xref: /openssh-portable/ge25519.c (revision f45f78ae)
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 "includes.h"
10 
11 #include "fe25519.h"
12 #include "sc25519.h"
13 #include "ge25519.h"
14 
15 /*
16  * Arithmetic on the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2
17  * with d = -(121665/121666) = 37095705934669439343138083508754565189542113879843219016388785533085940283555
18  * Base point: (15112221349535400772501151409588531511454012693041857206046113283949847762202,46316835694926478169428394003475163141307993866256225615783033603165251855960);
19  */
20 
21 /* d */
22 static const fe25519 ge25519_ecd = {{0xA3, 0x78, 0x59, 0x13, 0xCA, 0x4D, 0xEB, 0x75, 0xAB, 0xD8, 0x41, 0x41, 0x4D, 0x0A, 0x70, 0x00,
23                       0x98, 0xE8, 0x79, 0x77, 0x79, 0x40, 0xC7, 0x8C, 0x73, 0xFE, 0x6F, 0x2B, 0xEE, 0x6C, 0x03, 0x52}};
24 /* 2*d */
25 static const fe25519 ge25519_ec2d = {{0x59, 0xF1, 0xB2, 0x26, 0x94, 0x9B, 0xD6, 0xEB, 0x56, 0xB1, 0x83, 0x82, 0x9A, 0x14, 0xE0, 0x00,
26                        0x30, 0xD1, 0xF3, 0xEE, 0xF2, 0x80, 0x8E, 0x19, 0xE7, 0xFC, 0xDF, 0x56, 0xDC, 0xD9, 0x06, 0x24}};
27 /* sqrt(-1) */
28 static const fe25519 ge25519_sqrtm1 = {{0xB0, 0xA0, 0x0E, 0x4A, 0x27, 0x1B, 0xEE, 0xC4, 0x78, 0xE4, 0x2F, 0xAD, 0x06, 0x18, 0x43, 0x2F,
29                          0xA7, 0xD7, 0xFB, 0x3D, 0x99, 0x00, 0x4D, 0x2B, 0x0B, 0xDF, 0xC1, 0x4F, 0x80, 0x24, 0x83, 0x2B}};
30 
31 #define ge25519_p3 ge25519
32 
33 typedef struct
34 {
35   fe25519 x;
36   fe25519 z;
37   fe25519 y;
38   fe25519 t;
39 } ge25519_p1p1;
40 
41 typedef struct
42 {
43   fe25519 x;
44   fe25519 y;
45   fe25519 z;
46 } ge25519_p2;
47 
48 typedef struct
49 {
50   fe25519 x;
51   fe25519 y;
52 } ge25519_aff;
53 
54 
55 /* Packed coordinates of the base point */
56 const ge25519 ge25519_base = {{{0x1A, 0xD5, 0x25, 0x8F, 0x60, 0x2D, 0x56, 0xC9, 0xB2, 0xA7, 0x25, 0x95, 0x60, 0xC7, 0x2C, 0x69,
57                                 0x5C, 0xDC, 0xD6, 0xFD, 0x31, 0xE2, 0xA4, 0xC0, 0xFE, 0x53, 0x6E, 0xCD, 0xD3, 0x36, 0x69, 0x21}},
58                               {{0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
59                                 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}},
60                               {{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
61                                 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}},
62                               {{0xA3, 0xDD, 0xB7, 0xA5, 0xB3, 0x8A, 0xDE, 0x6D, 0xF5, 0x52, 0x51, 0x77, 0x80, 0x9F, 0xF0, 0x20,
63                                 0x7D, 0xE3, 0xAB, 0x64, 0x8E, 0x4E, 0xEA, 0x66, 0x65, 0x76, 0x8B, 0xD7, 0x0F, 0x5F, 0x87, 0x67}}};
64 
65 /* Multiples of the base point in affine representation */
66 static const ge25519_aff ge25519_base_multiples_affine[425] = {
67 #include "ge25519_base.data"
68 };
69 
p1p1_to_p2(ge25519_p2 * r,const ge25519_p1p1 * p)70 static void p1p1_to_p2(ge25519_p2 *r, const ge25519_p1p1 *p)
71 {
72   fe25519_mul(&r->x, &p->x, &p->t);
73   fe25519_mul(&r->y, &p->y, &p->z);
74   fe25519_mul(&r->z, &p->z, &p->t);
75 }
76 
p1p1_to_p3(ge25519_p3 * r,const ge25519_p1p1 * p)77 static void p1p1_to_p3(ge25519_p3 *r, const ge25519_p1p1 *p)
78 {
79   p1p1_to_p2((ge25519_p2 *)r, p);
80   fe25519_mul(&r->t, &p->x, &p->y);
81 }
82 
ge25519_mixadd2(ge25519_p3 * r,const ge25519_aff * q)83 static void ge25519_mixadd2(ge25519_p3 *r, const ge25519_aff *q)
84 {
85   fe25519 a,b,t1,t2,c,d,e,f,g,h,qt;
86   fe25519_mul(&qt, &q->x, &q->y);
87   fe25519_sub(&a, &r->y, &r->x); /* A = (Y1-X1)*(Y2-X2) */
88   fe25519_add(&b, &r->y, &r->x); /* B = (Y1+X1)*(Y2+X2) */
89   fe25519_sub(&t1, &q->y, &q->x);
90   fe25519_add(&t2, &q->y, &q->x);
91   fe25519_mul(&a, &a, &t1);
92   fe25519_mul(&b, &b, &t2);
93   fe25519_sub(&e, &b, &a); /* E = B-A */
94   fe25519_add(&h, &b, &a); /* H = B+A */
95   fe25519_mul(&c, &r->t, &qt); /* C = T1*k*T2 */
96   fe25519_mul(&c, &c, &ge25519_ec2d);
97   fe25519_add(&d, &r->z, &r->z); /* D = Z1*2 */
98   fe25519_sub(&f, &d, &c); /* F = D-C */
99   fe25519_add(&g, &d, &c); /* G = D+C */
100   fe25519_mul(&r->x, &e, &f);
101   fe25519_mul(&r->y, &h, &g);
102   fe25519_mul(&r->z, &g, &f);
103   fe25519_mul(&r->t, &e, &h);
104 }
105 
add_p1p1(ge25519_p1p1 * r,const ge25519_p3 * p,const ge25519_p3 * q)106 static void add_p1p1(ge25519_p1p1 *r, const ge25519_p3 *p, const ge25519_p3 *q)
107 {
108   fe25519 a, b, c, d, t;
109 
110   fe25519_sub(&a, &p->y, &p->x); /* A = (Y1-X1)*(Y2-X2) */
111   fe25519_sub(&t, &q->y, &q->x);
112   fe25519_mul(&a, &a, &t);
113   fe25519_add(&b, &p->x, &p->y); /* B = (Y1+X1)*(Y2+X2) */
114   fe25519_add(&t, &q->x, &q->y);
115   fe25519_mul(&b, &b, &t);
116   fe25519_mul(&c, &p->t, &q->t); /* C = T1*k*T2 */
117   fe25519_mul(&c, &c, &ge25519_ec2d);
118   fe25519_mul(&d, &p->z, &q->z); /* D = Z1*2*Z2 */
119   fe25519_add(&d, &d, &d);
120   fe25519_sub(&r->x, &b, &a); /* E = B-A */
121   fe25519_sub(&r->t, &d, &c); /* F = D-C */
122   fe25519_add(&r->z, &d, &c); /* G = D+C */
123   fe25519_add(&r->y, &b, &a); /* H = B+A */
124 }
125 
126 /* See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd */
dbl_p1p1(ge25519_p1p1 * r,const ge25519_p2 * p)127 static void dbl_p1p1(ge25519_p1p1 *r, const ge25519_p2 *p)
128 {
129   fe25519 a,b,c,d;
130   fe25519_square(&a, &p->x);
131   fe25519_square(&b, &p->y);
132   fe25519_square(&c, &p->z);
133   fe25519_add(&c, &c, &c);
134   fe25519_neg(&d, &a);
135 
136   fe25519_add(&r->x, &p->x, &p->y);
137   fe25519_square(&r->x, &r->x);
138   fe25519_sub(&r->x, &r->x, &a);
139   fe25519_sub(&r->x, &r->x, &b);
140   fe25519_add(&r->z, &d, &b);
141   fe25519_sub(&r->t, &r->z, &c);
142   fe25519_sub(&r->y, &d, &b);
143 }
144 
145 /* Constant-time version of: if(b) r = p */
cmov_aff(ge25519_aff * r,const ge25519_aff * p,unsigned char b)146 static void cmov_aff(ge25519_aff *r, const ge25519_aff *p, unsigned char b)
147 {
148   fe25519_cmov(&r->x, &p->x, b);
149   fe25519_cmov(&r->y, &p->y, b);
150 }
151 
equal(signed char b,signed char c)152 static unsigned char equal(signed char b,signed char c)
153 {
154   unsigned char ub = b;
155   unsigned char uc = c;
156   unsigned char x = ub ^ uc; /* 0: yes; 1..255: no */
157   crypto_uint32 y = x; /* 0: yes; 1..255: no */
158   y -= 1; /* 4294967295: yes; 0..254: no */
159   y >>= 31; /* 1: yes; 0: no */
160   return y;
161 }
162 
negative(signed char b)163 static unsigned char negative(signed char b)
164 {
165   unsigned long long x = b; /* 18446744073709551361..18446744073709551615: yes; 0..255: no */
166   x >>= 63; /* 1: yes; 0: no */
167   return x;
168 }
169 
choose_t(ge25519_aff * t,unsigned long long pos,signed char b)170 static void choose_t(ge25519_aff *t, unsigned long long pos, signed char b)
171 {
172   /* constant time */
173   fe25519 v;
174   *t = ge25519_base_multiples_affine[5*pos+0];
175   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+1],equal(b,1) | equal(b,-1));
176   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+2],equal(b,2) | equal(b,-2));
177   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+3],equal(b,3) | equal(b,-3));
178   cmov_aff(t, &ge25519_base_multiples_affine[5*pos+4],equal(b,-4));
179   fe25519_neg(&v, &t->x);
180   fe25519_cmov(&t->x, &v, negative(b));
181 }
182 
setneutral(ge25519 * r)183 static void setneutral(ge25519 *r)
184 {
185   fe25519_setzero(&r->x);
186   fe25519_setone(&r->y);
187   fe25519_setone(&r->z);
188   fe25519_setzero(&r->t);
189 }
190 
191 /* ********************************************************************
192  *                    EXPORTED FUNCTIONS
193  ******************************************************************** */
194 
195 /* return 0 on success, -1 otherwise */
ge25519_unpackneg_vartime(ge25519_p3 * r,const unsigned char p[32])196 int ge25519_unpackneg_vartime(ge25519_p3 *r, const unsigned char p[32])
197 {
198   unsigned char par;
199   fe25519 t, chk, num, den, den2, den4, den6;
200   fe25519_setone(&r->z);
201   par = p[31] >> 7;
202   fe25519_unpack(&r->y, p);
203   fe25519_square(&num, &r->y); /* x = y^2 */
204   fe25519_mul(&den, &num, &ge25519_ecd); /* den = dy^2 */
205   fe25519_sub(&num, &num, &r->z); /* x = y^2-1 */
206   fe25519_add(&den, &r->z, &den); /* den = dy^2+1 */
207 
208   /* Computation of sqrt(num/den) */
209   /* 1.: computation of num^((p-5)/8)*den^((7p-35)/8) = (num*den^7)^((p-5)/8) */
210   fe25519_square(&den2, &den);
211   fe25519_square(&den4, &den2);
212   fe25519_mul(&den6, &den4, &den2);
213   fe25519_mul(&t, &den6, &num);
214   fe25519_mul(&t, &t, &den);
215 
216   fe25519_pow2523(&t, &t);
217   /* 2. computation of r->x = t * num * den^3 */
218   fe25519_mul(&t, &t, &num);
219   fe25519_mul(&t, &t, &den);
220   fe25519_mul(&t, &t, &den);
221   fe25519_mul(&r->x, &t, &den);
222 
223   /* 3. Check whether sqrt computation gave correct result, multiply by sqrt(-1) if not: */
224   fe25519_square(&chk, &r->x);
225   fe25519_mul(&chk, &chk, &den);
226   if (!fe25519_iseq_vartime(&chk, &num))
227     fe25519_mul(&r->x, &r->x, &ge25519_sqrtm1);
228 
229   /* 4. Now we have one of the two square roots, except if input was not a square */
230   fe25519_square(&chk, &r->x);
231   fe25519_mul(&chk, &chk, &den);
232   if (!fe25519_iseq_vartime(&chk, &num))
233     return -1;
234 
235   /* 5. Choose the desired square root according to parity: */
236   if(fe25519_getparity(&r->x) != (1-par))
237     fe25519_neg(&r->x, &r->x);
238 
239   fe25519_mul(&r->t, &r->x, &r->y);
240   return 0;
241 }
242 
ge25519_pack(unsigned char r[32],const ge25519_p3 * p)243 void ge25519_pack(unsigned char r[32], const ge25519_p3 *p)
244 {
245   fe25519 tx, ty, zi;
246   fe25519_invert(&zi, &p->z);
247   fe25519_mul(&tx, &p->x, &zi);
248   fe25519_mul(&ty, &p->y, &zi);
249   fe25519_pack(r, &ty);
250   r[31] ^= fe25519_getparity(&tx) << 7;
251 }
252 
ge25519_isneutral_vartime(const ge25519_p3 * p)253 int ge25519_isneutral_vartime(const ge25519_p3 *p)
254 {
255   int ret = 1;
256   if(!fe25519_iszero(&p->x)) ret = 0;
257   if(!fe25519_iseq_vartime(&p->y, &p->z)) ret = 0;
258   return ret;
259 }
260 
261 /* computes [s1]p1 + [s2]p2 */
ge25519_double_scalarmult_vartime(ge25519_p3 * r,const ge25519_p3 * p1,const sc25519 * s1,const ge25519_p3 * p2,const sc25519 * s2)262 void ge25519_double_scalarmult_vartime(ge25519_p3 *r, const ge25519_p3 *p1, const sc25519 *s1, const ge25519_p3 *p2, const sc25519 *s2)
263 {
264   ge25519_p1p1 tp1p1;
265   ge25519_p3 pre[16];
266   unsigned char b[127];
267   int i;
268 
269   /* precomputation                                                        s2 s1 */
270   setneutral(pre);                                                      /* 00 00 */
271   pre[1] = *p1;                                                         /* 00 01 */
272   dbl_p1p1(&tp1p1,(ge25519_p2 *)p1);      p1p1_to_p3( &pre[2], &tp1p1); /* 00 10 */
273   add_p1p1(&tp1p1,&pre[1], &pre[2]);      p1p1_to_p3( &pre[3], &tp1p1); /* 00 11 */
274   pre[4] = *p2;                                                         /* 01 00 */
275   add_p1p1(&tp1p1,&pre[1], &pre[4]);      p1p1_to_p3( &pre[5], &tp1p1); /* 01 01 */
276   add_p1p1(&tp1p1,&pre[2], &pre[4]);      p1p1_to_p3( &pre[6], &tp1p1); /* 01 10 */
277   add_p1p1(&tp1p1,&pre[3], &pre[4]);      p1p1_to_p3( &pre[7], &tp1p1); /* 01 11 */
278   dbl_p1p1(&tp1p1,(ge25519_p2 *)p2);      p1p1_to_p3( &pre[8], &tp1p1); /* 10 00 */
279   add_p1p1(&tp1p1,&pre[1], &pre[8]);      p1p1_to_p3( &pre[9], &tp1p1); /* 10 01 */
280   dbl_p1p1(&tp1p1,(ge25519_p2 *)&pre[5]); p1p1_to_p3(&pre[10], &tp1p1); /* 10 10 */
281   add_p1p1(&tp1p1,&pre[3], &pre[8]);      p1p1_to_p3(&pre[11], &tp1p1); /* 10 11 */
282   add_p1p1(&tp1p1,&pre[4], &pre[8]);      p1p1_to_p3(&pre[12], &tp1p1); /* 11 00 */
283   add_p1p1(&tp1p1,&pre[1],&pre[12]);      p1p1_to_p3(&pre[13], &tp1p1); /* 11 01 */
284   add_p1p1(&tp1p1,&pre[2],&pre[12]);      p1p1_to_p3(&pre[14], &tp1p1); /* 11 10 */
285   add_p1p1(&tp1p1,&pre[3],&pre[12]);      p1p1_to_p3(&pre[15], &tp1p1); /* 11 11 */
286 
287   sc25519_2interleave2(b,s1,s2);
288 
289   /* scalar multiplication */
290   *r = pre[b[126]];
291   for(i=125;i>=0;i--)
292   {
293     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
294     p1p1_to_p2((ge25519_p2 *) r, &tp1p1);
295     dbl_p1p1(&tp1p1, (ge25519_p2 *)r);
296     if(b[i]!=0)
297     {
298       p1p1_to_p3(r, &tp1p1);
299       add_p1p1(&tp1p1, r, &pre[b[i]]);
300     }
301     if(i != 0) p1p1_to_p2((ge25519_p2 *)r, &tp1p1);
302     else p1p1_to_p3(r, &tp1p1);
303   }
304 }
305 
ge25519_scalarmult_base(ge25519_p3 * r,const sc25519 * s)306 void ge25519_scalarmult_base(ge25519_p3 *r, const sc25519 *s)
307 {
308   signed char b[85];
309   int i;
310   ge25519_aff t;
311   sc25519_window3(b,s);
312 
313   choose_t((ge25519_aff *)r, 0, b[0]);
314   fe25519_setone(&r->z);
315   fe25519_mul(&r->t, &r->x, &r->y);
316   for(i=1;i<85;i++)
317   {
318     choose_t(&t, (unsigned long long) i, b[i]);
319     ge25519_mixadd2(r, &t);
320   }
321 }
322