1 /*
2 * Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
3 *
4 * Permission is hereby granted, free of charge, to any person obtaining
5 * a copy of this software and associated documentation files (the
6 * "Software"), to deal in the Software without restriction, including
7 * without limitation the rights to use, copy, modify, merge, publish,
8 * distribute, sublicense, and/or sell copies of the Software, and to
9 * permit persons to whom the Software is furnished to do so, subject to
10 * the following conditions:
11 *
12 * The above copyright notice and this permission notice shall be
13 * included in all copies or substantial portions of the Software.
14 *
15 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
16 * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
17 * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
18 * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
19 * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
20 * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
21 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
22 * SOFTWARE.
23 */
24
25 #include "inner.h"
26
27 /* see bearssl_rsa.h */
28 size_t
br_rsa_i15_compute_privexp(void * d,const br_rsa_private_key * sk,uint32_t e)29 br_rsa_i15_compute_privexp(void *d,
30 const br_rsa_private_key *sk, uint32_t e)
31 {
32 /*
33 * We want to invert e modulo phi = (p-1)(q-1). This first
34 * requires computing phi, which is easy since we have the factors
35 * p and q in the private key structure.
36 *
37 * Since p = 3 mod 4 and q = 3 mod 4, phi/4 is an odd integer.
38 * We could invert e modulo phi/4 then patch the result to
39 * modulo phi, but this would involve assembling three modulus-wide
40 * values (phi/4, 1 and e) and calling moddiv, that requires
41 * three more temporaries, for a total of six big integers, or
42 * slightly more than 3 kB of stack space for RSA-4096. This
43 * exceeds our stack requirements.
44 *
45 * Instead, we first use one step of the extended GCD:
46 *
47 * - We compute phi = k*e + r (Euclidean division of phi by e).
48 * If public exponent e is correct, then r != 0 (e must be
49 * invertible modulo phi). We also have k != 0 since we
50 * enforce non-ridiculously-small factors.
51 *
52 * - We find small u, v such that u*e - v*r = 1 (using a
53 * binary GCD; we can arrange for u < r and v < e, i.e. all
54 * values fit on 32 bits).
55 *
56 * - Solution is: d = u + v*k
57 * This last computation is exact: since u < r and v < e,
58 * the above implies d < r + e*((phi-r)/e) = phi
59 */
60
61 uint16_t tmp[4 * ((BR_MAX_RSA_FACTOR + 14) / 15) + 12];
62 uint16_t *p, *q, *k, *m, *z, *phi;
63 const unsigned char *pbuf, *qbuf;
64 size_t plen, qlen, u, len, dlen;
65 uint32_t r, a, b, u0, v0, u1, v1, he, hr;
66 int i;
67
68 /*
69 * Check that e is correct.
70 */
71 if (e < 3 || (e & 1) == 0) {
72 return 0;
73 }
74
75 /*
76 * Check lengths of p and q, and that they are both odd.
77 */
78 pbuf = sk->p;
79 plen = sk->plen;
80 while (plen > 0 && *pbuf == 0) {
81 pbuf ++;
82 plen --;
83 }
84 if (plen < 5 || plen > (BR_MAX_RSA_FACTOR / 8)
85 || (pbuf[plen - 1] & 1) != 1)
86 {
87 return 0;
88 }
89 qbuf = sk->q;
90 qlen = sk->qlen;
91 while (qlen > 0 && *qbuf == 0) {
92 qbuf ++;
93 qlen --;
94 }
95 if (qlen < 5 || qlen > (BR_MAX_RSA_FACTOR / 8)
96 || (qbuf[qlen - 1] & 1) != 1)
97 {
98 return 0;
99 }
100
101 /*
102 * Output length is that of the modulus.
103 */
104 dlen = (sk->n_bitlen + 7) >> 3;
105 if (d == NULL) {
106 return dlen;
107 }
108
109 p = tmp;
110 br_i15_decode(p, pbuf, plen);
111 plen = (p[0] + 15) >> 4;
112 q = p + 1 + plen;
113 br_i15_decode(q, qbuf, qlen);
114 qlen = (q[0] + 15) >> 4;
115
116 /*
117 * Compute phi = (p-1)*(q-1), then move it over p-1 and q-1 (that
118 * we do not need anymore). The mulacc function sets the announced
119 * bit length of t to be the sum of the announced bit lengths of
120 * p-1 and q-1, which is usually exact but may overshoot by one 1
121 * bit in some cases; we readjust it to its true length.
122 */
123 p[1] --;
124 q[1] --;
125 phi = q + 1 + qlen;
126 br_i15_zero(phi, p[0]);
127 br_i15_mulacc(phi, p, q);
128 len = (phi[0] + 15) >> 4;
129 memmove(tmp, phi, (1 + len) * sizeof *phi);
130 phi = tmp;
131 phi[0] = br_i15_bit_length(phi + 1, len);
132 len = (phi[0] + 15) >> 4;
133
134 /*
135 * Divide phi by public exponent e. The final remainder r must be
136 * non-zero (otherwise, the key is invalid). The quotient is k,
137 * which we write over phi, since we don't need phi after that.
138 */
139 r = 0;
140 for (u = len; u >= 1; u --) {
141 /*
142 * Upon entry, r < e, and phi[u] < 2^15; hence,
143 * hi:lo < e*2^15. Thus, the produced word k[u]
144 * must be lower than 2^15, and the new remainder r
145 * is lower than e.
146 */
147 uint32_t hi, lo;
148
149 hi = r >> 17;
150 lo = (r << 15) + phi[u];
151 phi[u] = br_divrem(hi, lo, e, &r);
152 }
153 if (r == 0) {
154 return 0;
155 }
156 k = phi;
157
158 /*
159 * Compute u and v such that u*e - v*r = GCD(e,r). We use
160 * a binary GCD algorithm, with 6 extra integers a, b,
161 * u0, u1, v0 and v1. Initial values are:
162 * a = e u0 = 1 v0 = 0
163 * b = r u1 = r v1 = e-1
164 * The following invariants are maintained:
165 * a = u0*e - v0*r
166 * b = u1*e - v1*r
167 * 0 < a <= e
168 * 0 < b <= r
169 * 0 <= u0 <= r
170 * 0 <= v0 <= e
171 * 0 <= u1 <= r
172 * 0 <= v1 <= e
173 *
174 * At each iteration, we reduce either a or b by one bit, and
175 * adjust u0, u1, v0 and v1 to maintain the invariants:
176 * - if a is even, then a <- a/2
177 * - otherwise, if b is even, then b <- b/2
178 * - otherwise, if a > b, then a <- (a-b)/2
179 * - otherwise, if b > a, then b <- (b-a)/2
180 * Algorithm stops when a = b. At that point, the common value
181 * is the GCD of e and r; it must be 1 (otherwise, the private
182 * key or public exponent is not valid). The (u0,v0) or (u1,v1)
183 * pairs are the solution we are looking for.
184 *
185 * Since either a or b is reduced by at least 1 bit at each
186 * iteration, 62 iterations are enough to reach the end
187 * condition.
188 *
189 * To maintain the invariants, we must compute the same operations
190 * on the u* and v* values that we do on a and b:
191 * - When a is divided by 2, u0 and v0 must be divided by 2.
192 * - When b is divided by 2, u1 and v1 must be divided by 2.
193 * - When b is subtracted from a, u1 and v1 are subtracted from
194 * u0 and v0, respectively.
195 * - When a is subtracted from b, u0 and v0 are subtracted from
196 * u1 and v1, respectively.
197 *
198 * However, we want to keep the u* and v* values in their proper
199 * ranges. The following remarks apply:
200 *
201 * - When a is divided by 2, then a is even. Therefore:
202 *
203 * * If r is odd, then u0 and v0 must have the same parity;
204 * if they are both odd, then adding r to u0 and e to v0
205 * makes them both even, and the division by 2 brings them
206 * back to the proper range.
207 *
208 * * If r is even, then u0 must be even; if v0 is odd, then
209 * adding r to u0 and e to v0 makes them both even, and the
210 * division by 2 brings them back to the proper range.
211 *
212 * Thus, all we need to do is to look at the parity of v0,
213 * and add (r,e) to (u0,v0) when v0 is odd. In order to avoid
214 * a 32-bit overflow, we can add ((r+1)/2,(e/2)+1) after the
215 * division (r+1 does not overflow since r < e; and (e/2)+1
216 * is equal to (e+1)/2 since e is odd).
217 *
218 * - When we subtract b from a, three cases may occur:
219 *
220 * * u1 <= u0 and v1 <= v0: just do the subtractions
221 *
222 * * u1 > u0 and v1 > v0: compute:
223 * (u0, v0) <- (u0 + r - u1, v0 + e - v1)
224 *
225 * * u1 <= u0 and v1 > v0: compute:
226 * (u0, v0) <- (u0 + r - u1, v0 + e - v1)
227 *
228 * The fourth case (u1 > u0 and v1 <= v0) is not possible
229 * because it would contradict "b < a" (which is the reason
230 * why we subtract b from a).
231 *
232 * The tricky case is the third one: from the equations, it
233 * seems that u0 may go out of range. However, the invariants
234 * and ranges of other values imply that, in that case, the
235 * new u0 does not actually exceed the range.
236 *
237 * We can thus handle the subtraction by adding (r,e) based
238 * solely on the comparison between v0 and v1.
239 */
240 a = e;
241 b = r;
242 u0 = 1;
243 v0 = 0;
244 u1 = r;
245 v1 = e - 1;
246 hr = (r + 1) >> 1;
247 he = (e >> 1) + 1;
248 for (i = 0; i < 62; i ++) {
249 uint32_t oa, ob, agtb, bgta;
250 uint32_t sab, sba, da, db;
251 uint32_t ctl;
252
253 oa = a & 1; /* 1 if a is odd */
254 ob = b & 1; /* 1 if b is odd */
255 agtb = GT(a, b); /* 1 if a > b */
256 bgta = GT(b, a); /* 1 if b > a */
257
258 sab = oa & ob & agtb; /* 1 if a <- a-b */
259 sba = oa & ob & bgta; /* 1 if b <- b-a */
260
261 /* a <- a-b, u0 <- u0-u1, v0 <- v0-v1 */
262 ctl = GT(v1, v0);
263 a -= b & -sab;
264 u0 -= (u1 - (r & -ctl)) & -sab;
265 v0 -= (v1 - (e & -ctl)) & -sab;
266
267 /* b <- b-a, u1 <- u1-u0 mod r, v1 <- v1-v0 mod e */
268 ctl = GT(v0, v1);
269 b -= a & -sba;
270 u1 -= (u0 - (r & -ctl)) & -sba;
271 v1 -= (v0 - (e & -ctl)) & -sba;
272
273 da = NOT(oa) | sab; /* 1 if a <- a/2 */
274 db = (oa & NOT(ob)) | sba; /* 1 if b <- b/2 */
275
276 /* a <- a/2, u0 <- u0/2, v0 <- v0/2 */
277 ctl = v0 & 1;
278 a ^= (a ^ (a >> 1)) & -da;
279 u0 ^= (u0 ^ ((u0 >> 1) + (hr & -ctl))) & -da;
280 v0 ^= (v0 ^ ((v0 >> 1) + (he & -ctl))) & -da;
281
282 /* b <- b/2, u1 <- u1/2 mod r, v1 <- v1/2 mod e */
283 ctl = v1 & 1;
284 b ^= (b ^ (b >> 1)) & -db;
285 u1 ^= (u1 ^ ((u1 >> 1) + (hr & -ctl))) & -db;
286 v1 ^= (v1 ^ ((v1 >> 1) + (he & -ctl))) & -db;
287 }
288
289 /*
290 * Check that the GCD is indeed 1. If not, then the key is invalid
291 * (and there's no harm in leaking that piece of information).
292 */
293 if (a != 1) {
294 return 0;
295 }
296
297 /*
298 * Now we have u0*e - v0*r = 1. Let's compute the result as:
299 * d = u0 + v0*k
300 * We still have k in the tmp[] array, and its announced bit
301 * length is that of phi.
302 */
303 m = k + 1 + len;
304 m[0] = (2 << 4) + 2; /* bit length is 32 bits, encoded */
305 m[1] = v0 & 0x7FFF;
306 m[2] = (v0 >> 15) & 0x7FFF;
307 m[3] = v0 >> 30;
308 z = m + 4;
309 br_i15_zero(z, k[0]);
310 z[1] = u0 & 0x7FFF;
311 z[2] = (u0 >> 15) & 0x7FFF;
312 z[3] = u0 >> 30;
313 br_i15_mulacc(z, k, m);
314
315 /*
316 * Encode the result.
317 */
318 br_i15_encode(d, dlen, z);
319 return dlen;
320 }
321