=a->XES+b->XES;
	if (r->XES>FEXCESS_YYY)
	{
	#ifdef DEBUG_REDUCE
	printf("Sum too large - reducing it \n");
	#endif
	FP_reduce(r);
	}
	}

	/* Set r=a-b mod m */
	/* SU= 56 */
	void YYY::FP_sub(FP r,FP a,FP *b)
	{
	FP n;
	FP_neg(&n,b);
	FP_add(r,a,&n);
	}

	// https://graphics.stanford.edu/~seander/bithacks.html
	// constant time log to base 2 (or number of bits in)

	static int logb2(unsign32 v)
	{
	int r;
	v \|= v >> 1;
	v \|= v >> 2;
	v \|= v >> 4;
	v \|= v >> 8;
	v \|= v >> 16;

	v = v - ((v >> 1) & 0x55555555);
	v = (v & 0x33333333) + ((v >> 2) & 0x33333333);
	r = (((v + (v >> 4)) & 0xF0F0F0F) * 0x1010101) >> 24;
	return r;
	}

	// find appoximation to quotient of a/m
	// Out by at most 2.
	// Note that MAXXES is bounded to be 2-bits less than half a word
	static int quo(BIG n,BIG m)
	{
	int sh;
	chunk num,den;
	int hb=CHUNK/2;
	if (TBITS_YYY<hb)
	{
	sh=hb-TBITS_YYY;
	num=(n[NLEN_XXX-1]<<sh)\|(n[NLEN_XXX-2]>>(BASEBITS_XXX-sh));
	den=(m[NLEN_XXX-1]<<sh)\|(m[NLEN_XXX-2]>>(BASEBITS_XXX-sh));
	}
	else
	{
	num=n[NLEN_XXX-1];
	den=m[NLEN_XXX-1];
	}
	return (int)(num/(den+1));
	}

	/* SU= 48 */
	/* Fully reduce a mod Modulus */
	void YYY::FP_reduce(FP *a)
	{
	BIG m,r;
	int sr,sb,q;
	chunk carry;

	BIG_rcopy(m,Modulus);
	BIG_norm(a->g);

	if (a->XES>16)
	{
	q=quo(a->g,m);
	carry=BIG_pmul(r,m,q);
	r[NLEN_XXX-1]+=(carry<<BASEBITS_XXX); // correction - put any carry out back in again
	BIG_sub(a->g,a->g,r);
	BIG_norm(a->g);
	sb=2;
	}
	else sb=logb2(a->XES-1); // sb does not depend on the actual data

	BIG_fshl(m,sb);
	while (sb>0)
	{
	// constant time...
	sr=BIG_ssn(r,a->g,m); // optimized combined shift, subtract and norm
	BIG_cmove(a->g,r,1-sr);
	sb--;
	}

	a->XES=1;
	}

	void YYY::FP_norm(FP *x)
	{
	BIG_norm(x->g);
	}

	/* Set r=-a mod Modulus */
	/* SU= 64 */
	void YYY::FP_neg(FP r,FP a)
	{
	int sb;
	BIG m;

	BIG_rcopy(m,Modulus);

	sb=logb2(a->XES-1);
	BIG_fshl(m,sb);
	BIG_sub(r->g,m,a->g);
	r->XES=((sign32)1<<sb)+1; // +1 to cover case where a is zero ?

	if (r->XES>FEXCESS_YYY)
	{
	#ifdef DEBUG_REDUCE
	printf("Negation too large - reducing it \n");
	#endif
	FP_reduce(r);
	}

	}

	/* Set r=a/2. */
	/* SU= 56 */
	void YYY::FP_div2(FP r,FP a)
	{
	BIG m;
	BIG_rcopy(m,Modulus);
	FP_copy(r,a);
	if (BIG_parity(a->g)==0)
	{

	BIG_fshr(r->g,1);
	}
	else
	{
	BIG_add(r->g,r->g,m);
	BIG_norm(r->g);
	BIG_fshr(r->g,1);
	}
	}

	#if MODTYPE_YYY==PSEUDO_MERSENNE \|\| MODTYPE_YYY==GENERALISED_MERSENNE

	// See eprint paper https://eprint.iacr.org/2018/1038
	// If p=3 mod 4 r= x^{(p-3)/4}, if p=5 mod 8 r=x^{(p-5)/8}

	void YYY::FP_fpow(FP r,FP x)
	{
	int i,j,k,bw,w,nw,lo,m,n,c;
	FP xp[11],t,key;
	const int ac[]={1,2,3,6,12,15,30,60,120,240,255};
	// phase 1
	FP_copy(&xp[0],x); // 1
	FP_sqr(&xp[1],x); // 2
	FP_mul(&xp[2],&xp[1],x); //3
	FP_sqr(&xp[3],&xp[2]); // 6
	FP_sqr(&xp[4],&xp[3]); // 12
	FP_mul(&xp[5],&xp[4],&xp[2]); // 15
	FP_sqr(&xp[6],&xp[5]); // 30
	FP_sqr(&xp[7],&xp[6]); // 60
	FP_sqr(&xp[8],&xp[7]); // 120
	FP_sqr(&xp[9],&xp[8]); // 240
	FP_mul(&xp[10],&xp[9],&xp[5]); // 255

	#if MODTYPE_YYY==PSEUDO_MERSENNE
	n=MODBITS_YYY;
	#endif
	#if MODTYPE_YYY==GENERALISED_MERSENNE // Goldilocks ONLY
	n=MODBITS_YYY/2;
	#endif

	if (MOD8_YYY==5)
	{
	n-=3;
	c=(MConst+5)/8;
	} else {
	n-=2;
	c=(MConst+3)/4;
	}

	bw=0; w=1; while (w<c) {w*=2; bw+=1;}
	k=w-c;

	if (k!=0)
	{
	i=10; while (ac[i]>k) i--;
	FP_copy(&key,&xp[i]);
	k-=ac[i];
	}
	while (k!=0)
	{
	i--;
	if (ac[i]>k) continue;
	FP_mul(&key,&key,&xp[i]);
	k-=ac[i];
	}

	// phase 2
	FP_copy(&xp[1],&xp[2]);
	FP_copy(&xp[2],&xp[5]);
	FP_copy(&xp[3],&xp[10]);

	j=3; m=8;
	nw=n-bw;
	while (2*m<nw)
	{
	FP_copy(&t,&xp[j++]);
	for (i=0;i<m;i++)
	FP_sqr(&t,&t);
	FP_mul(&xp[j],&xp[j-1],&t);
	m*=2;
	}

	lo=nw-m;
	FP_copy(r,&xp[j]);

	while (lo!=0)
	{
	m/=2; j--;
	if (lo<m) continue;
	lo-=m;
	FP_copy(&t,r);
	for (i=0;i<m;i++)
	FP_sqr(&t,&t);
	FP_mul(r,&t,&xp[j]);
	}
	// phase 3

	if (bw!=0)
	{
	for (i=0;i<bw;i++ )
	FP_sqr(r,r);
	FP_mul(r,r,&key);
	}

	#if MODTYPE_YYY==GENERALISED_MERSENNE // Goldilocks ONLY
	FP_copy(&key,r);
	FP_sqr(&t,&key);
	FP_mul(r,&t,x);
	for (i=0;i<n+1;i++)
	FP_sqr(r,r);
	FP_mul(r,r,&key);
	#endif

	}

	void YYY::FP_inv(FP r,FP x)
	{
	FP y,t;
	FP_fpow(&y,x);
	if (MOD8_YYY==5)
	{ // r=x^3.y^8
	FP_sqr(&t,x);
	FP_mul(&t,&t,x);
	FP_sqr(&y,&y);
	FP_sqr(&y,&y);
	FP_sqr(&y,&y);
	FP_mul(r,&t,&y);
	} else {
	FP_sqr(&y,&y);
	FP_sqr(&y,&y);
	FP_mul(r,&y,x);
	}
	}

	#else

	void YYY::FP_pow(FP r,FP a,BIG b)
	{
	sign8 w[1+(NLEN_XXX*BASEBITS_XXX+3)/4];
	FP tb[16];
	BIG t;
	int i,nb;

	FP_norm(a);
	BIG_norm(b);
	BIG_copy(t,b);
	nb=1+(BIG_nbits(t)+3)/4;
	// convert exponent to 4-bit window
	for (i=0; i<nb; i++)
	{
	w[i]=BIG_lastbits(t,4);
	BIG_dec(t,w[i]);
	BIG_norm(t);
	BIG_fshr(t,4);
	}

	FP_one(&tb[0]);
	FP_copy(&tb[1],a);
	for (i=2;i<16;i++)
	FP_mul(&tb[i],&tb[i-1],a);

	FP_copy(r,&tb[w[nb-1]]);
	for (i=nb-2; i>=0; i--)
	{
	FP_sqr(r,r);
	FP_sqr(r,r);
	FP_sqr(r,r);
	FP_sqr(r,r);
	FP_mul(r,r,&tb[w[i]]);
	}
	FP_reduce(r);
	}

	/* set w=1/x */
	void YYY::FP_inv(FP w,FP x)
	{
	BIG m2;
	BIG_rcopy(m2,Modulus);
	BIG_dec(m2,2);
	BIG_norm(m2);
	FP_pow(w,x,m2);
	}

	#endif

	/* SU=8 */
	/* set n=1 */
	void YYY::FP_one(FP *n)
	{
	BIG b;
	BIG_one(b);
	FP_nres(n,b);
	}



	/* is r a QR? */
	int YYY::FP_qr(FP *r)
	{
	int j;
	BIG m;
	BIG b;
	BIG_rcopy(m,Modulus);
	FP_redc(b,r);
	j=BIG_jacobi(b,m);
	FP_nres(r,b);
	if (j==1) return 1;
	return 0;

	}

	/* Set a=sqrt(b) mod Modulus */
	/* SU= 160 */
	void YYY::FP_sqrt(FP r,FP a)
	{
	FP v,i;
	BIG b;
	BIG m;
	BIG_rcopy(m,Modulus);
	BIG_mod(a->g,m);
	BIG_copy(b,m);
	if (MOD8_YYY==5)
	{
	FP_copy(&i,a);
	BIG_fshl(i.g,1);
	#if MODTYPE_YYY == PSEUDO_MERSENNE \|\| MODTYPE_YYY==GENERALISED_MERSENNE
	FP_fpow(&v,&i);
	#else
	BIG_dec(b,5);
	BIG_norm(b);
	BIG_fshr(b,3); /* (p-5)/8 */
	FP_pow(&v,&i,b);
	#endif
	FP_mul(&i,&i,&v);
	FP_mul(&i,&i,&v);
	BIG_dec(i.g,1);
	FP_mul(r,a,&v);
	FP_mul(r,r,&i);
	FP_reduce(r);
	}
	if (MOD8_YYY==3 \|\| MOD8_YYY==7)
	{
	#if MODTYPE_YYY == PSEUDO_MERSENNE \|\| MODTYPE_YYY==GENERALISED_MERSENNE
	FP_fpow(r,a);
	FP_mul(r,r,a);
	#else
	BIG_inc(b,1);
	BIG_norm(b);
	BIG_fshr(b,2); /* (p+1)/4 */
	FP_pow(r,a,b);
	#endif
	}
	}