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 "fe25519.h" |
10 |
|
|
#include "sc25519.h" |
11 |
|
|
#include "ge25519.h" |
12 |
|
|
|
13 |
|
|
/* |
14 |
|
|
* Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 |
15 |
|
|
* with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555 |
16 |
|
|
* Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960); |
17 |
|
|
*/ |
18 |
|
|
|
19 |
|
|
/* d */ |
20 |
|
|
static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00, |
21 |
|
|
0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}}; |
22 |
|
|
/* 2*d */ |
23 |
|
|
static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00, |
24 |
|
|
0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}}; |
25 |
|
|
/* sqrt(-1) */ |
26 |
|
|
static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F, |
27 |
|
|
0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}}; |
28 |
|
|
|
29 |
|
|
#define ge25519_p3 ge25519 |
30 |
|
|
|
31 |
|
|
typedef struct |
32 |
|
|
{ |
33 |
|
|
fe25519 x; |
34 |
|
|
fe25519 z; |
35 |
|
|
fe25519 y; |
36 |
|
|
fe25519 t; |
37 |
|
|
} ge25519_p1p1; |
38 |
|
|
|
39 |
|
|
typedef struct |
40 |
|
|
{ |
41 |
|
|
fe25519 x; |
42 |
|
|
fe25519 y; |
43 |
|
|
fe25519 z; |
44 |
|
|
} ge25519_p2; |
45 |
|
|
|
46 |
|
|
typedef struct |
47 |
|
|
{ |
48 |
|
|
fe25519 x; |
49 |
|
|
fe25519 y; |
50 |
|
|
} ge25519_aff; |
51 |
|
|
|
52 |
|
|
|
53 |
|
|
/* Packed coordinates of the base point */ |
54 |
|
|
const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69, |
55 |
|
|
0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}}, |
56 |
|
|
{{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, |
57 |
|
|
0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}}, |
58 |
|
|
{{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, |
59 |
|
|
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}}, |
60 |
|
|
{{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20, |
61 |
|
|
0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}}; |
62 |
|
|
|
63 |
|
|
/* Multiples of the base point in affine representation */ |
64 |
|
|
static const ge25519_aff ge25519_base_multiples_affine[425] = { |
65 |
|
|
#include "ge25519_base.data" |
66 |
|
|
}; |
67 |
|
|
|
68 |
|
|
static void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p) |
69 |
|
|
{ |
70 |
|
|
fe25519_mul(&r->x, &p->x, &p->t); |
71 |
|
|
fe25519_mul(&r->y, &p->y, &p->z); |
72 |
|
|
fe25519_mul(&r->z, &p->z, &p->t); |
73 |
|
|
} |
74 |
|
|
|
75 |
|
|
static void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p) |
76 |
|
|
{ |
77 |
|
|
p1p1_to_p2((ge25519_p2 *)r, p); |
78 |
|
|
fe25519_mul(&r->t, &p->x, &p->y); |
79 |
|
|
} |
80 |
|
|
|
81 |
|
|
static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q) |
82 |
|
|
{ |
83 |
|
|
fe25519 a,b,t1,t2,c,d,e,f,g,h,qt; |
84 |
|
|
fe25519_mul(&qt, &q->x, &q->y); |
85 |
|
|
fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */ |
86 |
|
|
fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */ |
87 |
|
|
fe25519_sub(&t1, &q->y, &q->x); |
88 |
|
|
fe25519_add(&t2, &q->y, &q->x); |
89 |
|
|
fe25519_mul(&a, &a, &t1); |
90 |
|
|
fe25519_mul(&b, &b, &t2); |
91 |
|
|
fe25519_sub(&e, &b, &a); /* E = B-A */ |
92 |
|
|
fe25519_add(&h, &b, &a); /* H = B+A */ |
93 |
|
|
fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */ |
94 |
|
|
fe25519_mul(&c, &c, &ge25519_ec2d); |
95 |
|
|
fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */ |
96 |
|
|
fe25519_sub(&f, &d, &c); /* F = D-C */ |
97 |
|
|
fe25519_add(&g, &d, &c); /* G = D+C */ |
98 |
|
|
fe25519_mul(&r->x, &e, &f); |
99 |
|
|
fe25519_mul(&r->y, &h, &g); |
100 |
|
|
fe25519_mul(&r->z, &g, &f); |
101 |
|
|
fe25519_mul(&r->t, &e, &h); |
102 |
|
|
} |
103 |
|
|
|
104 |
|
|
static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q) |
105 |
|
|
{ |
106 |
|
|
fe25519 a, b, c, d, t; |
107 |
|
|
|
108 |
|
|
fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */ |
109 |
|
|
fe25519_sub(&t, &q->y, &q->x); |
110 |
|
|
fe25519_mul(&a, &a, &t); |
111 |
|
|
fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */ |
112 |
|
|
fe25519_add(&t, &q->x, &q->y); |
113 |
|
|
fe25519_mul(&b, &b, &t); |
114 |
|
|
fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */ |
115 |
|
|
fe25519_mul(&c, &c, &ge25519_ec2d); |
116 |
|
|
fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */ |
117 |
|
|
fe25519_add(&d, &d, &d); |
118 |
|
|
fe25519_sub(&r->x, &b, &a); /* E = B-A */ |
119 |
|
|
fe25519_sub(&r->t, &d, &c); /* F = D-C */ |
120 |
|
|
fe25519_add(&r->z, &d, &c); /* G = D+C */ |
121 |
|
|
fe25519_add(&r->y, &b, &a); /* H = B+A */ |
122 |
|
|
} |
123 |
|
|
|
124 |
|
|
/* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */ |
125 |
|
|
static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p) |
126 |
|
|
{ |
127 |
|
|
fe25519 a,b,c,d; |
128 |
|
|
fe25519_square(&a, &p->x); |
129 |
|
|
fe25519_square(&b, &p->y); |
130 |
|
|
fe25519_square(&c, &p->z); |
131 |
|
|
fe25519_add(&c, &c, &c); |
132 |
|
|
fe25519_neg(&d, &a); |
133 |
|
|
|
134 |
|
|
fe25519_add(&r->x, &p->x, &p->y); |
135 |
|
|
fe25519_square(&r->x, &r->x); |
136 |
|
|
fe25519_sub(&r->x, &r->x, &a); |
137 |
|
|
fe25519_sub(&r->x, &r->x, &b); |
138 |
|
|
fe25519_add(&r->z, &d, &b); |
139 |
|
|
fe25519_sub(&r->t, &r->z, &c); |
140 |
|
|
fe25519_sub(&r->y, &d, &b); |
141 |
|
|
} |
142 |
|
|
|
143 |
|
|
/* Constant-time version of: if(b) r = p */ |
144 |
|
|
static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b) |
145 |
|
|
{ |
146 |
|
|
fe25519_cmov(&r->x, &p->x, b); |
147 |
|
|
fe25519_cmov(&r->y, &p->y, b); |
148 |
|
|
} |
149 |
|
|
|
150 |
|
|
static unsigned char equal(signed char b,signed char c) |
151 |
|
|
{ |
152 |
|
|
unsigned char ub = b; |
153 |
|
|
unsigned char uc = c; |
154 |
|
|
unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */ |
155 |
|
|
crypto_uint32 y = x; /* 0: yes; 1..255: no */ |
156 |
|
|
y -= 1; /* 4294967295: yes; 0..254: no */ |
157 |
|
|
y >>= 31; /* 1: yes; 0: no */ |
158 |
|
|
return y; |
159 |
|
|
} |
160 |
|
|
|
161 |
|
|
static unsigned char negative(signed char b) |
162 |
|
|
{ |
163 |
|
|
unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */ |
164 |
|
|
x >>= 63; /* 1: yes; 0: no */ |
165 |
|
|
return x; |
166 |
|
|
} |
167 |
|
|
|
168 |
|
|
static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b) |
169 |
|
|
{ |
170 |
|
|
/* constant time */ |
171 |
|
|
fe25519 v; |
172 |
|
|
*t = ge25519_base_multiples_affine[5*pos+0]; |
173 |
|
|
cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1)); |
174 |
|
|
cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2)); |
175 |
|
|
cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3)); |
176 |
|
|
cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4)); |
177 |
|
|
fe25519_neg(&v, &t->x); |
178 |
|
|
fe25519_cmov(&t->x, &v, negative(b)); |
179 |
|
|
} |
180 |
|
|
|
181 |
|
|
static void setneutral(ge25519 *r) |
182 |
|
|
{ |
183 |
|
|
fe25519_setzero(&r->x); |
184 |
|
|
fe25519_setone(&r->y); |
185 |
|
|
fe25519_setone(&r->z); |
186 |
|
|
fe25519_setzero(&r->t); |
187 |
|
|
} |
188 |
|
|
|
189 |
|
|
/* ******************************************************************** |
190 |
|
|
* EXPORTED FUNCTIONS |
191 |
|
|
******************************************************************** */ |
192 |
|
|
|
193 |
|
|
/* return 0 on success, -1 otherwise */ |
194 |
|
|
int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32]) |
195 |
|
|
{ |
196 |
|
|
unsigned char par; |
197 |
|
|
fe25519 t, chk, num, den, den2, den4, den6; |
198 |
|
|
fe25519_setone(&r->z); |
199 |
|
|
par = p[31] >> 7; |
200 |
|
|
fe25519_unpack(&r->y, p); |
201 |
|
|
fe25519_square(&num, &r->y); /* x = y^2 */ |
202 |
|
|
fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */ |
203 |
|
|
fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */ |
204 |
|
|
fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */ |
205 |
|
|
|
206 |
|
|
/* Computation of sqrt(num/den) */ |
207 |
|
|
/* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */ |
208 |
|
|
fe25519_square(&den2, &den); |
209 |
|
|
fe25519_square(&den4, &den2); |
210 |
|
|
fe25519_mul(&den6, &den4, &den2); |
211 |
|
|
fe25519_mul(&t, &den6, &num); |
212 |
|
|
fe25519_mul(&t, &t, &den); |
213 |
|
|
|
214 |
|
|
fe25519_pow2523(&t, &t); |
215 |
|
|
/* 2. computation of r->x = t * num * den^3 */ |
216 |
|
|
fe25519_mul(&t, &t, &num); |
217 |
|
|
fe25519_mul(&t, &t, &den); |
218 |
|
|
fe25519_mul(&t, &t, &den); |
219 |
|
|
fe25519_mul(&r->x, &t, &den); |
220 |
|
|
|
221 |
|
|
/* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */ |
222 |
|
|
fe25519_square(&chk, &r->x); |
223 |
|
|
fe25519_mul(&chk, &chk, &den); |
224 |
|
|
if (!fe25519_iseq_vartime(&chk, &num)) |
225 |
|
|
fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1); |
226 |
|
|
|
227 |
|
|
/* 4. Now we have one of the two square roots, except if input was not a square */ |
228 |
|
|
fe25519_square(&chk, &r->x); |
229 |
|
|
fe25519_mul(&chk, &chk, &den); |
230 |
|
|
if (!fe25519_iseq_vartime(&chk, &num)) |
231 |
|
|
return -1; |
232 |
|
|
|
233 |
|
|
/* 5. Choose the desired square root according to parity: */ |
234 |
|
|
if(fe25519_getparity(&r->x) != (1-par)) |
235 |
|
|
fe25519_neg(&r->x, &r->x); |
236 |
|
|
|
237 |
|
|
fe25519_mul(&r->t, &r->x, &r->y); |
238 |
|
|
return 0; |
239 |
|
|
} |
240 |
|
|
|
241 |
|
|
void ge25519_pack(unsigned char r[32], const ge25519_p3 *p) |
242 |
|
|
{ |
243 |
|
|
fe25519 tx, ty, zi; |
244 |
|
|
fe25519_invert(&zi, &p->z); |
245 |
|
|
fe25519_mul(&tx, &p->x, &zi); |
246 |
|
|
fe25519_mul(&ty, &p->y, &zi); |
247 |
|
|
fe25519_pack(r, &ty); |
248 |
|
|
r[31] ^= fe25519_getparity(&tx) << 7; |
249 |
|
|
} |
250 |
|
|
|
251 |
|
|
int ge25519_isneutral_vartime(const ge25519_p3 *p) |
252 |
|
|
{ |
253 |
|
|
int ret = 1; |
254 |
|
|
if(!fe25519_iszero(&p->x)) ret = 0; |
255 |
|
|
if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0; |
256 |
|
|
return ret; |
257 |
|
|
} |
258 |
|
|
|
259 |
|
|
/* computes [s1]p1 + [s2]p2 */ |
260 |
|
|
void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2) |
261 |
|
|
{ |
262 |
|
|
ge25519_p1p1 tp1p1; |
263 |
|
|
ge25519_p3 pre[16]; |
264 |
|
|
unsigned char b[127]; |
265 |
|
|
int i; |
266 |
|
|
|
267 |
|
|
/* precomputation s2 s1 */ |
268 |
|
|
setneutral(pre); /* 00 00 */ |
269 |
|
|
pre[1] = *p1; /* 00 01 */ |
270 |
|
|
dbl_p1p1(&tp1p1,(ge25519_p2 *)p1); p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */ |
271 |
|
|
add_p1p1(&tp1p1,&pre[1], &pre[2]); p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */ |
272 |
|
|
pre[4] = *p2; /* 01 00 */ |
273 |
|
|
add_p1p1(&tp1p1,&pre[1], &pre[4]); p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */ |
274 |
|
|
add_p1p1(&tp1p1,&pre[2], &pre[4]); p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */ |
275 |
|
|
add_p1p1(&tp1p1,&pre[3], &pre[4]); p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */ |
276 |
|
|
dbl_p1p1(&tp1p1,(ge25519_p2 *)p2); p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */ |
277 |
|
|
add_p1p1(&tp1p1,&pre[1], &pre[8]); p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */ |
278 |
|
|
dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */ |
279 |
|
|
add_p1p1(&tp1p1,&pre[3], &pre[8]); p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */ |
280 |
|
|
add_p1p1(&tp1p1,&pre[4], &pre[8]); p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */ |
281 |
|
|
add_p1p1(&tp1p1,&pre[1],&pre[12]); p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */ |
282 |
|
|
add_p1p1(&tp1p1,&pre[2],&pre[12]); p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */ |
283 |
|
|
add_p1p1(&tp1p1,&pre[3],&pre[12]); p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */ |
284 |
|
|
|
285 |
|
|
sc25519_2interleave2(b,s1,s2); |
286 |
|
|
|
287 |
|
|
/* scalar multiplication */ |
288 |
|
|
*r = pre[b[126]]; |
289 |
|
|
for(i=125;i>=0;i--) |
290 |
|
|
{ |
291 |
|
|
dbl_p1p1(&tp1p1, (ge25519_p2 *)r); |
292 |
|
|
p1p1_to_p2((ge25519_p2 *) r, &tp1p1); |
293 |
|
|
dbl_p1p1(&tp1p1, (ge25519_p2 *)r); |
294 |
|
|
if(b[i]!=0) |
295 |
|
|
{ |
296 |
|
|
p1p1_to_p3(r, &tp1p1); |
297 |
|
|
add_p1p1(&tp1p1, r, &pre[b[i]]); |
298 |
|
|
} |
299 |
|
|
if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1); |
300 |
|
|
else p1p1_to_p3(r, &tp1p1); |
301 |
|
|
} |
302 |
|
|
} |
303 |
|
|
|
304 |
|
|
void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s) |
305 |
|
|
{ |
306 |
|
|
signed char b[85]; |
307 |
|
|
int i; |
308 |
|
|
ge25519_aff t; |
309 |
|
|
sc25519_window3(b,s); |
310 |
|
|
|
311 |
|
|
choose_t((ge25519_aff *)r, 0, b[0]); |
312 |
|
|
fe25519_setone(&r->z); |
313 |
|
|
fe25519_mul(&r->t, &r->x, &r->y); |
314 |
|
|
for(i=1;i<85;i++) |
315 |
|
|
{ |
316 |
|
|
choose_t(&t, (unsigned long long) i, b[i]); |
317 |
|
|
ge25519_mixadd2(r, &t); |
318 |
|
|
} |
319 |
|
|
} |