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 } 322
