Annotation of src/usr.bin/ssh/ge25519.c, Revision 1.2
1.2 ! djm 1: /* $OpenBSD$ */
1.1 markus 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: }