crypto/openssh/ge25519.c

   1 /* $OpenBSD: ge25519.c,v 1.3 2013/12/09 11:03:45 markus Exp $ */
   2
   3 /*
   4  * Public Domain, Authors: Daniel J. Bernstein, Niels Duif, Tanja Lange,
   5  * Peter Schwabe, Bo-Yin Yang.
   6  * Copied from supercop-20130419/crypto_sign/ed25519/ref/ge25519.c
   7  */
   8
   9 #include "includes.h"
  10
  11 #include "fe25519.h"
  12 #include "sc25519.h"
  13 #include "ge25519.h"
  14
  15 /*
  16  * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
  17  * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
  18  * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
  19  */
  20
  21 /* d */
  22 static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
  23                       0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
  24 /* 2*d */
  25 static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
  26                        0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
  27 /* sqrt(-1) */
  28 static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
  29                          0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};
  30
  31 #define ge25519_p3 ge25519
  32
  33 typedef struct
  34 {
  35   fe25519 x;
  36   fe25519 z;
  37   fe25519 y;
  38   fe25519 t;
  39 } ge25519_p1p1;
  40
  41 typedef struct
  42 {
  43   fe25519 x;
  44   fe25519 y;
  45   fe25519 z;
  46 } ge25519_p2;
  47
  48 typedef struct
  49 {
  50   fe25519 x;
  51   fe25519 y;
  52 } ge25519_aff;
  53
  54
  55 /* Packed coordinates of the base point */
  56 const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
  57                                 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
  58                               {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
  59                                 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
  60                               {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  61                                 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
  62                               {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
  63                                 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};
  64
  65 /* Multiples of the base point in affine representation */
  66 static const ge25519_aff ge25519_base_multiples_affine[425] = {
  67 #include "ge25519_base.data"
  68 };
  69
  70 static void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p)
  71 {
  72   fe25519_mul(&r->x, &p->x, &p->t);
  73   fe25519_mul(&r->y, &p->y, &p->z);
  74   fe25519_mul(&r->z, &p->z, &p->t);
  75 }
  76
  77 static void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p)
  78 {
  79   p1p1_to_p2((ge25519_p2 *)r, p);
  80   fe25519_mul(&r->t, &p->x, &p->y);
  81 }
  82
  83 static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q)
  84 {
  85   fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
  86   fe25519_mul(&qt, &q->x, &q->y);
  87   fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
  88   fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
  89   fe25519_sub(&t1, &q->y, &q->x);
  90   fe25519_add(&t2, &q->y, &q->x);
  91   fe25519_mul(&a, &a, &t1);
  92   fe25519_mul(&b, &b, &t2);
  93   fe25519_sub(&e, &b, &a); /* E = B-A */
  94   fe25519_add(&h, &b, &a); /* H = B+A */
  95   fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
  96   fe25519_mul(&c, &c, &ge25519_ec2d);
  97   fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
  98   fe25519_sub(&f, &d, &c); /* F = D-C */
  99   fe25519_add(&g, &d, &c); /* G = D+C */
 100   fe25519_mul(&r->x, &e, &f);
 101   fe25519_mul(&r->y, &h, &g);
 102   fe25519_mul(&r->z, &g, &f);
 103   fe25519_mul(&r->t, &e, &h);
 104 }
 105
 106 static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q)
 107 {
 108   fe25519 a, b, c, d, t;
 109
 110   fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
 111   fe25519_sub(&t, &q->y, &q->x);
 112   fe25519_mul(&a, &a, &t);
 113   fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
 114   fe25519_add(&t, &q->x, &q->y);
 115   fe25519_mul(&b, &b, &t);
 116   fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
 117   fe25519_mul(&c, &c, &ge25519_ec2d);
 118   fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
 119   fe25519_add(&d, &d, &d);
 120   fe25519_sub(&r->x, &b, &a); /* E = B-A */
 121   fe25519_sub(&r->t, &d, &c); /* F = D-C */
 122   fe25519_add(&r->z, &d, &c); /* G = D+C */
 123   fe25519_add(&r->y, &b, &a); /* H = B+A */
 124 }
 125
 126 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
 127 static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p)
 128 {
 129   fe25519 a,b,c,d;
 130   fe25519_square(&a, &p->x);
 131   fe25519_square(&b, &p->y);
 132   fe25519_square(&c, &p->z);
 133   fe25519_add(&c, &c, &c);
 134   fe25519_neg(&d, &a);
 135
 136   fe25519_add(&r->x, &p->x, &p->y);
 137   fe25519_square(&r->x, &r->x);
 138   fe25519_sub(&r->x, &r->x, &a);
 139   fe25519_sub(&r->x, &r->x, &b);
 140   fe25519_add(&r->z, &d, &b);
 141   fe25519_sub(&r->t, &r->z, &c);
 142   fe25519_sub(&r->y, &d, &b);
 143 }
 144
 145 /* Constant-time version of: if(b) r = p */
 146 static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b)
 147 {
 148   fe25519_cmov(&r->x, &p->x, b);
 149   fe25519_cmov(&r->y, &p->y, b);
 150 }
 151
 152 static unsigned char equal(signed char b,signed char c)
 153 {
 154   unsigned char ub = b;
 155   unsigned char uc = c;
 156   unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
 157   crypto_uint32 y = x; /* 0: yes; 1..255: no */
 158   y -= 1; /* 4294967295: yes; 0..254: no */
 159   y >>= 31; /* 1: yes; 0: no */
 160   return y;
 161 }
 162
 163 static unsigned char negative(signed char b)
 164 {
 165   unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
 166   x >>= 63; /* 1: yes; 0: no */
 167   return x;
 168 }
 169
 170 static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
 171 {
 172   /* constant time */
 173   fe25519 v;
 174   *t = ge25519_base_multiples_affine[5*pos+0];
 175   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1));
 176   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2));
 177   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3));
 178   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
 179   fe25519_neg(&v, &t->x);
 180   fe25519_cmov(&t->x, &v, negative(b));
 181 }
 182
 183 static void setneutral(ge25519 *r)
 184 {
 185   fe25519_setzero(&r->x);
 186   fe25519_setone(&r->y);
 187   fe25519_setone(&r->z);
 188   fe25519_setzero(&r->t);
 189 }
 190
 191 /* ********************************************************************
 192  *                    EXPORTED FUNCTIONS
 193  ******************************************************************** */
 194
 195 /* return 0 on success, -1 otherwise */
 196 int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
 197 {
 198   unsigned char par;
 199   fe25519 t, chk, num, den, den2, den4, den6;
 200   fe25519_setone(&r->z);
 201   par = p[31] >> 7;
 202   fe25519_unpack(&r->y, p);
 203   fe25519_square(&num, &r->y); /* x = y^2 */
 204   fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
 205   fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
 206   fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */
 207
 208   /* Computation of sqrt(num/den) */
 209   /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
 210   fe25519_square(&den2, &den);
 211   fe25519_square(&den4, &den2);
 212   fe25519_mul(&den6, &den4, &den2);
 213   fe25519_mul(&t, &den6, &num);
 214   fe25519_mul(&t, &t, &den);
 215
 216   fe25519_pow2523(&t, &t);
 217   /* 2. computation of r->x = t * num * den^3 */
 218   fe25519_mul(&t, &t, &num);
 219   fe25519_mul(&t, &t, &den);
 220   fe25519_mul(&t, &t, &den);
 221   fe25519_mul(&r->x, &t, &den);
 222
 223   /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
 224   fe25519_square(&chk, &r->x);
 225   fe25519_mul(&chk, &chk, &den);
 226   if (!fe25519_iseq_vartime(&chk, &num))
 227     fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);
 228
 229   /* 4. Now we have one of the two square roots, except if input was not a square */
 230   fe25519_square(&chk, &r->x);
 231   fe25519_mul(&chk, &chk, &den);
 232   if (!fe25519_iseq_vartime(&chk, &num))
 233     return -1;
 234
 235   /* 5. Choose the desired square root according to parity: */
 236   if(fe25519_getparity(&r->x) != (1-par))
 237     fe25519_neg(&r->x, &r->x);
 238
 239   fe25519_mul(&r->t, &r->x, &r->y);
 240   return 0;
 241 }
 242
 243 void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
 244 {
 245   fe25519 tx, ty, zi;
 246   fe25519_invert(&zi, &p->z);
 247   fe25519_mul(&tx, &p->x, &zi);
 248   fe25519_mul(&ty, &p->y, &zi);
 249   fe25519_pack(r, &ty);
 250   r[31] ^= fe25519_getparity(&tx) << 7;
 251 }
 252
 253 int ge25519_isneutral_vartime(const ge25519_p3 *p)
 254 {
 255   int ret = 1;
 256   if(!fe25519_iszero(&p->x)) ret = 0;
 257   if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0;
 258   return ret;
 259 }
 260
 261 /* computes [s1]p1 + [s2]p2 */
 262 void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2)
 263 {
 264   ge25519_p1p1 tp1p1;
 265   ge25519_p3 pre[16];
 266   unsigned char b[127];
 267   int i;
 268
 269   /* precomputation                                                        s2 s1 */
 270   setneutral(pre);                                                      /* 00 00 */
 271   pre[1] = *p1;                                                         /* 00 01 */
 272   dbl_p1p1(&tp1p1,(ge25519_p2 *)p1);      p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */
 273   add_p1p1(&tp1p1,&pre[1], &pre[2]);      p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
 274   pre[4] = *p2;                                                         /* 01 00 */
 275   add_p1p1(&tp1p1,&pre[1], &pre[4]);      p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
 276   add_p1p1(&tp1p1,&pre[2], &pre[4]);      p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
 277   add_p1p1(&tp1p1,&pre[3], &pre[4]);      p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
 278   dbl_p1p1(&tp1p1,(ge25519_p2 *)p2);      p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */
 279   add_p1p1(&tp1p1,&pre[1], &pre[8]);      p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
 280   dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
 281   add_p1p1(&tp1p1,&pre[3], &pre[8]);      p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
 282   add_p1p1(&tp1p1,&pre[4], &pre[8]);      p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
 283   add_p1p1(&tp1p1,&pre[1],&pre[12]);      p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
 284   add_p1p1(&tp1p1,&pre[2],&pre[12]);      p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
 285   add_p1p1(&tp1p1,&pre[3],&pre[12]);      p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */
 286
 287   sc25519_2interleave2(b,s1,s2);
 288
 289   /* scalar multiplication */
 290   *r = pre[b[126]];
 291   for(i=125;i>=0;i--)
 292   {
 293     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
 294     p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
 295     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
 296     if(b[i]!=0)
 297     {
 298       p1p1_to_p3(r, &tp1p1);
 299       add_p1p1(&tp1p1, r, &pre[b[i]]);
 300     }
 301     if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
 302     else p1p1_to_p3(r, &tp1p1);
 303   }
 304 }
 305
 306 void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
 307 {
 308   signed char b[85];
 309   int i;
 310   ge25519_aff t;
 311   sc25519_window3(b,s);
 312
 313   choose_t((ge25519_aff *)r, 0, b[0]);
 314   fe25519_setone(&r->z);
 315   fe25519_mul(&r->t, &r->x, &r->y);
 316   for(i=1;i<85;i++)
 317   {
 318     choose_t(&t, (unsigned long long) i, b[i]);
 319     ge25519_mixadd2(r, &t);
 320   }
 321 }