ge25519.c - security/openssh-library - Rivoreo Source Code Repositories

 /* $OpenBSD: ge25519.c,v 1.2 2013/12/07 00:26:37 djm Exp $ */

 /* Public Domain, from supercop-20130419/crypto_sign/ed25519/ref/ge25519.c */

 #include "fe25519.h"
 #include "sc25519.h"
 #include "ge25519.h"

 /*
  * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
  * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
  * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
  */

 /* d */
 static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
                       0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
 /* 2*d */
 static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
                        0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
 /* sqrt(-1) */
 static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
                          0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};

 #define ge25519_p3 ge25519

 typedef struct
 {
   fe25519 x;
   fe25519 z;
   fe25519 y;
   fe25519 t;
 } ge25519_p1p1;

 typedef struct
 {
   fe25519 x;
   fe25519 y;
   fe25519 z;
 } ge25519_p2;

 typedef struct
 {
   fe25519 x;
   fe25519 y;
 } ge25519_aff;


 /* Packed coordinates of the base point */
 const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
                                 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
                               {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
                                 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
                               {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
                                 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
                               {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
                                 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};

 /* Multiples of the base point in affine representation */
 static const ge25519_aff ge25519_base_multiples_affine[425] = {
 #include "ge25519_base.data"
 };

 static void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p)
 {
   fe25519_mul(&r->x, &p->x, &p->t);
   fe25519_mul(&r->y, &p->y, &p->z);
   fe25519_mul(&r->z, &p->z, &p->t);
 }

 static void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p)
 {
   p1p1_to_p2((ge25519_p2 *)r, p);
   fe25519_mul(&r->t, &p->x, &p->y);
 }

 static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q)
 {
   fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
   fe25519_mul(&qt, &q->x, &q->y);
   fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
   fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
   fe25519_sub(&t1, &q->y, &q->x);
   fe25519_add(&t2, &q->y, &q->x);
   fe25519_mul(&a, &a, &t1);
   fe25519_mul(&b, &b, &t2);
   fe25519_sub(&e, &b, &a); /* E = B-A */
   fe25519_add(&h, &b, &a); /* H = B+A */
   fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
   fe25519_mul(&c, &c, &ge25519_ec2d);
   fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
   fe25519_sub(&f, &d, &c); /* F = D-C */
   fe25519_add(&g, &d, &c); /* G = D+C */
   fe25519_mul(&r->x, &e, &f);
   fe25519_mul(&r->y, &h, &g);
   fe25519_mul(&r->z, &g, &f);
   fe25519_mul(&r->t, &e, &h);
 }

 static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q)
 {
   fe25519 a, b, c, d, t;

   fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
   fe25519_sub(&t, &q->y, &q->x);
   fe25519_mul(&a, &a, &t);
   fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
   fe25519_add(&t, &q->x, &q->y);
   fe25519_mul(&b, &b, &t);
   fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
   fe25519_mul(&c, &c, &ge25519_ec2d);
   fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
   fe25519_add(&d, &d, &d);
   fe25519_sub(&r->x, &b, &a); /* E = B-A */
   fe25519_sub(&r->t, &d, &c); /* F = D-C */
   fe25519_add(&r->z, &d, &c); /* G = D+C */
   fe25519_add(&r->y, &b, &a); /* H = B+A */
 }

 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
 static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p)
 {
   fe25519 a,b,c,d;
   fe25519_square(&a, &p->x);
   fe25519_square(&b, &p->y);
   fe25519_square(&c, &p->z);
   fe25519_add(&c, &c, &c);
   fe25519_neg(&d, &a);

   fe25519_add(&r->x, &p->x, &p->y);
   fe25519_square(&r->x, &r->x);
   fe25519_sub(&r->x, &r->x, &a);
   fe25519_sub(&r->x, &r->x, &b);
   fe25519_add(&r->z, &d, &b);
   fe25519_sub(&r->t, &r->z, &c);
   fe25519_sub(&r->y, &d, &b);
 }

 /* Constant-time version of: if(b) r = p */
 static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b)
 {
   fe25519_cmov(&r->x, &p->x, b);
   fe25519_cmov(&r->y, &p->y, b);
 }

 static unsigned char equal(signed char b,signed char c)
 {
   unsigned char ub = b;
   unsigned char uc = c;
   unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
   crypto_uint32 y = x; /* 0: yes; 1..255: no */
   y -= 1; /* 4294967295: yes; 0..254: no */
   y >>= 31; /* 1: yes; 0: no */
   return y;
 }

 static unsigned char negative(signed char b)
 {
   unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
   x >>= 63; /* 1: yes; 0: no */
   return x;
 }

 static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
 {
   /* constant time */
   fe25519 v;
   *t = ge25519_base_multiples_affine[5*pos+0];
   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1));
   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2));
   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3));
   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
   fe25519_neg(&v, &t->x);
   fe25519_cmov(&t->x, &v, negative(b));
 }

 static void setneutral(ge25519 *r)
 {
   fe25519_setzero(&r->x);
   fe25519_setone(&r->y);
   fe25519_setone(&r->z);
   fe25519_setzero(&r->t);
 }

 /* ********************************************************************
  *                    EXPORTED FUNCTIONS
  ******************************************************************** */

 /* return 0 on success, -1 otherwise */
 int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
 {
   unsigned char par;
   fe25519 t, chk, num, den, den2, den4, den6;
   fe25519_setone(&r->z);
   par = p[31] >> 7;
   fe25519_unpack(&r->y, p);
   fe25519_square(&num, &r->y); /* x = y^2 */
   fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
   fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
   fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */

   /* Computation of sqrt(num/den) */
   /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
   fe25519_square(&den2, &den);
   fe25519_square(&den4, &den2);
   fe25519_mul(&den6, &den4, &den2);
   fe25519_mul(&t, &den6, &num);
   fe25519_mul(&t, &t, &den);

   fe25519_pow2523(&t, &t);
   /* 2. computation of r->x = t * num * den^3 */
   fe25519_mul(&t, &t, &num);
   fe25519_mul(&t, &t, &den);
   fe25519_mul(&t, &t, &den);
   fe25519_mul(&r->x, &t, &den);

   /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
   fe25519_square(&chk, &r->x);
   fe25519_mul(&chk, &chk, &den);
   if (!fe25519_iseq_vartime(&chk, &num))
     fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);

   /* 4. Now we have one of the two square roots, except if input was not a square */
   fe25519_square(&chk, &r->x);
   fe25519_mul(&chk, &chk, &den);
   if (!fe25519_iseq_vartime(&chk, &num))
     return -1;

   /* 5. Choose the desired square root according to parity: */
   if(fe25519_getparity(&r->x) != (1-par))
     fe25519_neg(&r->x, &r->x);

   fe25519_mul(&r->t, &r->x, &r->y);
   return 0;
 }

 void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
 {
   fe25519 tx, ty, zi;
   fe25519_invert(&zi, &p->z);
   fe25519_mul(&tx, &p->x, &zi);
   fe25519_mul(&ty, &p->y, &zi);
   fe25519_pack(r, &ty);
   r[31] ^= fe25519_getparity(&tx) << 7;
 }

 int ge25519_isneutral_vartime(const ge25519_p3 *p)
 {
   int ret = 1;
   if(!fe25519_iszero(&p->x)) ret = 0;
   if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0;
   return ret;
 }

 /* computes [s1]p1 + [s2]p2 */
 void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2)
 {
   ge25519_p1p1 tp1p1;
   ge25519_p3 pre[16];
   unsigned char b[127];
   int i;

   /* precomputation                                                        s2 s1 */
   setneutral(pre);                                                      /* 00 00 */
   pre[1] = *p1;                                                         /* 00 01 */
   dbl_p1p1(&tp1p1,(ge25519_p2 *)p1);      p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */
   add_p1p1(&tp1p1,&pre[1], &pre[2]);      p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
   pre[4] = *p2;                                                         /* 01 00 */
   add_p1p1(&tp1p1,&pre[1], &pre[4]);      p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
   add_p1p1(&tp1p1,&pre[2], &pre[4]);      p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
   add_p1p1(&tp1p1,&pre[3], &pre[4]);      p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
   dbl_p1p1(&tp1p1,(ge25519_p2 *)p2);      p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */
   add_p1p1(&tp1p1,&pre[1], &pre[8]);      p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
   dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
   add_p1p1(&tp1p1,&pre[3], &pre[8]);      p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
   add_p1p1(&tp1p1,&pre[4], &pre[8]);      p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
   add_p1p1(&tp1p1,&pre[1],&pre[12]);      p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
   add_p1p1(&tp1p1,&pre[2],&pre[12]);      p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
   add_p1p1(&tp1p1,&pre[3],&pre[12]);      p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */

   sc25519_2interleave2(b,s1,s2);

   /* scalar multiplication */
   *r = pre[b[126]];
   for(i=125;i>=0;i--)
   {
     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
     p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
     if(b[i]!=0)
     {
       p1p1_to_p3(r, &tp1p1);
       add_p1p1(&tp1p1, r, &pre[b[i]]);
     }
     if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
     else p1p1_to_p3(r, &tp1p1);
   }
 }

 void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
 {
   signed char b[85];
   int i;
   ge25519_aff t;
   sc25519_window3(b,s);

   choose_t((ge25519_aff *)r, 0, b[0]);
   fe25519_setone(&r->z);
   fe25519_mul(&r->t, &r->x, &r->y);
   for(i=1;i<85;i++)
   {
     choose_t(&t, (unsigned long long) i, b[i]);
     ge25519_mixadd2(r, &t);
   }
 }
	/* $OpenBSD: ge25519.c,v 1.2 2013/12/07 00:26:37 djm Exp $ */

	/* Public Domain, from supercop-20130419/crypto_sign/ed25519/ref/ge25519.c */

	#include "fe25519.h"
	#include "sc25519.h"
	#include "ge25519.h"

	/*
	* Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
	* with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
	* Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
	*/

	/* d */
	static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
	0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
	/* 2d /
	static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
	0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
	/* sqrt(-1) */
	static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
	0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};

	#define ge25519_p3 ge25519

	typedef struct
	{
	fe25519 x;
	fe25519 z;
	fe25519 y;
	fe25519 t;
	} ge25519_p1p1;

	typedef struct
	{
	fe25519 x;
	fe25519 y;
	fe25519 z;
	} ge25519_p2;

	typedef struct
	{
	fe25519 x;
	fe25519 y;
	} ge25519_aff;


	/* Packed coordinates of the base point */
	const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
	0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
	{{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
	0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
	{{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
	0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
	{{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
	0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};

	/* Multiples of the base point in affine representation */
	static const ge25519_aff ge25519_base_multiples_affine[425] = {
	#include "ge25519_base.data"
	};

	static void p1p1_to_p2(ge25519_p2 r, const ge25519_p1p1 p)
	{
	fe25519_mul(&r->x, &p->x, &p->t);
	fe25519_mul(&r->y, &p->y, &p->z);
	fe25519_mul(&r->z, &p->z, &p->t);
	}

	static void p1p1_to_p3(ge25519_p3 r, const ge25519_p1p1 p)
	{
	p1p1_to_p2((ge25519_p2 *)r, p);
	fe25519_mul(&r->t, &p->x, &p->y);
	}

	static void ge25519_mixadd2(ge25519_p3 r, const ge25519_aff q)
	{
	fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
	fe25519_mul(&qt, &q->x, &q->y);
	fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)(Y2-X2) /
	fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)(Y2+X2) /
	fe25519_sub(&t1, &q->y, &q->x);
	fe25519_add(&t2, &q->y, &q->x);
	fe25519_mul(&a, &a, &t1);
	fe25519_mul(&b, &b, &t2);
	fe25519_sub(&e, &b, &a); /* E = B-A */
	fe25519_add(&h, &b, &a); /* H = B+A */
	fe25519_mul(&c, &r->t, &qt); /* C = T1kT2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_add(&d, &r->z, &r->z); /* D = Z12 /
	fe25519_sub(&f, &d, &c); /* F = D-C */
	fe25519_add(&g, &d, &c); /* G = D+C */
	fe25519_mul(&r->x, &e, &f);
	fe25519_mul(&r->y, &h, &g);
	fe25519_mul(&r->z, &g, &f);
	fe25519_mul(&r->t, &e, &h);
	}

	static void add_p1p1(ge25519_p1p1 r, const ge25519_p3 p, const ge25519_p3 *q)
	{
	fe25519 a, b, c, d, t;

	fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)(Y2-X2) /
	fe25519_sub(&t, &q->y, &q->x);
	fe25519_mul(&a, &a, &t);
	fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)(Y2+X2) /
	fe25519_add(&t, &q->x, &q->y);
	fe25519_mul(&b, &b, &t);
	fe25519_mul(&c, &p->t, &q->t); /* C = T1kT2 */
	fe25519_mul(&c, &c, &ge25519_ec2d);
	fe25519_mul(&d, &p->z, &q->z); /* D = Z12Z2 */
	fe25519_add(&d, &d, &d);
	fe25519_sub(&r->x, &b, &a); /* E = B-A */
	fe25519_sub(&r->t, &d, &c); /* F = D-C */
	fe25519_add(&r->z, &d, &c); /* G = D+C */
	fe25519_add(&r->y, &b, &a); /* H = B+A */
	}

	/* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
	static void dbl_p1p1(ge25519_p1p1 r, const ge25519_p2 p)
	{
	fe25519 a,b,c,d;
	fe25519_square(&a, &p->x);
	fe25519_square(&b, &p->y);
	fe25519_square(&c, &p->z);
	fe25519_add(&c, &c, &c);
	fe25519_neg(&d, &a);

	fe25519_add(&r->x, &p->x, &p->y);
	fe25519_square(&r->x, &r->x);
	fe25519_sub(&r->x, &r->x, &a);
	fe25519_sub(&r->x, &r->x, &b);
	fe25519_add(&r->z, &d, &b);
	fe25519_sub(&r->t, &r->z, &c);
	fe25519_sub(&r->y, &d, &b);
	}

	/* Constant-time version of: if(b) r = p */
	static void cmov_aff(ge25519_aff r, const ge25519_aff p, unsigned char b)
	{
	fe25519_cmov(&r->x, &p->x, b);
	fe25519_cmov(&r->y, &p->y, b);
	}

	static unsigned char equal(signed char b,signed char c)
	{
	unsigned char ub = b;
	unsigned char uc = c;
	unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
	crypto_uint32 y = x; /* 0: yes; 1..255: no */
	y -= 1; /* 4294967295: yes; 0..254: no */
	y >>= 31; /* 1: yes; 0: no */
	return y;
	}

	static unsigned char negative(signed char b)
	{
	unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
	x >>= 63; /* 1: yes; 0: no */
	return x;
	}

	static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
	{
	/* constant time */
	fe25519 v;
	t = ge25519_base_multiples_affine[5pos+0];
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) \| equal(b,-1));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) \| equal(b,-2));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) \| equal(b,-3));
	cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
	fe25519_neg(&v, &t->x);
	fe25519_cmov(&t->x, &v, negative(b));
	}

	static void setneutral(ge25519 *r)
	{
	fe25519_setzero(&r->x);
	fe25519_setone(&r->y);
	fe25519_setone(&r->z);
	fe25519_setzero(&r->t);
	}

	/* ********************************************************************
	* EXPORTED FUNCTIONS
	******************************************************************** */

	/* return 0 on success, -1 otherwise */
	int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
	{
	unsigned char par;
	fe25519 t, chk, num, den, den2, den4, den6;
	fe25519_setone(&r->z);
	par = p[31] >> 7;
	fe25519_unpack(&r->y, p);
	fe25519_square(&num, &r->y); /* x = y^2 */
	fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
	fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
	fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */

	/* Computation of sqrt(num/den) */
	/* 1.: computation of num^((p-5)/8)den^((7p-35)/8) = (numden^7)^((p-5)/8) */
	fe25519_square(&den2, &den);
	fe25519_square(&den4, &den2);
	fe25519_mul(&den6, &den4, &den2);
	fe25519_mul(&t, &den6, &num);
	fe25519_mul(&t, &t, &den);

	fe25519_pow2523(&t, &t);
	/* 2. computation of r->x = t * num * den^3 */
	fe25519_mul(&t, &t, &num);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&t, &t, &den);
	fe25519_mul(&r->x, &t, &den);

	/* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (!fe25519_iseq_vartime(&chk, &num))
	fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);

	/* 4. Now we have one of the two square roots, except if input was not a square */
	fe25519_square(&chk, &r->x);
	fe25519_mul(&chk, &chk, &den);
	if (!fe25519_iseq_vartime(&chk, &num))
	return -1;

	/* 5. Choose the desired square root according to parity: */
	if(fe25519_getparity(&r->x) != (1-par))
	fe25519_neg(&r->x, &r->x);

	fe25519_mul(&r->t, &r->x, &r->y);
	return 0;
	}

	void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
	{
	fe25519 tx, ty, zi;
	fe25519_invert(&zi, &p->z);
	fe25519_mul(&tx, &p->x, &zi);
	fe25519_mul(&ty, &p->y, &zi);
	fe25519_pack(r, &ty);
	r[31] ^= fe25519_getparity(&tx) << 7;
	}

	int ge25519_isneutral_vartime(const ge25519_p3 *p)
	{
	int ret = 1;
	if(!fe25519_iszero(&p->x)) ret = 0;
	if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0;
	return ret;
	}

	/* computes [s1]p1 + [s2]p2 */
	void ge25519_double_scalarmult_vartime(ge25519_p3 r, const ge25519_p3 p1, const sc25519 s1, const ge25519_p3 p2, const sc25519 *s2)
	{
	ge25519_p1p1 tp1p1;
	ge25519_p3 pre[16];
	unsigned char b[127];
	int i;

	/* precomputation s2 s1 */
	setneutral(pre); /* 00 00 */
	pre[1] = p1; / 00 01 */
	dbl_p1p1(&tp1p1,(ge25519_p2 )p1); p1p1_to_p3( &pre[2], &tp1p1); / 00 10 */
	add_p1p1(&tp1p1,&pre[1], &pre[2]); p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
	pre[4] = p2; / 01 00 */
	add_p1p1(&tp1p1,&pre[1], &pre[4]); p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
	add_p1p1(&tp1p1,&pre[2], &pre[4]); p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
	add_p1p1(&tp1p1,&pre[3], &pre[4]); p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
	dbl_p1p1(&tp1p1,(ge25519_p2 )p2); p1p1_to_p3( &pre[8], &tp1p1); / 10 00 */
	add_p1p1(&tp1p1,&pre[1], &pre[8]); p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
	dbl_p1p1(&tp1p1,(ge25519_p2 )&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); / 10 10 */
	add_p1p1(&tp1p1,&pre[3], &pre[8]); p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
	add_p1p1(&tp1p1,&pre[4], &pre[8]); p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
	add_p1p1(&tp1p1,&pre[1],&pre[12]); p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
	add_p1p1(&tp1p1,&pre[2],&pre[12]); p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
	add_p1p1(&tp1p1,&pre[3],&pre[12]); p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */

	sc25519_2interleave2(b,s1,s2);

	/* scalar multiplication */
	*r = pre[b[126]];
	for(i=125;i>=0;i--)
	{
	dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
	p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
	dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
	if(b[i]!=0)
	{
	p1p1_to_p3(r, &tp1p1);
	add_p1p1(&tp1p1, r, &pre[b[i]]);
	}
	if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
	else p1p1_to_p3(r, &tp1p1);
	}
	}

	void ge25519_scalarmult_base(ge25519_p3 r, const sc25519 s)
	{
	signed char b[85];
	int i;
	ge25519_aff t;
	sc25519_window3(b,s);

	choose_t((ge25519_aff *)r, 0, b[0]);
	fe25519_setone(&r->z);
	fe25519_mul(&r->t, &r->x, &r->y);
	for(i=1;i<85;i++)
	{
	choose_t(&t, (unsigned long long) i, b[i]);
	ge25519_mixadd2(r, &t);
	}
	}