Line data Source code
1 : #include "./fd_bn254.h"
2 :
3 : /* G2 */
4 :
5 : /* COV: unlike g1, g2 operations are not exposed to users.
6 : So many edge cases and checks for zero are never triggered, e.g. by syscall tests. */
7 :
8 : static inline int
9 14709 : fd_bn254_g2_is_zero( fd_bn254_g2_t const * p ) {
10 14709 : return fd_bn254_fp2_is_zero( &p->Z );
11 14709 : }
12 :
13 : static inline int
14 : fd_bn254_g2_eq( fd_bn254_g2_t const * p,
15 127 : fd_bn254_g2_t const * q ) {
16 127 : if( fd_bn254_g2_is_zero( p ) ) {
17 13 : return fd_bn254_g2_is_zero( q );
18 13 : }
19 114 : if( fd_bn254_g2_is_zero( q ) ) {
20 0 : return 0;
21 0 : }
22 :
23 114 : fd_bn254_fp2_t pz2[1], qz2[1];
24 114 : fd_bn254_fp2_t l[1], r[1];
25 :
26 114 : fd_bn254_fp2_sqr( pz2, &p->Z );
27 114 : fd_bn254_fp2_sqr( qz2, &q->Z );
28 :
29 114 : fd_bn254_fp2_mul( l, &p->X, qz2 );
30 114 : fd_bn254_fp2_mul( r, &q->X, pz2 );
31 114 : if( !fd_bn254_fp2_eq( l, r ) ) {
32 8 : return 0;
33 8 : }
34 :
35 106 : fd_bn254_fp2_mul( l, &p->Y, qz2 );
36 106 : fd_bn254_fp2_mul( l, l, &q->Z );
37 106 : fd_bn254_fp2_mul( r, &q->Y, pz2 );
38 106 : fd_bn254_fp2_mul( r, r, &p->Z );
39 106 : return fd_bn254_fp2_eq( l, r );
40 114 : }
41 :
42 : static inline fd_bn254_g2_t *
43 : fd_bn254_g2_set( fd_bn254_g2_t * r,
44 351 : fd_bn254_g2_t const * p ) {
45 351 : fd_bn254_fp2_set( &r->X, &p->X );
46 351 : fd_bn254_fp2_set( &r->Y, &p->Y );
47 351 : fd_bn254_fp2_set( &r->Z, &p->Z );
48 351 : return r;
49 351 : }
50 :
51 : static inline fd_bn254_g2_t *
52 : fd_bn254_g2_neg( fd_bn254_g2_t * r,
53 84 : fd_bn254_g2_t const * p ) {
54 84 : fd_bn254_fp2_set( &r->X, &p->X );
55 84 : fd_bn254_fp2_neg( &r->Y, &p->Y );
56 84 : fd_bn254_fp2_set( &r->Z, &p->Z );
57 84 : return r;
58 84 : }
59 :
60 : static inline fd_bn254_g2_t *
61 52 : fd_bn254_g2_set_zero( fd_bn254_g2_t * r ) {
62 : // fd_bn254_fp2_set_zero( &r->X );
63 : // fd_bn254_fp2_set_zero( &r->Y );
64 52 : fd_bn254_fp2_set_zero( &r->Z );
65 52 : return r;
66 52 : }
67 :
68 : static inline fd_bn254_g2_t *
69 : fd_bn254_g2_to_affine( fd_bn254_g2_t * r,
70 0 : fd_bn254_g2_t const * p ) {
71 0 : if( FD_UNLIKELY( fd_bn254_fp2_is_zero( &p->Z ) || fd_bn254_fp2_is_one( &p->Z ) ) ) {
72 0 : return fd_bn254_g2_set( r, p );
73 0 : }
74 :
75 0 : fd_bn254_fp2_t iz[1], iz2[1];
76 0 : fd_bn254_fp2_inv( iz, &p->Z );
77 0 : fd_bn254_fp2_sqr( iz2, iz );
78 :
79 : /* X / Z^2, Y / Z^3 */
80 0 : fd_bn254_fp2_mul( &r->X, &p->X, iz2 );
81 0 : fd_bn254_fp2_mul( &r->Y, &p->Y, iz2 );
82 0 : fd_bn254_fp2_mul( &r->Y, &r->Y, iz );
83 0 : fd_bn254_fp2_set_one( &r->Z );
84 0 : return r;
85 0 : }
86 :
87 : uchar *
88 : fd_bn254_g2_tobytes( uchar out[128],
89 : fd_bn254_g2_t const * p,
90 0 : int big_endian ) {
91 0 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
92 0 : fd_memset( out, 0, 128UL );
93 : /* no flags */
94 0 : return out;
95 0 : }
96 :
97 0 : fd_bn254_g2_t r[1];
98 0 : fd_bn254_g2_to_affine( r, p );
99 :
100 0 : fd_bn254_fp2_from_mont( &r->X, &r->X );
101 0 : fd_bn254_fp2_from_mont( &r->Y, &r->Y );
102 :
103 0 : fd_bn254_fp2_tobytes_nm( &out[ 0], &r->X, big_endian );
104 0 : fd_bn254_fp2_tobytes_nm( &out[64], &r->Y, big_endian );
105 : /* no flags */
106 0 : return out;
107 0 : }
108 :
109 : static inline fd_bn254_g2_t *
110 : fd_bn254_g2_frob( fd_bn254_g2_t * r,
111 338 : fd_bn254_g2_t const * p ) {
112 338 : fd_bn254_fp2_conj( &r->X, &p->X );
113 338 : fd_bn254_fp2_mul ( &r->X, &r->X, &fd_bn254_const_frob_gamma1_mont[1] );
114 338 : fd_bn254_fp2_conj( &r->Y, &p->Y );
115 338 : fd_bn254_fp2_mul ( &r->Y, &r->Y, &fd_bn254_const_frob_gamma1_mont[2] );
116 338 : fd_bn254_fp2_conj( &r->Z, &p->Z );
117 338 : return r;
118 338 : }
119 :
120 : static inline fd_bn254_g2_t *
121 : fd_bn254_g2_frob2( fd_bn254_g2_t * r,
122 211 : fd_bn254_g2_t const * p ) {
123 : /* X */
124 211 : fd_bn254_fp_mul( &r->X.el[0], &p->X.el[0], &fd_bn254_const_frob_gamma2_mont[1] );
125 211 : fd_bn254_fp_mul( &r->X.el[1], &p->X.el[1], &fd_bn254_const_frob_gamma2_mont[1] );
126 : /* Y */
127 211 : fd_bn254_fp_mul( &r->Y.el[0], &p->Y.el[0], &fd_bn254_const_frob_gamma2_mont[2] );
128 211 : fd_bn254_fp_mul( &r->Y.el[1], &p->Y.el[1], &fd_bn254_const_frob_gamma2_mont[2] );
129 : /* Z=1 */
130 211 : fd_bn254_fp2_set( &r->Z, &p->Z );
131 211 : return r;
132 211 : }
133 :
134 : /* fd_bn254_g2_dbl computes r = 2p.
135 : https://hyperelliptic.org/efd/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l */
136 : fd_bn254_g2_t *
137 : fd_bn254_g2_dbl( fd_bn254_g2_t * r,
138 7130 : fd_bn254_g2_t const * p ) {
139 : /* p==0, return 0 */
140 7130 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
141 13 : return fd_bn254_g2_set_zero( r );
142 13 : }
143 :
144 7117 : fd_bn254_fp2_t a[1], b[1], c[1];
145 7117 : fd_bn254_fp2_t d[1], e[1], f[1];
146 :
147 : /* A = X1^2 */
148 7117 : fd_bn254_fp2_sqr( a, &p->X );
149 : /* B = Y1^2 */
150 7117 : fd_bn254_fp2_sqr( b, &p->Y );
151 : /* C = B^2 */
152 7117 : fd_bn254_fp2_sqr( c, b );
153 : /* D = 2*((X1+B)^2-A-C)
154 : (X1+B)^2 = X1^2 + 2*X1*B + B^2
155 : D = 2*(X1^2 + 2*X1*B + B^2 - A - C)
156 : D = 2*(X1^2 + 2*X1*B + B^2 - X1^2 - B^2)
157 : ^ ^ ^ ^
158 : |---------------|-----| |
159 : |------------|
160 : These terms cancel each other out, and we're left with:
161 : D = 2*(2*X1*B) */
162 7117 : fd_bn254_fp2_mul( d, &p->X, b );
163 7117 : fd_bn254_fp2_add( d, d, d );
164 7117 : fd_bn254_fp2_add( d, d, d );
165 : /* E = 3*A */
166 7117 : fd_bn254_fp2_add( e, a, a );
167 7117 : fd_bn254_fp2_add( e, a, e );
168 : /* F = E^2 */
169 7117 : fd_bn254_fp2_sqr( f, e );
170 : /* X3 = F-2*D */
171 7117 : fd_bn254_fp2_add( &r->X, d, d );
172 7117 : fd_bn254_fp2_sub( &r->X, f, &r->X );
173 : /* Z3 = (Y1+Z1)^2-YY-ZZ
174 : note: compute Z3 before Y3 because it depends on p->Y,
175 : that might be overwritten if r==p. */
176 : /* Z3 = 2*Y1*Z1 */
177 7117 : fd_bn254_fp2_mul( &r->Z, &p->Y, &p->Z );
178 7117 : fd_bn254_fp2_add( &r->Z, &r->Z, &r->Z );
179 : /* Y3 = E*(D-X3)-8*C */
180 7117 : fd_bn254_fp2_sub( &r->Y, d, &r->X );
181 7117 : fd_bn254_fp2_mul( &r->Y, e, &r->Y );
182 7117 : fd_bn254_fp2_add( c, c, c ); /* 2*c */
183 7117 : fd_bn254_fp2_add( c, c, c ); /* 4*y */
184 7117 : fd_bn254_fp2_add( c, c, c ); /* 8*y */
185 7117 : fd_bn254_fp2_sub( &r->Y, &r->Y, c );
186 7117 : return r;
187 7130 : }
188 :
189 : /* fd_bn254_g2_add_mixed computes r = p + q, when q->Z==1.
190 : http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl */
191 : fd_bn254_g2_t *
192 : fd_bn254_g2_add_mixed( fd_bn254_g2_t * r,
193 : fd_bn254_g2_t const * p,
194 3184 : fd_bn254_g2_t const * q ) {
195 : /* p==0, return q */
196 3184 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
197 13 : return fd_bn254_g2_set( r, q );
198 13 : }
199 : /* q==0, return p */
200 3171 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( q ) ) ) {
201 0 : return fd_bn254_g2_set( r, p );
202 0 : }
203 3171 : fd_bn254_fp2_t zz[1], u2[1], s2[1];
204 3171 : fd_bn254_fp2_t h[1], hh[1];
205 3171 : fd_bn254_fp2_t i[1], j[1];
206 3171 : fd_bn254_fp2_t rr[1], v[1];
207 : /* Z1Z1 = Z1^2 */
208 3171 : fd_bn254_fp2_sqr( zz, &p->Z );
209 : /* U2 = X2*Z1Z1 */
210 3171 : fd_bn254_fp2_mul( u2, &q->X, zz );
211 : /* S2 = Y2*Z1*Z1Z1 */
212 3171 : fd_bn254_fp2_mul( s2, &q->Y, &p->Z );
213 3171 : fd_bn254_fp2_mul( s2, s2, zz );
214 :
215 : /* if p==q, call fd_bn254_g2_dbl */
216 3171 : if( FD_UNLIKELY( fd_bn254_fp2_eq( u2, &p->X ) && fd_bn254_fp2_eq( s2, &p->Y ) ) ) {
217 0 : return fd_bn254_g2_dbl( r, p );
218 0 : }
219 :
220 : /* H = U2-X1 */
221 3171 : fd_bn254_fp2_sub( h, u2, &p->X );
222 : /* HH = H^2 */
223 3171 : fd_bn254_fp2_sqr( hh, h );
224 : /* I = 4*HH */
225 3171 : fd_bn254_fp2_add( i, hh, hh );
226 3171 : fd_bn254_fp2_add( i, i, i );
227 : /* J = H*I */
228 3171 : fd_bn254_fp2_mul( j, h, i );
229 : /* r = 2*(S2-Y1) */
230 3171 : fd_bn254_fp2_sub( rr, s2, &p->Y );
231 3171 : fd_bn254_fp2_add( rr, rr, rr );
232 : /* V = X1*I */
233 3171 : fd_bn254_fp2_mul( v, &p->X, i );
234 : /* X3 = r^2-J-2*V */
235 3171 : fd_bn254_fp2_sqr( &r->X, rr );
236 3171 : fd_bn254_fp2_sub( &r->X, &r->X, j );
237 3171 : fd_bn254_fp2_sub( &r->X, &r->X, v );
238 3171 : fd_bn254_fp2_sub( &r->X, &r->X, v );
239 : /* Y3 = r*(V-X3)-2*Y1*J
240 : note: i no longer used */
241 3171 : fd_bn254_fp2_mul( i, &p->Y, j ); /* i = Y1*J */
242 3171 : fd_bn254_fp2_add( i, i, i ); /* i = 2*Y1*J */
243 3171 : fd_bn254_fp2_sub( &r->Y, v, &r->X );
244 3171 : fd_bn254_fp2_mul( &r->Y, &r->Y, rr );
245 3171 : fd_bn254_fp2_sub( &r->Y, &r->Y, i );
246 : /* Z3 = (Z1+H)^2-Z1Z1-HH */
247 3171 : fd_bn254_fp2_add( &r->Z, &p->Z, h );
248 3171 : fd_bn254_fp2_sqr( &r->Z, &r->Z );
249 3171 : fd_bn254_fp2_sub( &r->Z, &r->Z, zz );
250 3171 : fd_bn254_fp2_sub( &r->Z, &r->Z, hh );
251 3171 : return r;
252 3171 : }
253 :
254 : /* fd_bn254_g2_add computes r = p + q.
255 : http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl */
256 : fd_bn254_g2_t *
257 : fd_bn254_g2_add( fd_bn254_g2_t * r,
258 : fd_bn254_g2_t const * p,
259 253 : fd_bn254_g2_t const * q ) {
260 : /* p==0, return q */
261 253 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
262 26 : return fd_bn254_g2_set( r, q );
263 26 : }
264 : /* q==0, return p */
265 227 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( q ) ) ) {
266 0 : return fd_bn254_g2_set( r, p );
267 0 : }
268 227 : fd_bn254_fp2_t zz1[1], zz2[1];
269 227 : fd_bn254_fp2_t u1[1], s1[1];
270 227 : fd_bn254_fp2_t u2[1], s2[1];
271 227 : fd_bn254_fp2_t h[1];
272 227 : fd_bn254_fp2_t i[1], j[1];
273 227 : fd_bn254_fp2_t rr[1], v[1];
274 : /* Z1Z1 = Z1^2 */
275 227 : fd_bn254_fp2_sqr( zz1, &p->Z );
276 : /* Z2Z2 = Z2^2 */
277 227 : fd_bn254_fp2_sqr( zz2, &q->Z );
278 : /* U1 = X1*Z2Z2 */
279 227 : fd_bn254_fp2_mul( u1, &p->X, zz2 );
280 : /* U2 = X2*Z1Z1 */
281 227 : fd_bn254_fp2_mul( u2, &q->X, zz1 );
282 : /* S1 = Y1*Z2*Z2Z2 */
283 227 : fd_bn254_fp2_mul( s1, &p->Y, &q->Z );
284 227 : fd_bn254_fp2_mul( s1, s1, zz2 );
285 : /* S2 = Y2*Z1*Z1Z1 */
286 227 : fd_bn254_fp2_mul( s2, &q->Y, &p->Z );
287 227 : fd_bn254_fp2_mul( s2, s2, zz1 );
288 :
289 : /* if p==q, call fd_bn254_g2_dbl */
290 : // if( FD_UNLIKELY( fd_bn254_fp2_eq( u2, &p->X ) && fd_bn254_fp2_eq( s2, &p->Y ) ) ) {
291 : // return fd_bn254_g2_dbl( r, p );
292 : // }
293 :
294 : /* H = U2-U1 */
295 227 : fd_bn254_fp2_sub( h, u2, u1 );
296 : /* HH = (2*H)^2 */
297 227 : fd_bn254_fp2_add( i, h, h );
298 227 : fd_bn254_fp2_sqr( i, i );
299 : /* J = H*I */
300 227 : fd_bn254_fp2_mul( j, h, i );
301 : /* r = 2*(S2-S1) */
302 227 : fd_bn254_fp2_sub( rr, s2, s1 );
303 227 : fd_bn254_fp2_add( rr, rr, rr );
304 : /* V = U1*I */
305 227 : fd_bn254_fp2_mul( v, u1, i );
306 : /* X3 = r^2-J-2*V */
307 227 : fd_bn254_fp2_sqr( &r->X, rr );
308 227 : fd_bn254_fp2_sub( &r->X, &r->X, j );
309 227 : fd_bn254_fp2_sub( &r->X, &r->X, v );
310 227 : fd_bn254_fp2_sub( &r->X, &r->X, v );
311 : /* Y3 = r*(V-X3)-2*S1*J
312 : note: i no longer used */
313 227 : fd_bn254_fp2_mul( i, s1, j ); /* i = S1*J */
314 227 : fd_bn254_fp2_add( i, i, i ); /* i = 2*S1*J */
315 227 : fd_bn254_fp2_sub( &r->Y, v, &r->X );
316 227 : fd_bn254_fp2_mul( &r->Y, &r->Y, rr );
317 227 : fd_bn254_fp2_sub( &r->Y, &r->Y, i );
318 : /* Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H */
319 227 : fd_bn254_fp2_add( &r->Z, &p->Z, &q->Z );
320 227 : fd_bn254_fp2_sqr( &r->Z, &r->Z );
321 227 : fd_bn254_fp2_sub( &r->Z, &r->Z, zz1 );
322 227 : fd_bn254_fp2_sub( &r->Z, &r->Z, zz2 );
323 227 : fd_bn254_fp2_mul( &r->Z, &r->Z, h );
324 227 : return r;
325 227 : }
326 :
327 : /* fd_bn254_g2_affine_add computes r = p + q.
328 : Both p, q are affine, i.e. Z==1. */
329 : static fd_bn254_g2_t *
330 : fd_bn254_g2_affine_add( fd_bn254_g2_t * r,
331 : fd_bn254_g2_t const * p,
332 114 : fd_bn254_g2_t const * q ) {
333 : /* p==0, return q */
334 114 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
335 0 : return fd_bn254_g2_set( r, q );
336 0 : }
337 : /* q==0, return p */
338 114 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( q ) ) ) {
339 0 : return fd_bn254_g2_set( r, p );
340 0 : }
341 :
342 114 : fd_bn254_fp2_t lambda[1], x[1], y[1];
343 :
344 : /* same X, either the points are equal or opposite */
345 114 : if( fd_bn254_fp2_eq( &p->X, &q->X ) ) {
346 0 : if( fd_bn254_fp2_eq( &p->Y, &q->Y ) ) {
347 : /* p==q => point double: lambda = 3 * x1^2 / (2 * y1) */
348 0 : fd_bn254_fp2_sqr( x, &p->X ); /* x = x1^2 */
349 0 : fd_bn254_fp2_add( y, x, x ); /* y = 2 x1^2 */
350 0 : fd_bn254_fp2_add( x, x, y ); /* x = 3 x1^2 */
351 0 : fd_bn254_fp2_add( y, &p->Y, &p->Y );
352 0 : fd_bn254_fp2_inv( lambda, y );
353 0 : fd_bn254_fp2_mul( lambda, lambda, x );
354 0 : } else {
355 : /* p==-q => r=0 */
356 0 : return fd_bn254_g2_set_zero( r );
357 0 : }
358 114 : } else {
359 : /* point add: lambda = (y1 - y2) / (x1 - x2) */
360 114 : fd_bn254_fp2_sub( x, &p->X, &q->X );
361 114 : fd_bn254_fp2_sub( y, &p->Y, &q->Y );
362 114 : fd_bn254_fp2_inv( lambda, x );
363 114 : fd_bn254_fp2_mul( lambda, lambda, y );
364 114 : }
365 :
366 : /* x3 = lambda^2 - x1 - x2 */
367 114 : fd_bn254_fp2_sqr( x, lambda );
368 114 : fd_bn254_fp2_sub( x, x, &p->X );
369 114 : fd_bn254_fp2_sub( x, x, &q->X );
370 :
371 : /* y3 = lambda * (x1 - x3) - y1 */
372 114 : fd_bn254_fp2_sub( y, &p->X, x );
373 114 : fd_bn254_fp2_mul( y, y, lambda );
374 114 : fd_bn254_fp2_sub( y, y, &p->Y );
375 :
376 114 : fd_bn254_fp2_set( &r->X, x );
377 114 : fd_bn254_fp2_set( &r->Y, y );
378 114 : fd_bn254_fp2_set_one( &r->Z );
379 114 : return r;
380 114 : }
381 :
382 : /* fd_bn254_g2_scalar_mul computes r = [s]P.
383 : p must be in affine form (p->Z == 1).
384 : The result is in projective coordinates over Fp2. */
385 : fd_bn254_g2_t *
386 : fd_bn254_g2_scalar_mul( fd_bn254_g2_t * r,
387 : fd_bn254_g2_t const * p,
388 127 : fd_bn254_scalar_t const * s ) {
389 127 : if( FD_UNLIKELY( fd_uint256_is_zero( s ) || fd_bn254_g2_is_zero( p ) ) ) {
390 13 : return fd_bn254_g2_set_zero( r );
391 13 : }
392 :
393 114 : const ulong g1_const[ 3 ] = { 0x7a7bd9d4391eb18eUL, 0x4ccef014a773d2cfUL, 0x0000000000000002UL };
394 114 : ulong b1[ 3 ], b2[ 2 ];
395 114 : fd_bn254_glv_sxg3( b1, s, g1_const );
396 114 : fd_bn254_glv_sxg2( b2, s, g2_const );
397 :
398 : /* k1 = s - b1*N_C - b2*N_B (may be negative for G2) */
399 114 : fd_uint256_t k1_abs[1];
400 114 : int k1_neg = 0;
401 114 : {
402 114 : ulong p_nc[ 4 ];
403 : /* b2*nb will produce at most 3 limbs, so we want the 4th zeroed for the addition. */
404 114 : ulong p_nb[ 4 ] = {0};
405 114 : ulong t[ 4 ];
406 114 : fd_bn254_glv_mul3x2( p_nc, b1, nc );
407 114 : fd_bn254_glv_mul2x1( p_nb, b2, nb );
408 114 : fd_bn254_glv_add4( t, p_nc, p_nb );
409 114 : ulong borrow = fd_bn254_glv_sub4( k1_abs->limbs, s->limbs, t );
410 114 : if( borrow ) {
411 0 : k1_neg = 1;
412 0 : fd_bn254_glv_negate4( k1_abs->limbs );
413 0 : }
414 114 : }
415 :
416 : /* k2 = b2*N_A - b1*N_B (usually negative for G2) */
417 114 : fd_uint256_t k2_abs[1];
418 114 : int k2_neg = 0;
419 114 : {
420 114 : ulong pos[ 4 ], neg[ 4 ];
421 114 : fd_bn254_glv_mul2x2( pos, b2, na );
422 114 : fd_bn254_glv_mul3x1( neg, b1, nb );
423 114 : ulong borrow = fd_bn254_glv_sub4( k2_abs->limbs, pos, neg );
424 114 : if( borrow ) {
425 0 : k2_neg = 1;
426 0 : fd_bn254_glv_negate4( k2_abs->limbs );
427 0 : }
428 114 : }
429 :
430 : /* pt1 = P, pt2 = phi(P) = (beta * P.x, P.y).
431 : If k1 < 0, negate pt1. If k2 < 0, negate pt2. */
432 114 : fd_bn254_g2_t pt1[1], pt2[1];
433 114 : fd_bn254_g2_set( pt1, p );
434 114 : fd_bn254_fp_mul( &pt2->X.el[0], &p->X.el[0], fd_bn254_const_beta_mont );
435 114 : fd_bn254_fp_mul( &pt2->X.el[1], &p->X.el[1], fd_bn254_const_beta_mont );
436 114 : fd_bn254_fp2_set( &pt2->Y, &p->Y );
437 114 : fd_bn254_fp2_set_one( &pt2->Z );
438 114 : if( k1_neg ) {
439 0 : fd_bn254_fp2_neg( &pt1->Y, &pt1->Y );
440 0 : }
441 114 : if( k2_neg ) {
442 0 : fd_bn254_fp2_neg( &pt2->Y, &pt2->Y );
443 0 : }
444 :
445 114 : fd_bn254_g2_t pt12[1];
446 114 : fd_bn254_g2_affine_add( pt12, pt1, pt2 );
447 :
448 : /* Shamir's trick: simultaneous double-and-add on k1, k2. */
449 114 : int i = 255;
450 22116 : for( ; i>=0; i-- ) {
451 22116 : int k1b = !!fd_uint256_bit( k1_abs, i );
452 22116 : int k2b = !!fd_uint256_bit( k2_abs, i );
453 22116 : if( k1b || k2b ) {
454 114 : fd_bn254_g2_set( r, ( k1b && k2b ) ? pt12 : ( k1b ? pt1 : pt2 ) );
455 114 : break;
456 114 : }
457 22116 : }
458 114 : if( FD_UNLIKELY( i<0 ) ) {
459 0 : return fd_bn254_g2_set_zero( r );
460 0 : }
461 7120 : for( i--; i >= 0; i-- ) {
462 7006 : fd_bn254_g2_dbl( r, r );
463 7006 : int k1b = !!fd_uint256_bit( k1_abs, i );
464 7006 : int k2b = !!fd_uint256_bit( k2_abs, i );
465 7006 : if( k1b && k2b ) {
466 0 : fd_bn254_g2_add_mixed( r, r, pt12 );
467 7006 : } else if( k1b ) {
468 3058 : fd_bn254_g2_add_mixed( r, r, pt1 );
469 3948 : } else if( k2b ) {
470 0 : fd_bn254_g2_add_mixed( r, r, pt2 );
471 0 : }
472 7006 : }
473 :
474 114 : return r;
475 114 : }
476 :
477 : /* fd_bn254_g2_frombytes_internal extracts (x, y) and performs basic checks.
478 : This is used by fd_bn254_g2_compress() and fd_bn254_g2_frombytes_check_subgroup(). */
479 : static inline fd_bn254_g2_t *
480 : fd_bn254_g2_frombytes_internal( fd_bn254_g2_t * p,
481 : uchar const in[128],
482 617 : int big_endian ) {
483 : /* Special case: all zeros => point at infinity */
484 617 : const uchar zero[128] = { 0 };
485 617 : if( FD_UNLIKELY( fd_memeq( in, zero, 128 ) ) ) {
486 2 : return fd_bn254_g2_set_zero( p );
487 2 : }
488 :
489 : /* Check x < p */
490 615 : if( FD_UNLIKELY( !fd_bn254_fp2_frombytes_nm( &p->X, &in[0], big_endian, NULL, NULL ) ) ) {
491 294 : return NULL;
492 294 : }
493 :
494 : /* Check flags and y < p */
495 321 : int is_inf, is_neg;
496 321 : if( FD_UNLIKELY( !fd_bn254_fp2_frombytes_nm( &p->Y, &in[64], big_endian, &is_inf, &is_neg ) ) ) {
497 152 : return NULL;
498 152 : }
499 :
500 169 : if( FD_UNLIKELY( is_inf ) ) {
501 24 : return fd_bn254_g2_set_zero( p );
502 24 : }
503 :
504 145 : fd_bn254_fp2_set_one( &p->Z );
505 145 : return p;
506 169 : }
507 :
508 : /* fd_bn254_g2_frombytes_check_eq_only performs frombytes, checks the curve
509 : equation, but does NOT check subgroup membership. */
510 : static inline fd_bn254_g2_t *
511 : fd_bn254_g2_frombytes_check_eq_only( fd_bn254_g2_t * p,
512 : uchar const in[128],
513 355 : int big_endian ) {
514 355 : if( FD_UNLIKELY( !fd_bn254_g2_frombytes_internal( p, in, big_endian ) ) ) {
515 222 : return NULL;
516 222 : }
517 133 : if( FD_UNLIKELY( fd_bn254_g2_is_zero( p ) ) ) {
518 13 : return p;
519 13 : }
520 :
521 120 : fd_bn254_fp2_to_mont( &p->X, &p->X );
522 120 : fd_bn254_fp2_to_mont( &p->Y, &p->Y );
523 120 : fd_bn254_fp2_set_one( &p->Z );
524 :
525 : /* Check that y^2 = x^3 + b */
526 120 : fd_bn254_fp2_t y2[1], x3b[1];
527 120 : fd_bn254_fp2_sqr( y2, &p->Y );
528 120 : fd_bn254_fp2_sqr( x3b, &p->X );
529 120 : fd_bn254_fp2_mul( x3b, x3b, &p->X );
530 120 : fd_bn254_fp2_add( x3b, x3b, fd_bn254_const_twist_b_mont );
531 120 : if( FD_UNLIKELY( !fd_bn254_fp2_eq( y2, x3b ) ) ) {
532 8 : return NULL;
533 8 : }
534 112 : return p;
535 120 : }
536 :
537 : /* fd_bn254_g2_frombytes_check_subgroup performs frombytes AND checks subgroup membership. */
538 : static inline fd_bn254_g2_t *
539 : fd_bn254_g2_frombytes_check_subgroup( fd_bn254_g2_t * p,
540 : uchar const in[128],
541 355 : int big_endian ) {
542 355 : if( FD_UNLIKELY( fd_bn254_g2_frombytes_check_eq_only( p, in, big_endian )==NULL ) ) {
543 230 : return NULL;
544 230 : }
545 :
546 : /* G2 does NOT have prime order, so we have to check group membership. */
547 :
548 : /* We use the fast subgroup membership check, that requires a single 64-bit scalar mul.
549 : https://eprint.iacr.org/2022/348, Sec 3.1.
550 : [r]P == 0 <==> [x+1]P + ψ([x]P) + ψ²([x]P) = ψ³([2x]P)
551 : See also: https://github.com/Consensys/gnark-crypto/blob/v0.12.1/ecc/bn254/g2.go#L404
552 :
553 : For reference, the followings also work:
554 :
555 : 1) very slow: 256-bit scalar mul
556 :
557 : fd_bn254_g2_t r[1];
558 : fd_bn254_g2_scalar_mul( r, p, fd_bn254_const_r );
559 : if( !fd_bn254_g2_is_zero( r ) ) return NULL;
560 :
561 : 2) slow: 128-bit scalar mul
562 :
563 : fd_bn254_g2_t a[1], b[1];
564 : const fd_bn254_scalar_t six_x_sqr[1] = {{{ 0xf83e9682e87cfd46, 0x6f4d8248eeb859fb, 0x0, 0x0, }}};
565 : fd_bn254_g2_scalar_mul( a, p, six_x_sqr );
566 : fd_bn254_g2_frob( b, p );
567 : if( !fd_bn254_g2_eq( a, b ) ) return NULL; */
568 :
569 125 : fd_bn254_g2_t xp[1], l[1], psi[1], r[1];
570 125 : fd_bn254_g2_scalar_mul( xp, p, fd_bn254_const_x ); /* 64-bit */
571 125 : fd_bn254_g2_add_mixed( l, xp, p );
572 :
573 125 : fd_bn254_g2_frob( psi, xp );
574 125 : fd_bn254_g2_add( l, l, psi );
575 :
576 125 : fd_bn254_g2_frob2( psi, xp ); /* faster than frob( psi, psi ) */
577 125 : fd_bn254_g2_add( l, l, psi );
578 :
579 125 : fd_bn254_g2_frob( psi, psi );
580 125 : fd_bn254_g2_dbl( r, psi );
581 125 : if( FD_UNLIKELY( !fd_bn254_g2_eq( l, r ) ) ) {
582 8 : return NULL;
583 8 : }
584 117 : return p;
585 125 : }
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