1 /* ecc-25519.c
2
3 Arithmetic and tables for curve25519,
4
5 Copyright (C) 2014 Niels Möller
6
7 This file is part of GNU Nettle.
8
9 GNU Nettle is free software: you can redistribute it and/or
10 modify it under the terms of either:
11
12 * the GNU Lesser General Public License as published by the Free
13 Software Foundation; either version 3 of the License, or (at your
14 option) any later version.
15
16 or
17
18 * the GNU General Public License as published by the Free
19 Software Foundation; either version 2 of the License, or (at your
20 option) any later version.
21
22 or both in parallel, as here.
23
24 GNU Nettle is distributed in the hope that it will be useful,
25 but WITHOUT ANY WARRANTY; without even the implied warranty of
26 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
27 General Public License for more details.
28
29 You should have received copies of the GNU General Public License and
30 the GNU Lesser General Public License along with this program. If
31 not, see http://www.gnu.org/licenses/.
32 */
33
34 #if HAVE_CONFIG_H
35 # include "config.h"
36 #endif
37
38 #include <assert.h>
39
40 #include "ecc.h"
41 #include "ecc-internal.h"
42
43 #define USE_REDC 0
44
45 #include "ecc-25519.h"
46
47 #define PHIGH_BITS (GMP_NUMB_BITS * ECC_LIMB_SIZE - 255)
48
49 #if HAVE_NATIVE_ecc_25519_modp
50
51 #define ecc_25519_modp nettle_ecc_25519_modp
52 void
53 ecc_25519_modp (const struct ecc_modulo *m, mp_limb_t *rp);
54 #else
55
56 #if PHIGH_BITS == 0
57 #error Unsupported limb size */
58 #endif
59
60 static void
ecc_25519_modp(const struct ecc_modulo * m UNUSED,mp_limb_t * rp)61 ecc_25519_modp(const struct ecc_modulo *m UNUSED, mp_limb_t *rp)
62 {
63 mp_limb_t hi, cy;
64
65 cy = mpn_addmul_1 (rp, rp + ECC_LIMB_SIZE, ECC_LIMB_SIZE,
66 (mp_limb_t) 19 << PHIGH_BITS);
67 hi = rp[ECC_LIMB_SIZE-1];
68 cy = (cy << PHIGH_BITS) + (hi >> (GMP_NUMB_BITS - PHIGH_BITS));
69 rp[ECC_LIMB_SIZE-1] = (hi & (GMP_NUMB_MASK >> PHIGH_BITS))
70 + sec_add_1 (rp, rp, ECC_LIMB_SIZE - 1, 19 * cy);
71 }
72 #endif /* HAVE_NATIVE_ecc_25519_modp */
73
74 #define QHIGH_BITS (GMP_NUMB_BITS * ECC_LIMB_SIZE - 252)
75
76 #if QHIGH_BITS == 0
77 #error Unsupported limb size */
78 #endif
79
80 static void
ecc_25519_modq(const struct ecc_modulo * q,mp_limb_t * rp)81 ecc_25519_modq (const struct ecc_modulo *q, mp_limb_t *rp)
82 {
83 mp_size_t n;
84 mp_limb_t cy;
85
86 /* n is the offset where we add in the next term */
87 for (n = ECC_LIMB_SIZE; n-- > 0;)
88 {
89 cy = mpn_submul_1 (rp + n,
90 q->B_shifted, ECC_LIMB_SIZE,
91 rp[n + ECC_LIMB_SIZE]);
92 /* Top limb of mBmodq_shifted is zero, so we get cy == 0 or 1 */
93 assert (cy < 2);
94 cnd_add_n (cy, rp+n, q->m, ECC_LIMB_SIZE);
95 }
96
97 cy = mpn_submul_1 (rp, q->m, ECC_LIMB_SIZE,
98 rp[ECC_LIMB_SIZE-1] >> (GMP_NUMB_BITS - QHIGH_BITS));
99 assert (cy < 2);
100 cnd_add_n (cy, rp, q->m, ECC_LIMB_SIZE);
101 }
102
103 /* Needs 2*ecc->size limbs at rp, and 2*ecc->size additional limbs of
104 scratch space. No overlap allowed. */
105 static void
ecc_mod_pow_2kp1(const struct ecc_modulo * m,mp_limb_t * rp,const mp_limb_t * xp,unsigned k,mp_limb_t * tp)106 ecc_mod_pow_2kp1 (const struct ecc_modulo *m,
107 mp_limb_t *rp, const mp_limb_t *xp,
108 unsigned k, mp_limb_t *tp)
109 {
110 if (k & 1)
111 {
112 ecc_mod_sqr (m, tp, xp);
113 k--;
114 }
115 else
116 {
117 ecc_mod_sqr (m, rp, xp);
118 ecc_mod_sqr (m, tp, rp);
119 k -= 2;
120 }
121 while (k > 0)
122 {
123 ecc_mod_sqr (m, rp, tp);
124 ecc_mod_sqr (m, tp, rp);
125 k -= 2;
126 }
127 ecc_mod_mul (m, rp, tp, xp);
128 }
129
130 /* Computes a^{(p-5)/8} = a^{2^{252}-3} mod m. Needs 5 * n scratch
131 space. */
132 static void
ecc_mod_pow_252m3(const struct ecc_modulo * m,mp_limb_t * rp,const mp_limb_t * ap,mp_limb_t * scratch)133 ecc_mod_pow_252m3 (const struct ecc_modulo *m,
134 mp_limb_t *rp, const mp_limb_t *ap, mp_limb_t *scratch)
135 {
136 #define a7 scratch
137 #define t0 (scratch + ECC_LIMB_SIZE)
138 #define t1 (scratch + 3*ECC_LIMB_SIZE)
139
140 /* a^{2^252 - 3} = a^{(p-5)/8}, using the addition chain
141 2^252 - 3
142 = 1 + (2^252-4)
143 = 1 + 4 (2^250-1)
144 = 1 + 4 (2^125+1)(2^125-1)
145 = 1 + 4 (2^125+1)(1+2(2^124-1))
146 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^62-1))
147 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(2^31-1))
148 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^28-1)))
149 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^14-1)))
150 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(2^7-1)))
151 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(1+2(2^6-1))))
152 = 1 + 4 (2^125+1)(1+2(2^62+1)(2^31+1)(7+8(2^14+1)(2^7+1)(1+2(2^3+1)*7)))
153 */
154
155 ecc_mod_pow_2kp1 (m, t0, ap, 1, t1); /* a^3 */
156 ecc_mod_sqr (m, rp, t0); /* a^6 */
157 ecc_mod_mul (m, a7, rp, ap); /* a^7 */
158 ecc_mod_pow_2kp1 (m, rp, a7, 3, t0); /* a^63 = a^{2^6-1} */
159 ecc_mod_sqr (m, t0, rp); /* a^{2^7-2} */
160 ecc_mod_mul (m, rp, t0, ap); /* a^{2^7-1} */
161 ecc_mod_pow_2kp1 (m, t0, rp, 7, t1); /* a^{2^14-1}*/
162 ecc_mod_pow_2kp1 (m, rp, t0, 14, t1); /* a^{2^28-1} */
163 ecc_mod_sqr (m, t0, rp); /* a^{2^29-2} */
164 ecc_mod_sqr (m, t1, t0); /* a^{2^30-4} */
165 ecc_mod_sqr (m, t0, t1); /* a^{2^31-8} */
166 ecc_mod_mul (m, rp, t0, a7); /* a^{2^31-1} */
167 ecc_mod_pow_2kp1 (m, t0, rp, 31, t1); /* a^{2^62-1} */
168 ecc_mod_pow_2kp1 (m, rp, t0, 62, t1); /* a^{2^124-1}*/
169 ecc_mod_sqr (m, t0, rp); /* a^{2^125-2} */
170 ecc_mod_mul (m, rp, t0, ap); /* a^{2^125-1} */
171 ecc_mod_pow_2kp1 (m, t0, rp, 125, t1);/* a^{2^250-1} */
172 ecc_mod_sqr (m, rp, t0); /* a^{2^251-2} */
173 ecc_mod_sqr (m, t0, rp); /* a^{2^252-4} */
174 ecc_mod_mul (m, rp, t0, ap); /* a^{2^252-3} */
175 #undef t0
176 #undef t1
177 #undef a7
178 }
179
180 /* Needs 5*ECC_LIMB_SIZE scratch space. */
181 #define ECC_25519_INV_ITCH (5*ECC_LIMB_SIZE)
182
ecc_25519_inv(const struct ecc_modulo * p,mp_limb_t * rp,const mp_limb_t * ap,mp_limb_t * scratch)183 static void ecc_25519_inv (const struct ecc_modulo *p,
184 mp_limb_t *rp, const mp_limb_t *ap,
185 mp_limb_t *scratch)
186 {
187 #define t0 scratch
188
189 /* Addition chain
190
191 p - 2 = 2^{255} - 21
192 = 1 + 2 (1 + 4 (2^{252}-3))
193 */
194 ecc_mod_pow_252m3 (p, rp, ap, t0);
195 ecc_mod_sqr (p, t0, rp);
196 ecc_mod_sqr (p, rp, t0);
197 ecc_mod_mul (p, t0, ap, rp);
198 ecc_mod_sqr (p, rp, t0);
199 ecc_mod_mul (p, t0, ap, rp);
200 mpn_copyi (rp, t0, ECC_LIMB_SIZE); /* FIXME: Eliminate copy? */
201 #undef t0
202 }
203
204 /* First, do a canonical reduction, then check if zero */
205 static int
ecc_25519_zero_p(const struct ecc_modulo * p,mp_limb_t * xp)206 ecc_25519_zero_p (const struct ecc_modulo *p, mp_limb_t *xp)
207 {
208 mp_limb_t cy;
209 mp_limb_t w;
210 mp_size_t i;
211 #if PHIGH_BITS > 0
212 mp_limb_t hi = xp[ECC_LIMB_SIZE-1];
213 xp[ECC_LIMB_SIZE-1] = (hi & (GMP_NUMB_MASK >> PHIGH_BITS))
214 + sec_add_1 (xp, xp, ECC_LIMB_SIZE - 1, 19 * (hi >> (GMP_NUMB_BITS - PHIGH_BITS)));
215 #endif
216 cy = mpn_sub_n (xp, xp, p->m, ECC_LIMB_SIZE);
217 cnd_add_n (cy, xp, p->m, ECC_LIMB_SIZE);
218
219 for (i = 0, w = 0; i < ECC_LIMB_SIZE; i++)
220 w |= xp[i];
221 return w == 0;
222 }
223
224 /* Compute x such that x^2 = u/v (mod p). Returns one on success, zero
225 on failure. We use the e = 2 special case of the Shanks-Tonelli
226 algorithm (see http://www.math.vt.edu/people/brown/doc/sqrts.pdf,
227 or Henri Cohen, Computational Algebraic Number Theory, 1.5.1).
228
229 To avoid a separate inversion, we also use a trick of djb's, to
230 compute the candidate root as
231
232 x = (u/v)^{(p+3)/8} = u v^3 (u v^7)^{(p-5)/8}.
233 */
234 #if ECC_SQRT_E != 2
235 #error Broken curve25519 parameters
236 #endif
237
238 /* Needs 4*n space + scratch for ecc_mod_pow_252m3. */
239 #define ECC_25519_SQRT_ITCH (9*ECC_LIMB_SIZE)
240
241 static int
ecc_25519_sqrt(const struct ecc_modulo * p,mp_limb_t * rp,const mp_limb_t * up,const mp_limb_t * vp,mp_limb_t * scratch)242 ecc_25519_sqrt(const struct ecc_modulo *p, mp_limb_t *rp,
243 const mp_limb_t *up, const mp_limb_t *vp,
244 mp_limb_t *scratch)
245 {
246 int pos, neg;
247
248 #define uv3 scratch
249 #define uv7 (scratch + ECC_LIMB_SIZE)
250 #define uv7p (scratch + 2*ECC_LIMB_SIZE)
251 #define v2 (scratch + 2*ECC_LIMB_SIZE)
252 #define uv (scratch + 3*ECC_LIMB_SIZE)
253 #define v4 (scratch + 3*ECC_LIMB_SIZE)
254
255 #define scratch_out (scratch + 4 * ECC_LIMB_SIZE)
256
257 #define x2 scratch
258 #define vx2 (scratch + ECC_LIMB_SIZE)
259 #define t0 (scratch + 2*ECC_LIMB_SIZE)
260
261 /* Live values */
262 ecc_mod_sqr (p, v2, vp); /* v2 */
263 ecc_mod_mul (p, uv, up, vp); /* uv, v2 */
264 ecc_mod_mul (p, uv3, uv, v2); /* uv3, v2 */
265 ecc_mod_sqr (p, v4, v2); /* uv3, v4 */
266 ecc_mod_mul (p, uv7, uv3, v4); /* uv3, uv7 */
267 ecc_mod_pow_252m3 (p, uv7p, uv7, scratch_out); /* uv3, uv7p */
268 ecc_mod_mul (p, rp, uv7p, uv3); /* none */
269
270 /* Check sign. If square root exists, have v x^2 = ±u */
271 ecc_mod_sqr (p, x2, rp);
272 ecc_mod_mul (p, vx2, x2, vp);
273 ecc_mod_add (p, t0, vx2, up);
274 neg = ecc_25519_zero_p (p, t0);
275 ecc_mod_sub (p, t0, up, vx2);
276 pos = ecc_25519_zero_p (p, t0);
277
278 ecc_mod_mul (p, t0, rp, ecc_sqrt_z);
279 cnd_copy (neg, rp, t0, ECC_LIMB_SIZE);
280 return pos | neg;
281
282 #undef uv3
283 #undef uv7
284 #undef uv7p
285 #undef v2
286 #undef v4
287 #undef scratch_out
288 #undef x2
289 #undef vx2
290 #undef t0
291 }
292
293 const struct ecc_curve _nettle_curve25519 =
294 {
295 {
296 255,
297 ECC_LIMB_SIZE,
298 ECC_BMODP_SIZE,
299 0,
300 ECC_25519_INV_ITCH,
301 ECC_25519_SQRT_ITCH,
302
303 ecc_p,
304 ecc_Bmodp,
305 ecc_Bmodp_shifted,
306 NULL,
307 ecc_pp1h,
308
309 ecc_25519_modp,
310 ecc_25519_modp,
311 ecc_25519_inv,
312 ecc_25519_sqrt,
313 },
314 {
315 253,
316 ECC_LIMB_SIZE,
317 ECC_BMODQ_SIZE,
318 0,
319 ECC_MOD_INV_ITCH (ECC_LIMB_SIZE),
320 0,
321
322 ecc_q,
323 ecc_Bmodq,
324 ecc_mBmodq_shifted, /* Use q - 2^{252} instead. */
325 NULL,
326 ecc_qp1h,
327
328 ecc_25519_modq,
329 ecc_25519_modq,
330 ecc_mod_inv,
331 NULL,
332 },
333
334 0, /* No redc */
335 ECC_PIPPENGER_K,
336 ECC_PIPPENGER_C,
337
338 ECC_ADD_EHH_ITCH (ECC_LIMB_SIZE),
339 ECC_MUL_A_EH_ITCH (ECC_LIMB_SIZE),
340 ECC_MUL_G_EH_ITCH (ECC_LIMB_SIZE),
341 ECC_EH_TO_A_ITCH (ECC_LIMB_SIZE, ECC_25519_INV_ITCH),
342
343 ecc_add_ehh,
344 ecc_mul_a_eh,
345 ecc_mul_g_eh,
346 ecc_eh_to_a,
347
348 ecc_d, /* Use the Edwards curve constant. */
349 ecc_g,
350 ecc_edwards,
351 ecc_unit,
352 ecc_table
353 };
354