1 /* $OpenBSD: mod_ge25519.c,v 1.2 2014/01/08 05:51:35 deraadt 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 "fe25519.h" 10 #include "sc25519.h" 11 #include "ge25519.h" 12 13 /* 14 * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 15 * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555 16 * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960); 17 */ 18 19 /* d */ 20 static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00, 21 0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}}; 22 /* 2*d */ 23 static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00, 24 0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}}; 25 /* sqrt(-1) */ 26 static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F, 27 0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}}; 28 29 #define ge25519_p3 ge25519 30 31 typedef struct 32 { 33 fe25519 x; 34 fe25519 z; 35 fe25519 y; 36 fe25519 t; 37 } ge25519_p1p1; 38 39 typedef struct 40 { 41 fe25519 x; 42 fe25519 y; 43 fe25519 z; 44 } ge25519_p2; 45 46 typedef struct 47 { 48 fe25519 x; 49 fe25519 y; 50 } ge25519_aff; 51 52 53 /* Packed coordinates of the base point */ 54 const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69, 55 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}}, 56 {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 57 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}}, 58 {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 59 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}, 60 {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20, 61 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}}; 62 63 #ifndef VERIFYONLY 64 /* Multiples of the base point in affine representation */ 65 static const ge25519_aff ge25519_base_multiples_affine[425] = { 66 #include "ge25519_base.data" 67 }; 68 #endif 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 #ifndef VERIFYONLY 84 static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q) 85 { 86 fe25519 a,b,t1,t2,c,d,e,f,g,h,qt; 87 fe25519_mul(&qt, &q->x, &q->y); 88 fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */ 89 fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */ 90 fe25519_sub(&t1, &q->y, &q->x); 91 fe25519_add(&t2, &q->y, &q->x); 92 fe25519_mul(&a, &a, &t1); 93 fe25519_mul(&b, &b, &t2); 94 fe25519_sub(&e, &b, &a); /* E = B-A */ 95 fe25519_add(&h, &b, &a); /* H = B+A */ 96 fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */ 97 fe25519_mul(&c, &c, &ge25519_ec2d); 98 fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */ 99 fe25519_sub(&f, &d, &c); /* F = D-C */ 100 fe25519_add(&g, &d, &c); /* G = D+C */ 101 fe25519_mul(&r->x, &e, &f); 102 fe25519_mul(&r->y, &h, &g); 103 fe25519_mul(&r->z, &g, &f); 104 fe25519_mul(&r->t, &e, &h); 105 } 106 #endif 107 108 static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q) 109 { 110 fe25519 a, b, c, d, t; 111 112 fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */ 113 fe25519_sub(&t, &q->y, &q->x); 114 fe25519_mul(&a, &a, &t); 115 fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */ 116 fe25519_add(&t, &q->x, &q->y); 117 fe25519_mul(&b, &b, &t); 118 fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */ 119 fe25519_mul(&c, &c, &ge25519_ec2d); 120 fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */ 121 fe25519_add(&d, &d, &d); 122 fe25519_sub(&r->x, &b, &a); /* E = B-A */ 123 fe25519_sub(&r->t, &d, &c); /* F = D-C */ 124 fe25519_add(&r->z, &d, &c); /* G = D+C */ 125 fe25519_add(&r->y, &b, &a); /* H = B+A */ 126 } 127 128 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */ 129 static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p) 130 { 131 fe25519 a,b,c,d; 132 fe25519_square(&a, &p->x); 133 fe25519_square(&b, &p->y); 134 fe25519_square(&c, &p->z); 135 fe25519_add(&c, &c, &c); 136 fe25519_neg(&d, &a); 137 138 fe25519_add(&r->x, &p->x, &p->y); 139 fe25519_square(&r->x, &r->x); 140 fe25519_sub(&r->x, &r->x, &a); 141 fe25519_sub(&r->x, &r->x, &b); 142 fe25519_add(&r->z, &d, &b); 143 fe25519_sub(&r->t, &r->z, &c); 144 fe25519_sub(&r->y, &d, &b); 145 } 146 147 #ifndef VERIFYONLY 148 /* Constant-time version of: if(b) r = p */ 149 static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b) 150 { 151 fe25519_cmov(&r->x, &p->x, b); 152 fe25519_cmov(&r->y, &p->y, b); 153 } 154 155 static unsigned char equal(signed char b,signed char c) 156 { 157 unsigned char ub = b; 158 unsigned char uc = c; 159 unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */ 160 crypto_uint32 y = x; /* 0: yes; 1..255: no */ 161 y -= 1; /* 4294967295: yes; 0..254: no */ 162 y >>= 31; /* 1: yes; 0: no */ 163 return y; 164 } 165 166 static unsigned char negative(signed char b) 167 { 168 unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */ 169 x >>= 63; /* 1: yes; 0: no */ 170 return x; 171 } 172 173 static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b) 174 { 175 /* constant time */ 176 fe25519 v; 177 *t = ge25519_base_multiples_affine[5*pos+0]; 178 cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1)); 179 cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2)); 180 cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3)); 181 cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4)); 182 fe25519_neg(&v, &t->x); 183 fe25519_cmov(&t->x, &v, negative(b)); 184 } 185 #endif 186 187 static void setneutral(ge25519 *r) 188 { 189 fe25519_setzero(&r->x); 190 fe25519_setone(&r->y); 191 fe25519_setone(&r->z); 192 fe25519_setzero(&r->t); 193 } 194 195 /* ******************************************************************** 196 * EXPORTED FUNCTIONS 197 ******************************************************************** */ 198 199 /* return 0 on success, -1 otherwise */ 200 int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32]) 201 { 202 unsigned char par; 203 fe25519 t, chk, num, den, den2, den4, den6; 204 fe25519_setone(&r->z); 205 par = p[31] >> 7; 206 fe25519_unpack(&r->y, p); 207 fe25519_square(&num, &r->y); /* x = y^2 */ 208 fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */ 209 fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */ 210 fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */ 211 212 /* Computation of sqrt(num/den) */ 213 /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */ 214 fe25519_square(&den2, &den); 215 fe25519_square(&den4, &den2); 216 fe25519_mul(&den6, &den4, &den2); 217 fe25519_mul(&t, &den6, &num); 218 fe25519_mul(&t, &t, &den); 219 220 fe25519_pow2523(&t, &t); 221 /* 2. computation of r->x = t * num * den^3 */ 222 fe25519_mul(&t, &t, &num); 223 fe25519_mul(&t, &t, &den); 224 fe25519_mul(&t, &t, &den); 225 fe25519_mul(&r->x, &t, &den); 226 227 /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */ 228 fe25519_square(&chk, &r->x); 229 fe25519_mul(&chk, &chk, &den); 230 if (!fe25519_iseq_vartime(&chk, &num)) 231 fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1); 232 233 /* 4. Now we have one of the two square roots, except if input was not a square */ 234 fe25519_square(&chk, &r->x); 235 fe25519_mul(&chk, &chk, &den); 236 if (!fe25519_iseq_vartime(&chk, &num)) 237 return -1; 238 239 /* 5. Choose the desired square root according to parity: */ 240 if(fe25519_getparity(&r->x) != (1-par)) 241 fe25519_neg(&r->x, &r->x); 242 243 fe25519_mul(&r->t, &r->x, &r->y); 244 return 0; 245 } 246 247 void ge25519_pack(unsigned char r[32], const ge25519_p3 *p) 248 { 249 fe25519 tx, ty, zi; 250 fe25519_invert(&zi, &p->z); 251 fe25519_mul(&tx, &p->x, &zi); 252 fe25519_mul(&ty, &p->y, &zi); 253 fe25519_pack(r, &ty); 254 r[31] ^= fe25519_getparity(&tx) << 7; 255 } 256 257 int ge25519_isneutral_vartime(const ge25519_p3 *p) 258 { 259 int ret = 1; 260 if(!fe25519_iszero(&p->x)) ret = 0; 261 if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0; 262 return ret; 263 } 264 265 /* computes [s1]p1 + [s2]p2 */ 266 void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2) 267 { 268 ge25519_p1p1 tp1p1; 269 ge25519_p3 pre[16]; 270 unsigned char b[127]; 271 int i; 272 273 /* precomputation s2 s1 */ 274 setneutral(pre); /* 00 00 */ 275 pre[1] = *p1; /* 00 01 */ 276 dbl_p1p1(&tp1p1,(ge25519_p2 *)p1); p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */ 277 add_p1p1(&tp1p1,&pre[1], &pre[2]); p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */ 278 pre[4] = *p2; /* 01 00 */ 279 add_p1p1(&tp1p1,&pre[1], &pre[4]); p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */ 280 add_p1p1(&tp1p1,&pre[2], &pre[4]); p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */ 281 add_p1p1(&tp1p1,&pre[3], &pre[4]); p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */ 282 dbl_p1p1(&tp1p1,(ge25519_p2 *)p2); p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */ 283 add_p1p1(&tp1p1,&pre[1], &pre[8]); p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */ 284 dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */ 285 add_p1p1(&tp1p1,&pre[3], &pre[8]); p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */ 286 add_p1p1(&tp1p1,&pre[4], &pre[8]); p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */ 287 add_p1p1(&tp1p1,&pre[1],&pre[12]); p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */ 288 add_p1p1(&tp1p1,&pre[2],&pre[12]); p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */ 289 add_p1p1(&tp1p1,&pre[3],&pre[12]); p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */ 290 291 sc25519_2interleave2(b,s1,s2); 292 293 /* scalar multiplication */ 294 *r = pre[b[126]]; 295 for(i=125;i>=0;i--) 296 { 297 dbl_p1p1(&tp1p1, (ge25519_p2 *)r); 298 p1p1_to_p2((ge25519_p2 *) r, &tp1p1); 299 dbl_p1p1(&tp1p1, (ge25519_p2 *)r); 300 if(b[i]!=0) 301 { 302 p1p1_to_p3(r, &tp1p1); 303 add_p1p1(&tp1p1, r, &pre[b[i]]); 304 } 305 if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1); 306 else p1p1_to_p3(r, &tp1p1); 307 } 308 } 309 310 #ifndef VERIFYONLY 311 void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s) 312 { 313 signed char b[85]; 314 int i; 315 ge25519_aff t; 316 sc25519_window3(b,s); 317 318 choose_t((ge25519_aff *)r, 0, b[0]); 319 fe25519_setone(&r->z); 320 fe25519_mul(&r->t, &r->x, &r->y); 321 for(i=1;i<85;i++) 322 { 323 choose_t(&t, (unsigned long long) i, b[i]); 324 ge25519_mixadd2(r, &t); 325 } 326 } 327 #endif 328