Annotation of src/usr.bin/ssh/ge25519.c, Revision 1.1
1.1 ! markus 1: /* $OpenBSD: */
! 2:
! 3: /* Public Domain, from supercop-20130419/crypto_sign/ed25519/ref/ge25519.c */
! 4:
! 5: #include "fe25519.h"
! 6: #include "sc25519.h"
! 7: #include "ge25519.h"
! 8:
! 9: /*
! 10: * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
! 11: * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
! 12: * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
! 13: */
! 14:
! 15: /* d */
! 16: static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
! 17: 0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
! 18: /* 2*d */
! 19: static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
! 20: 0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
! 21: /* sqrt(-1) */
! 22: static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
! 23: 0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};
! 24:
! 25: #define ge25519_p3 ge25519
! 26:
! 27: typedef struct
! 28: {
! 29: fe25519 x;
! 30: fe25519 z;
! 31: fe25519 y;
! 32: fe25519 t;
! 33: } ge25519_p1p1;
! 34:
! 35: typedef struct
! 36: {
! 37: fe25519 x;
! 38: fe25519 y;
! 39: fe25519 z;
! 40: } ge25519_p2;
! 41:
! 42: typedef struct
! 43: {
! 44: fe25519 x;
! 45: fe25519 y;
! 46: } ge25519_aff;
! 47:
! 48:
! 49: /* Packed coordinates of the base point */
! 50: const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
! 51: 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
! 52: {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
! 53: 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
! 54: {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
! 55: 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
! 56: {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
! 57: 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};
! 58:
! 59: /* Multiples of the base point in affine representation */
! 60: static const ge25519_aff ge25519_base_multiples_affine[425] = {
! 61: #include "ge25519_base.data"
! 62: };
! 63:
! 64: static void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p)
! 65: {
! 66: fe25519_mul(&r->x, &p->x, &p->t);
! 67: fe25519_mul(&r->y, &p->y, &p->z);
! 68: fe25519_mul(&r->z, &p->z, &p->t);
! 69: }
! 70:
! 71: static void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p)
! 72: {
! 73: p1p1_to_p2((ge25519_p2 *)r, p);
! 74: fe25519_mul(&r->t, &p->x, &p->y);
! 75: }
! 76:
! 77: static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q)
! 78: {
! 79: fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
! 80: fe25519_mul(&qt, &q->x, &q->y);
! 81: fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
! 82: fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
! 83: fe25519_sub(&t1, &q->y, &q->x);
! 84: fe25519_add(&t2, &q->y, &q->x);
! 85: fe25519_mul(&a, &a, &t1);
! 86: fe25519_mul(&b, &b, &t2);
! 87: fe25519_sub(&e, &b, &a); /* E = B-A */
! 88: fe25519_add(&h, &b, &a); /* H = B+A */
! 89: fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
! 90: fe25519_mul(&c, &c, &ge25519_ec2d);
! 91: fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
! 92: fe25519_sub(&f, &d, &c); /* F = D-C */
! 93: fe25519_add(&g, &d, &c); /* G = D+C */
! 94: fe25519_mul(&r->x, &e, &f);
! 95: fe25519_mul(&r->y, &h, &g);
! 96: fe25519_mul(&r->z, &g, &f);
! 97: fe25519_mul(&r->t, &e, &h);
! 98: }
! 99:
! 100: static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q)
! 101: {
! 102: fe25519 a, b, c, d, t;
! 103:
! 104: fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
! 105: fe25519_sub(&t, &q->y, &q->x);
! 106: fe25519_mul(&a, &a, &t);
! 107: fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
! 108: fe25519_add(&t, &q->x, &q->y);
! 109: fe25519_mul(&b, &b, &t);
! 110: fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
! 111: fe25519_mul(&c, &c, &ge25519_ec2d);
! 112: fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
! 113: fe25519_add(&d, &d, &d);
! 114: fe25519_sub(&r->x, &b, &a); /* E = B-A */
! 115: fe25519_sub(&r->t, &d, &c); /* F = D-C */
! 116: fe25519_add(&r->z, &d, &c); /* G = D+C */
! 117: fe25519_add(&r->y, &b, &a); /* H = B+A */
! 118: }
! 119:
! 120: /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
! 121: static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p)
! 122: {
! 123: fe25519 a,b,c,d;
! 124: fe25519_square(&a, &p->x);
! 125: fe25519_square(&b, &p->y);
! 126: fe25519_square(&c, &p->z);
! 127: fe25519_add(&c, &c, &c);
! 128: fe25519_neg(&d, &a);
! 129:
! 130: fe25519_add(&r->x, &p->x, &p->y);
! 131: fe25519_square(&r->x, &r->x);
! 132: fe25519_sub(&r->x, &r->x, &a);
! 133: fe25519_sub(&r->x, &r->x, &b);
! 134: fe25519_add(&r->z, &d, &b);
! 135: fe25519_sub(&r->t, &r->z, &c);
! 136: fe25519_sub(&r->y, &d, &b);
! 137: }
! 138:
! 139: /* Constant-time version of: if(b) r = p */
! 140: static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b)
! 141: {
! 142: fe25519_cmov(&r->x, &p->x, b);
! 143: fe25519_cmov(&r->y, &p->y, b);
! 144: }
! 145:
! 146: static unsigned char equal(signed char b,signed char c)
! 147: {
! 148: unsigned char ub = b;
! 149: unsigned char uc = c;
! 150: unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
! 151: crypto_uint32 y = x; /* 0: yes; 1..255: no */
! 152: y -= 1; /* 4294967295: yes; 0..254: no */
! 153: y >>= 31; /* 1: yes; 0: no */
! 154: return y;
! 155: }
! 156:
! 157: static unsigned char negative(signed char b)
! 158: {
! 159: unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
! 160: x >>= 63; /* 1: yes; 0: no */
! 161: return x;
! 162: }
! 163:
! 164: static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
! 165: {
! 166: /* constant time */
! 167: fe25519 v;
! 168: *t = ge25519_base_multiples_affine[5*pos+0];
! 169: cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1));
! 170: cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2));
! 171: cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3));
! 172: cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
! 173: fe25519_neg(&v, &t->x);
! 174: fe25519_cmov(&t->x, &v, negative(b));
! 175: }
! 176:
! 177: static void setneutral(ge25519 *r)
! 178: {
! 179: fe25519_setzero(&r->x);
! 180: fe25519_setone(&r->y);
! 181: fe25519_setone(&r->z);
! 182: fe25519_setzero(&r->t);
! 183: }
! 184:
! 185: /* ********************************************************************
! 186: * EXPORTED FUNCTIONS
! 187: ******************************************************************** */
! 188:
! 189: /* return 0 on success, -1 otherwise */
! 190: int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
! 191: {
! 192: unsigned char par;
! 193: fe25519 t, chk, num, den, den2, den4, den6;
! 194: fe25519_setone(&r->z);
! 195: par = p[31] >> 7;
! 196: fe25519_unpack(&r->y, p);
! 197: fe25519_square(&num, &r->y); /* x = y^2 */
! 198: fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
! 199: fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
! 200: fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */
! 201:
! 202: /* Computation of sqrt(num/den) */
! 203: /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
! 204: fe25519_square(&den2, &den);
! 205: fe25519_square(&den4, &den2);
! 206: fe25519_mul(&den6, &den4, &den2);
! 207: fe25519_mul(&t, &den6, &num);
! 208: fe25519_mul(&t, &t, &den);
! 209:
! 210: fe25519_pow2523(&t, &t);
! 211: /* 2. computation of r->x = t * num * den^3 */
! 212: fe25519_mul(&t, &t, &num);
! 213: fe25519_mul(&t, &t, &den);
! 214: fe25519_mul(&t, &t, &den);
! 215: fe25519_mul(&r->x, &t, &den);
! 216:
! 217: /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
! 218: fe25519_square(&chk, &r->x);
! 219: fe25519_mul(&chk, &chk, &den);
! 220: if (!fe25519_iseq_vartime(&chk, &num))
! 221: fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);
! 222:
! 223: /* 4. Now we have one of the two square roots, except if input was not a square */
! 224: fe25519_square(&chk, &r->x);
! 225: fe25519_mul(&chk, &chk, &den);
! 226: if (!fe25519_iseq_vartime(&chk, &num))
! 227: return -1;
! 228:
! 229: /* 5. Choose the desired square root according to parity: */
! 230: if(fe25519_getparity(&r->x) != (1-par))
! 231: fe25519_neg(&r->x, &r->x);
! 232:
! 233: fe25519_mul(&r->t, &r->x, &r->y);
! 234: return 0;
! 235: }
! 236:
! 237: void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
! 238: {
! 239: fe25519 tx, ty, zi;
! 240: fe25519_invert(&zi, &p->z);
! 241: fe25519_mul(&tx, &p->x, &zi);
! 242: fe25519_mul(&ty, &p->y, &zi);
! 243: fe25519_pack(r, &ty);
! 244: r[31] ^= fe25519_getparity(&tx) << 7;
! 245: }
! 246:
! 247: int ge25519_isneutral_vartime(const ge25519_p3 *p)
! 248: {
! 249: int ret = 1;
! 250: if(!fe25519_iszero(&p->x)) ret = 0;
! 251: if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0;
! 252: return ret;
! 253: }
! 254:
! 255: /* computes [s1]p1 + [s2]p2 */
! 256: void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2)
! 257: {
! 258: ge25519_p1p1 tp1p1;
! 259: ge25519_p3 pre[16];
! 260: unsigned char b[127];
! 261: int i;
! 262:
! 263: /* precomputation s2 s1 */
! 264: setneutral(pre); /* 00 00 */
! 265: pre[1] = *p1; /* 00 01 */
! 266: dbl_p1p1(&tp1p1,(ge25519_p2 *)p1); p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */
! 267: add_p1p1(&tp1p1,&pre[1], &pre[2]); p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
! 268: pre[4] = *p2; /* 01 00 */
! 269: add_p1p1(&tp1p1,&pre[1], &pre[4]); p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
! 270: add_p1p1(&tp1p1,&pre[2], &pre[4]); p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
! 271: add_p1p1(&tp1p1,&pre[3], &pre[4]); p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
! 272: dbl_p1p1(&tp1p1,(ge25519_p2 *)p2); p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */
! 273: add_p1p1(&tp1p1,&pre[1], &pre[8]); p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
! 274: dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
! 275: add_p1p1(&tp1p1,&pre[3], &pre[8]); p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
! 276: add_p1p1(&tp1p1,&pre[4], &pre[8]); p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
! 277: add_p1p1(&tp1p1,&pre[1],&pre[12]); p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
! 278: add_p1p1(&tp1p1,&pre[2],&pre[12]); p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
! 279: add_p1p1(&tp1p1,&pre[3],&pre[12]); p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */
! 280:
! 281: sc25519_2interleave2(b,s1,s2);
! 282:
! 283: /* scalar multiplication */
! 284: *r = pre[b[126]];
! 285: for(i=125;i>=0;i--)
! 286: {
! 287: dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
! 288: p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
! 289: dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
! 290: if(b[i]!=0)
! 291: {
! 292: p1p1_to_p3(r, &tp1p1);
! 293: add_p1p1(&tp1p1, r, &pre[b[i]]);
! 294: }
! 295: if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
! 296: else p1p1_to_p3(r, &tp1p1);
! 297: }
! 298: }
! 299:
! 300: void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
! 301: {
! 302: signed char b[85];
! 303: int i;
! 304: ge25519_aff t;
! 305: sc25519_window3(b,s);
! 306:
! 307: choose_t((ge25519_aff *)r, 0, b[0]);
! 308: fe25519_setone(&r->z);
! 309: fe25519_mul(&r->t, &r->x, &r->y);
! 310: for(i=1;i<85;i++)
! 311: {
! 312: choose_t(&t, (unsigned long long) i, b[i]);
! 313: ge25519_mixadd2(r, &t);
! 314: }
! 315: }
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