Line data Source code
1 : /* Copyright (C) 2000-2005 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 : #include "pari.h"
15 : #include "paripriv.h"
16 : /*******************************************************************/
17 : /* */
18 : /* QUADRATIC POLYNOMIAL ASSOCIATED TO A DISCRIMINANT */
19 : /* */
20 : /*******************************************************************/
21 :
22 : void
23 151560 : check_quaddisc(GEN x, long *s, long *pr, const char *f)
24 : {
25 : long r;
26 151560 : if (typ(x) != t_INT) pari_err_TYPE(f,x);
27 151546 : *s = signe(x);
28 151546 : if (Z_issquare(x)) pari_err_DOMAIN(f,"issquare(disc)","=", gen_1,x);
29 151547 : r = mod4(x); if (*s < 0 && r) r = 4 - r;
30 151547 : if (r > 1) pari_err_DOMAIN(f,"disc % 4",">", gen_1,x);
31 151533 : if (pr) *pr = r;
32 151533 : }
33 : void
34 6916 : check_quaddisc_real(GEN x, long *r, const char *f)
35 : {
36 6916 : long sx; check_quaddisc(x, &sx, r, f);
37 6916 : if (sx < 0) pari_err_DOMAIN(f, "disc","<",gen_0,x);
38 6916 : }
39 : void
40 2177 : check_quaddisc_imag(GEN x, long *r, const char *f)
41 : {
42 2177 : long sx; check_quaddisc(x, &sx, r, f);
43 2170 : if (sx > 0) pari_err_DOMAIN(f, "disc",">",gen_0,x);
44 2170 : }
45 :
46 : /* X^2 + b X + c is the canonical quadratic t_POL of discriminant D.
47 : * Dodd is nonzero iff D is odd */
48 : static void
49 974299 : quadpoly_bc(GEN D, long Dodd, GEN *b, GEN *c)
50 : {
51 974299 : if (Dodd)
52 : {
53 875935 : pari_sp av = avma;
54 875935 : *b = gen_m1;
55 875935 : *c = gc_INT(av, shifti(subui(1,D), -2));
56 : }
57 : else
58 : {
59 98364 : *b = gen_0;
60 98364 : *c = shifti(D,-2); togglesign(*c);
61 : }
62 974299 : }
63 : /* X^2 - X - (D-1)/4 or X^2 - D/4 */
64 : static GEN
65 243341 : quadpoly_ii(GEN D, long Dmod4)
66 : {
67 243341 : GEN b, c, y = cgetg(5,t_POL);
68 243341 : y[1] = evalsigne(1) | evalvarn(0);
69 243341 : quadpoly_bc(D, Dmod4, &b,&c);
70 243341 : gel(y,2) = c;
71 243341 : gel(y,3) = b;
72 243341 : gel(y,4) = gen_1; return y;
73 : }
74 : GEN
75 2044 : quadpoly(GEN D)
76 : {
77 : long s, Dmod4;
78 2044 : check_quaddisc(D, &s, &Dmod4, "quadpoly");
79 2037 : return quadpoly_ii(D, Dmod4);
80 : }
81 : GEN /* no checks */
82 241304 : quadpoly_i(GEN D) { return quadpoly_ii(D, Mod4(D)); }
83 :
84 : GEN
85 1029 : quadpoly0(GEN x, long v)
86 : {
87 1029 : GEN T = quadpoly(x);
88 1022 : if (v > 0) setvarn(T, v);
89 1022 : return T;
90 : }
91 :
92 : GEN
93 0 : quadgen(GEN x)
94 0 : { retmkquad(quadpoly(x), gen_0, gen_1); }
95 :
96 : GEN
97 574 : quadgen0(GEN x, long v)
98 : {
99 574 : if (v==-1) v = fetch_user_var("w");
100 574 : retmkquad(quadpoly0(x, v), gen_0, gen_1);
101 : }
102 :
103 : /***********************************************************************/
104 : /** **/
105 : /** BINARY QUADRATIC FORMS **/
106 : /** **/
107 : /***********************************************************************/
108 : static int
109 814212 : is_qfi(GEN q) { return typ(q)==t_QFB && qfb_is_qfi(q); }
110 :
111 : static GEN
112 1826004 : check_qfbext(const char *fun, GEN x)
113 : {
114 1826004 : long t = typ(x);
115 1826004 : if (t == t_QFB) return x;
116 196 : if (t == t_VEC && lg(x)==3)
117 : {
118 196 : GEN q = gel(x,1);
119 196 : if (!is_qfi(q) && typ(gel(x,2))==t_REAL) return q;
120 : }
121 0 : pari_err_TYPE(fun, x);
122 : return NULL;/* LCOV_EXCL_LINE */
123 : }
124 :
125 : static GEN
126 139081 : qfb3(GEN x, GEN y, GEN z)
127 139081 : { retmkqfb(icopy(x), icopy(y), icopy(z), qfb_disc3(x,y,z)); }
128 :
129 : static int
130 23783631 : qfb_equal(GEN x, GEN y)
131 : {
132 23783631 : return equalii(gel(x,1),gel(y,1))
133 1592910 : && equalii(gel(x,2),gel(y,2))
134 25376541 : && equalii(gel(x,3),gel(y,3));
135 : }
136 :
137 : /* valid for t_QFB, qfr3, qfr5; shallow */
138 : static GEN
139 888164 : qfb_inv(GEN x) {
140 888164 : GEN z = shallowcopy(x);
141 888160 : gel(z,2) = negi(gel(z,2));
142 888160 : return z;
143 : }
144 : /* valid for t_QFB, GC clean */
145 : static GEN
146 7 : qfbinv(GEN x)
147 7 : { retmkqfb(icopy(gel(x,1)),negi(gel(x,2)),icopy(gel(x,3)), icopy(gel(x,4))); }
148 :
149 : GEN
150 77231 : Qfb0(GEN a, GEN b, GEN c)
151 : {
152 : GEN q, D;
153 77231 : if (!b)
154 : {
155 49 : if (c) pari_err_TYPE("Qfb",c);
156 42 : if (typ(a) == t_VEC && lg(a) == 4)
157 21 : { b = gel(a,2); c = gel(a,3); a = gel(a,1); }
158 21 : else if (typ(a) == t_POL && degpol(a) == 2)
159 7 : { b = gel(a,3); c = gel(a,2); a = gel(a,4); }
160 14 : else if (typ(a) == t_MAT && lg(a)==3 && lgcols(a)==3)
161 : {
162 7 : b = gadd(gcoeff(a,2,1), gcoeff(a,1,2));
163 7 : c = gcoeff(a,2,2); a = gcoeff(a,1,1);
164 : }
165 : else
166 7 : pari_err_TYPE("Qfb",a);
167 : }
168 77182 : else if (!c)
169 7 : pari_err_TYPE("Qfb",b);
170 77210 : if (typ(a)!=t_INT) pari_err_TYPE("Qfb",a);
171 77203 : if (typ(b)!=t_INT) pari_err_TYPE("Qfb",b);
172 77203 : if (typ(c)!=t_INT) pari_err_TYPE("Qfb",c);
173 77203 : q = qfb3(a, b, c); D = qfb_disc(q);
174 77203 : if (signe(D) < 0)
175 42392 : { if (signe(a) < 0) pari_err_IMPL("negative definite t_QFB"); }
176 34811 : else if (Z_issquare(D)) pari_err_DOMAIN("Qfb","issquare(disc)","=", gen_1,q);
177 77196 : return q;
178 : }
179 :
180 : /***********************************************************************/
181 : /** **/
182 : /** Reduction **/
183 : /** **/
184 : /***********************************************************************/
185 :
186 : /* assume a > 0. Write b = q*2a + r, with -a < r <= a */
187 : static GEN
188 16740811 : dvmdii_round(GEN b, GEN a, GEN *r)
189 : {
190 16740811 : GEN a2 = shifti(a, 1), q = dvmdii(b, a2, r);
191 16740597 : if (signe(b) >= 0) {
192 9198917 : if (abscmpii(*r, a) > 0) { q = addiu(q, 1); *r = subii(*r, a2); }
193 : } else { /* r <= 0 */
194 7541680 : if (abscmpii(*r, a) >= 0){ q = subiu(q, 1); *r = addii(*r, a2); }
195 : }
196 16740250 : return q;
197 : }
198 : /* Assume 0 < a <= LONG_MAX. Ensure no overflow */
199 : static long
200 111307704 : dvmdsu_round(long b, ulong a, long *r)
201 : {
202 111307704 : ulong a2 = a << 1, q, ub, ur;
203 111307704 : if (b >= 0) {
204 70952193 : ub = b;
205 70952193 : q = ub / a2;
206 70952193 : ur = ub % a2;
207 70952193 : if (ur > a) { ur -= a; q++; *r = (long)ur; *r -= (long)a; }
208 25971335 : else *r = (long)ur;
209 70952193 : return (long)q;
210 : } else { /* r <= 0 */
211 40355511 : ub = (ulong)-b; /* |b| */
212 40355511 : q = ub / a2;
213 40355511 : ur = ub % a2;
214 40355511 : if (ur >= a) { ur -= a; q++; *r = (long)ur; *r = (long)a - *r; }
215 21513212 : else *r = -(long)ur;
216 40355511 : return -(long)q;
217 : }
218 : }
219 : /* reduce b mod 2*a. Update b,c */
220 : static void
221 2783988 : REDB(GEN a, GEN *b, GEN *c)
222 : {
223 2783988 : GEN r, q = dvmdii_round(*b, a, &r);
224 2783988 : if (!signe(q)) return;
225 2714529 : *c = subii(*c, mulii(q, shifti(addii(*b, r),-1)));
226 2714529 : *b = r;
227 : }
228 : /* Assume a > 0. Reduce b mod 2*a. Update b,c */
229 : static void
230 111295458 : sREDB(ulong a, long *b, ulong *c)
231 : {
232 : long r, q;
233 : ulong uz;
234 120825959 : if (a > LONG_MAX) return; /* b already reduced */
235 111295458 : q = dvmdsu_round(*b, a, &r);
236 111319359 : if (q == 0) return;
237 : /* Final (a,r,c2) satisfies |r| <= |b| hence c2 <= c, c2 = c - q*z,
238 : * where z = (b+r) / 2, representable as long, has the same sign as q. */
239 101788858 : if (*b < 0)
240 : { /* uz = -z >= 0, q < 0 */
241 35550596 : if (r >= 0) /* different signs=>no overflow, exact division */
242 18897352 : uz = (ulong)-((*b + r)>>1);
243 : else
244 : {
245 16653244 : ulong ub = (ulong)-*b, ur = (ulong)-r;
246 16653244 : uz = (ub + ur) >> 1;
247 : }
248 35550596 : *c -= (-q) * uz; /* c -= qz */
249 : }
250 : else
251 : { /* uz = z >= 0, q > 0 */
252 66238262 : if (r <= 0)
253 45039033 : uz = (*b + r)>>1;
254 : else
255 : {
256 21199229 : ulong ub = (ulong)*b, ur = (ulong)r;
257 21199229 : uz = ((ub + ur) >> 1);
258 : }
259 66238262 : *c -= q * uz; /* c -= qz */
260 : }
261 101788858 : *b = r;
262 : }
263 : static void
264 13956821 : REDBU(GEN a, GEN *b, GEN *c, GEN u1, GEN *u2)
265 : { /* REDB(a,b,c) */
266 13956821 : GEN r, q = dvmdii_round(*b, a, &r);
267 13956254 : *c = subii(*c, mulii(q, shifti(addii(*b, r),-1)));
268 13956091 : *b = r;
269 13956091 : *u2 = subii(*u2, mulii(q, u1));
270 13956111 : }
271 :
272 : /* q t_QFB, return reduced representative and set base change U in Sl2(Z) */
273 : static GEN
274 6634383 : qfbredsl2_imag_basecase(GEN q, GEN *U)
275 : {
276 6634383 : pari_sp av = avma;
277 : GEN z, u1,u2,v1,v2,Q;
278 6634383 : GEN a = gel(q,1), b = gel(q,2), c = gel(q,3);
279 : long cmp;
280 6634383 : u1 = gen_1; u2 = gen_0;
281 6634383 : cmp = abscmpii(a, b);
282 6634391 : if (cmp < 0)
283 2110639 : REDBU(a,&b,&c, u1,&u2);
284 4523752 : else if (cmp == 0 && signe(b) < 0)
285 : { /* b = -a */
286 11781 : b = negi(b);
287 11781 : u2 = gen_1;
288 : }
289 : for(;;)
290 : {
291 18479997 : cmp = abscmpii(a, c); if (cmp <= 0) break;
292 11845244 : swap(a,c); b = negi(b);
293 11846172 : z = u1; u1 = u2; u2 = negi(z);
294 11846184 : REDBU(a,&b,&c, u1,&u2);
295 11845614 : if (gc_needed(av, 1)) {
296 7 : if (DEBUGMEM>1) pari_warn(warnmem, "qfbredsl2");
297 7 : (void)gc_all(av, 5, &a,&b,&c, &u1,&u2);
298 : }
299 : }
300 6634363 : if (cmp == 0 && signe(b) < 0)
301 : {
302 17679 : b = negi(b);
303 17679 : z = u1; u1 = u2; u2 = negi(z);
304 : }
305 : /* Let q = (A,B,C). q o [u1,u2; v1,v2] = Q implies
306 : * [v1,v2] = (1/C) [(b-B)/2 u1 - a u2, c u1 - (b+B)/2 u2] */
307 6634363 : z = shifti(subii(b, gel(q,2)), -1);
308 6634362 : v1 = subii(mulii(z, u1), mulii(a, u2)); v1 = diviiexact(v1, gel(q,3));
309 6634381 : z = subii(z, b);
310 6634383 : v2 = addii(mulii(z, u2), mulii(c, u1)); v2 = diviiexact(v2, gel(q,3));
311 6634363 : *U = mkmat2(mkcol2(u1,v1), mkcol2(u2,v2));
312 6634418 : Q = mkqfb(a,b,c,gel(q,4));
313 6634422 : return gc_all(av, 2, &Q, U);
314 : }
315 :
316 : static GEN
317 1058098 : setq_b0(ulong a, ulong c, GEN D)
318 1058098 : { retmkqfb(utoipos(a), gen_0, utoipos(c), icopy(D)); }
319 : /* assume |sb| = 1 */
320 : static GEN
321 94617435 : setq(ulong a, ulong b, ulong c, long sb, GEN D)
322 94617435 : { retmkqfb(utoipos(a), sb==1? utoipos(b): utoineg(b), utoipos(c), icopy(D)); }
323 : /* 0 < a, c < 2^BIL, b = 0 */
324 : static GEN
325 946027 : qfbred_imag_1_b0(ulong a, ulong c, GEN D)
326 946027 : { return (a <= c)? setq_b0(a, c, D): setq_b0(c, a, D); }
327 :
328 : /* 0 < a, c < 2^BIL: single word affair */
329 : static GEN
330 95767427 : qfbred_imag_1(pari_sp av, GEN a, GEN b, GEN c, GEN D)
331 : {
332 : ulong ua, ub, uc;
333 : long sb;
334 : for(;;)
335 144743 : { /* at most twice */
336 95767427 : long lb = lgefint(b); /* <= 3 after first loop */
337 95767427 : if (lb == 2) return qfbred_imag_1_b0(a[2],c[2], D);
338 94821400 : if (lb == 3 && uel(b,2) <= (ulong)LONG_MAX) break;
339 140511 : REDB(a,&b,&c);
340 140504 : if (uel(a,2) <= uel(c,2))
341 : { /* lg(b) <= 3 but may be too large for itos */
342 0 : long s = signe(b);
343 0 : set_avma(av);
344 0 : if (!s) return qfbred_imag_1_b0(a[2], c[2], D);
345 0 : if (a[2] == c[2]) s = 1;
346 0 : return setq(a[2], b[2], c[2], s, D);
347 : }
348 140504 : swap(a,c); b = negi(b);
349 : }
350 : /* b != 0 */
351 94680889 : set_avma(av);
352 94680319 : ua = a[2];
353 94680319 : ub = sb = b[2]; if (signe(b) < 0) sb = -sb;
354 94680319 : uc = c[2];
355 94680319 : if (ua < ub)
356 35744054 : sREDB(ua, &sb, &uc);
357 58936265 : else if (ua == ub && sb < 0) sb = (long)ub;
358 170305227 : while(ua > uc)
359 : {
360 75579648 : lswap(ua,uc); sb = -sb;
361 75579648 : sREDB(ua, &sb, &uc);
362 : }
363 94725579 : if (!sb) return setq_b0(ua, uc, D);
364 : else
365 : {
366 94613506 : long s = 1;
367 94613506 : if (sb < 0)
368 : {
369 35674492 : sb = -sb;
370 35674492 : if (ua != uc) s = -1;
371 : }
372 94613506 : return setq(ua, sb, uc, s, D);
373 : }
374 : }
375 :
376 : static GEN
377 7 : rhoimag(GEN x)
378 : {
379 7 : pari_sp av = avma;
380 7 : GEN a = gel(x,1), b = gel(x,2), c = gel(x,3);
381 7 : int fl = abscmpii(a, c);
382 7 : if (fl <= 0)
383 : {
384 7 : int fg = abscmpii(a, b);
385 7 : if (fg >= 0)
386 : {
387 7 : x = gcopy(x);
388 7 : if ((!fl || !fg) && signe(gel(x,2)) < 0) setsigne(gel(x,2), 1);
389 7 : return x;
390 : }
391 : }
392 0 : swap(a,c); b = negi(b);
393 0 : REDB(a, &b, &c);
394 0 : return gc_GEN(av, mkqfb(a,b,c, qfb_disc(x)));
395 : }
396 :
397 : /* qfr3 / qfr5 */
398 :
399 : /* t_QFB are unusable: D, sqrtD, isqrtD are recomputed all the time and the
400 : * logarithmic Shanks's distance is costly and hard to control.
401 : * qfr3 / qfr5 routines take a container of t_INTs (e.g a t_VEC) as argument,
402 : * at least 3 (resp. 5) components [it is a feature that they do not check the
403 : * precise type or length of the input]. They return a vector of length 3
404 : * (resp. 5). A qfr3 [a,b,c] contains the form coeffs, in a qfr5 [a,b,c, e,d]
405 : * the t_INT e is a binary exponent, d a t_REAL, coding the distance in
406 : * multiplicative form: the true distance is obtained from qfr5_dist.
407 : * All other qfr routines are obsolete (inefficient) wrappers */
408 :
409 : /* static functions are not stack-clean. Unless mentionned otherwise, public
410 : * functions are. */
411 :
412 : #define EMAX 22
413 : static void
414 10217928 : fix_expo(GEN x)
415 : {
416 10217928 : if (expo(gel(x,5)) >= (1L << EMAX)) {
417 0 : gel(x,4) = addiu(gel(x,4), 1);
418 0 : shiftr_inplace(gel(x,5), - (1L << EMAX));
419 : }
420 10217928 : }
421 :
422 : /* (1/2) log (|d| * 2^{e * 2^EMAX}). Not stack clean if e != 0 */
423 : GEN
424 184688 : qfr5_dist(GEN e, GEN d, long prec)
425 : {
426 184688 : GEN t = logr_abs(d);
427 184688 : if (signe(e)) {
428 0 : GEN u = mulir(e, mplog2(prec));
429 0 : shiftr_inplace(u, EMAX); t = addrr(t, u);
430 : }
431 184688 : shiftr_inplace(t, -1); return t;
432 : }
433 :
434 : static void
435 14152531 : rho_get_BC(GEN *B, GEN *C, GEN a, GEN b, GEN c, struct qfr_data *S)
436 : {
437 : GEN t, u, q;
438 14152531 : t = (abscmpii(S->isqrtD,c) >= 0)? S->isqrtD: absi_shallow(c);
439 14152531 : q = truedvmdii(addii(t, b), shifti(c,1), &u);
440 14152531 : *B = subii(t, u); /* |t| - ((|t|+b) % 2c) */
441 14152531 : *C = subii(a, mulii(q, subii(b, mulii(q,c))));
442 14152531 : }
443 : /* Not stack-clean */
444 : GEN
445 1139446 : qfr3_rho(GEN x, struct qfr_data *S)
446 : {
447 1139446 : GEN B, C, a = gel(x,1), b = gel(x,2), c = gel(x,3);
448 1139446 : rho_get_BC(&B, &C, a, b, c, S);
449 1139446 : return mkvec3(c, B, C);
450 : }
451 :
452 : /* Not stack-clean */
453 : GEN
454 8486681 : qfr5_rho(GEN x, struct qfr_data *S)
455 : {
456 8486681 : GEN B, C, a = gel(x,1), b = gel(x,2), c = gel(x,3), y;
457 8486681 : long sb = signe(b);
458 8486681 : rho_get_BC(&B, &C, a, b, c, S);
459 8486681 : y = mkvec5(c, B, C, gel(x,4), gel(x,5));
460 8486681 : if (sb) {
461 8482698 : GEN t = subii(sqri(b), S->D);
462 8482698 : if (sb < 0)
463 2509500 : t = divir(t, sqrr(subir(b,S->sqrtD)));
464 : else
465 5973198 : t = divri(sqrr(addir(b,S->sqrtD)), t);
466 : /* t = (b + sqrt(D)) / (b - sqrt(D)), evaluated stably */
467 8482698 : gel(y,5) = mulrr(t, gel(y,5)); fix_expo(y);
468 3983 : } else gel(y,5) = negr(gel(y,5));
469 8486681 : return y;
470 : }
471 :
472 : /* Not stack-clean */
473 : GEN
474 217728 : qfr_to_qfr5(GEN x, long prec)
475 217728 : { return mkvec5(gel(x,1),gel(x,2),gel(x,3),gen_0,real_1(prec)); }
476 :
477 : /* d0 = initial distance, x = [a,b,c, expo(d), d], d = exp(2*distance) */
478 : GEN
479 483 : qfr5_to_qfr(GEN x, GEN D, GEN d0)
480 : {
481 483 : if (d0)
482 : {
483 140 : GEN n = gel(x,4), d = absr(gel(x,5));
484 140 : if (signe(n))
485 : {
486 0 : n = addis(shifti(n, EMAX), expo(d));
487 0 : setexpo(d, 0); d = logr_abs(d);
488 0 : if (signe(n)) d = addrr(d, mulir(n, mplog2(lg(d0))));
489 0 : shiftr_inplace(d, -1);
490 0 : d0 = addrr(d0, d);
491 : }
492 140 : else if (!gequal1(d)) /* avoid loss of precision */
493 : {
494 91 : d = logr_abs(d);
495 91 : shiftr_inplace(d, -1);
496 91 : d0 = addrr(d0, d);
497 : }
498 : }
499 483 : x = qfr3_to_qfr(x, D);
500 483 : return d0 ? mkvec2(x,d0): x;
501 : }
502 :
503 : /* Not stack-clean */
504 : GEN
505 31920 : qfr3_to_qfr(GEN x, GEN d) { retmkqfb(gel(x,1), gel(x,2), gel(x,3), d); }
506 :
507 : static int
508 17712973 : ab_isreduced(GEN a, GEN b, GEN isqrtD)
509 : {
510 : GEN t;
511 17712973 : if (signe(b) <= 0 || abscmpii(b, isqrtD) > 0) return 0;
512 5271643 : t = addii_sign(isqrtD,1, shifti(a,1),-1); /* floor(sqrt(D)) - |2a| */
513 1099055 : return signe(t) < 0 ? abscmpii(b, t) >= 0
514 6370698 : : abscmpii(b, t) > 0;
515 : }
516 :
517 : /* Not stack-clean */
518 : GEN
519 1952895 : qfr5_red(GEN x, struct qfr_data *S)
520 : {
521 1952895 : pari_sp av = avma;
522 8465338 : while (!ab_isreduced(gel(x,1), gel(x,2), S->isqrtD))
523 : {
524 6512443 : x = qfr5_rho(x, S);
525 6512443 : if (gc_needed(av,2))
526 : {
527 0 : if (DEBUGMEM>1) pari_warn(warnmem,"qfr5_red");
528 0 : x = gc_GEN(av, x);
529 : }
530 : }
531 1952895 : return x;
532 : }
533 : /* Not stack-clean */
534 : GEN
535 1172847 : qfr3_red(GEN x, struct qfr_data *S)
536 : {
537 1172847 : pari_sp av = avma;
538 1172847 : GEN a = gel(x,1), b = gel(x,2), c = gel(x,3);
539 5699251 : while (!ab_isreduced(a, b, S->isqrtD))
540 : {
541 : GEN B, C;
542 4526404 : rho_get_BC(&B, &C, a, b, c, S);
543 4526404 : a = c; b = B; c = C;
544 4526404 : if (gc_needed(av,2))
545 : {
546 0 : if (DEBUGMEM>1) pari_warn(warnmem,"qfr3_red");
547 0 : (void)gc_all(av, 3, &a, &b, &c);
548 : }
549 : }
550 1172847 : return mkvec3(a, b, c);
551 : }
552 :
553 : void
554 2170 : qfr_data_init(GEN D, long prec, struct qfr_data *S)
555 : {
556 2170 : S->D = D;
557 2170 : S->sqrtD = sqrtr(itor(S->D,prec));
558 2170 : S->isqrtD = truncr(S->sqrtD);
559 2170 : }
560 :
561 : static GEN
562 140 : qfr5_init(GEN x, GEN d, struct qfr_data *S)
563 : {
564 140 : long prec = realprec(d), l = -expo(d);
565 140 : if (l < BITS_IN_LONG) l = BITS_IN_LONG;
566 140 : prec = maxss(prec, nbits2prec(l));
567 140 : S->D = qfb_disc(x);
568 140 : x = qfr_to_qfr5(x,prec);
569 140 : if (!S->sqrtD) S->sqrtD = sqrtr(itor(S->D,prec));
570 0 : else if (typ(S->sqrtD) != t_REAL) pari_err_TYPE("qfr_init",S->sqrtD);
571 :
572 140 : if (!S->isqrtD)
573 : {
574 126 : pari_sp av=avma;
575 : long e;
576 126 : S->isqrtD = gcvtoi(S->sqrtD,&e);
577 126 : if (e>-2) { set_avma(av); S->isqrtD = sqrti(S->D); }
578 : }
579 14 : else if (typ(S->isqrtD) != t_INT) pari_err_TYPE("qfr_init",S->isqrtD);
580 140 : return x;
581 : }
582 : static GEN
583 371 : qfr3_init(GEN x, struct qfr_data *S)
584 : {
585 371 : S->D = qfb_disc(x);
586 371 : if (!S->isqrtD) S->isqrtD = sqrti(S->D);
587 280 : else if (typ(S->isqrtD) != t_INT) pari_err_TYPE("qfr_init",S->isqrtD);
588 371 : return x;
589 : }
590 :
591 : #define qf_NOD 2
592 : #define qf_STEP 1
593 :
594 : static GEN
595 427 : qfbred_real_basecase_i(GEN x, long flag, GEN isqrtD, GEN sqrtD)
596 : {
597 : struct qfr_data S;
598 427 : GEN d = NULL, y;
599 427 : if (typ(x)==t_VEC) { d = gel(x,2); x = gel(x,1); } else flag |= qf_NOD;
600 427 : S.sqrtD = sqrtD;
601 427 : S.isqrtD = isqrtD;
602 427 : y = (flag & qf_NOD)? qfr3_init(x, &S): qfr5_init(x, d, &S);
603 427 : switch(flag) {
604 63 : case 0: y = qfr5_red(y,&S); break;
605 336 : case qf_NOD: y = qfr3_red(y,&S); break;
606 21 : case qf_STEP: y = qfr5_rho(y,&S); break;
607 7 : case qf_STEP|qf_NOD: y = qfr3_rho(y,&S); break;
608 0 : default: pari_err_FLAG("qfbred");
609 : }
610 427 : return qfr5_to_qfr(y, qfb_disc(x), d);
611 : }
612 :
613 : static void
614 13379357 : _rhorealsl2(GEN *pa, GEN *pb, GEN *pc, GEN *pu1, GEN *pu2, GEN *pv1,
615 : GEN *pv2, GEN rd)
616 : {
617 13379357 : GEN C = mpabs_shallow(*pc), t = addii(*pb, gmax_shallow(rd,C));
618 13379357 : GEN r, q = truedvmdii(t, shifti(C,1), &r);
619 13379357 : GEN a = *pa, b= *pb, c = *pc;
620 13379357 : if (signe(c) < 0) togglesign(q);
621 13379357 : *pa = *pc;
622 13379357 : *pb = subii(t, addii(r, *pb));
623 13379357 : *pc = subii(a,mulii(q,subii(b, mulii(q,c))));
624 13379357 : r = *pu1; *pu1 = *pv1; *pv1 = subii(mulii(q, *pv1), r);
625 13379357 : r = *pu2; *pu2 = *pv2; *pv2 = subii(mulii(q, *pv2), r);
626 13379357 : }
627 :
628 : static GEN
629 10810674 : rhorealsl2(GEN A, GEN rd)
630 : {
631 10810674 : GEN V = gel(A,1), M = gel(A,2);
632 10810674 : GEN a = gel(V,1), b = gel(V,2), c = gel(V,3), d = qfb_disc(V);
633 10810674 : GEN u1 = gcoeff(M,1,1), v1 = gcoeff(M,1,2);
634 10810674 : GEN u2 = gcoeff(M,2,1), v2 = gcoeff(M,2,2);
635 10810674 : _rhorealsl2(&a,&b,&c, &u1,&u2,&v1,&v2, rd);
636 10810674 : return mkvec2(mkqfb(a,b,c,d), mkmat22(u1,v1,u2,v2));
637 : }
638 :
639 : static GEN
640 979701 : qfbredsl2_real_basecase(GEN V, GEN rd)
641 : {
642 979701 : pari_sp av = avma;
643 979701 : GEN u1 = gen_1, u2 = gen_0, v1 = gen_0, v2 = gen_1;
644 979701 : GEN a = gel(V,1), b = gel(V,2), c = gel(V,3), d = qfb_disc(V);
645 3548384 : while (!ab_isreduced(a,b,rd))
646 : {
647 2568683 : _rhorealsl2(&a,&b,&c, &u1,&u2,&v1,&v2, rd);
648 2568683 : if (gc_needed(av, 1))
649 : {
650 0 : if (DEBUGMEM>1) pari_warn(warnmem,"qfbredsl2");
651 0 : (void)gc_all(av, 7, &a,&b,&c,&u1,&u2,&v1,&v2);
652 : }
653 : }
654 979701 : return gc_GEN(av, mkvec2(mkqfb(a,b,c,d), mkmat22(u1,v1,u2,v2)));
655 : }
656 :
657 : /* fast reduction of qfb with positive coefficients, based on
658 : Arnold Schoenhage, Fast reduction and composition of binary quadratic forms,
659 : Proc. of Intern. Symp. on Symbolic and Algebraic Computation (Bonn) (S. M.
660 : Watt, ed.), ACM Press, 1991, pp. 128-133.
661 : <https://dl.acm.org/doi/pdf/10.1145/120694.120711>
662 : With thanks to Keegan Ryan
663 : BA20230927
664 : */
665 :
666 : #define QFBRED_LIMIT 9000
667 :
668 : /* pqfb: qf with positive coefficients */
669 :
670 : static int
671 4940497 : lti2n(GEN a, long m)
672 : {
673 4940497 : return signe(a) < 0 || expi(a) < m ? 1 : 0;
674 : }
675 :
676 : static GEN
677 1921185 : pqfbred_1(GEN Q, long m, GEN U)
678 : {
679 1921185 : GEN a = gel(Q,1), b = gel(Q,2), c = gel(Q,3), d = gel(Q,4);
680 1921185 : if (abscmpii(a, c) < 0)
681 : {
682 : GEN t, at, r;
683 960317 : GEN r2 = addii(shifti(a, m + 2), d);
684 960317 : long e2 = expi(r2);
685 960317 : r = int2n(signe(r2) < 0 || e2 < 2*m+2 ? m+1 : e2>>1);
686 960317 : t = truedivii(subii(b, r), shifti(a,1));
687 960317 : if (signe(t)==0) pari_err_BUG("pqfbred_1");
688 960317 : at = mulii(a,t);
689 960317 : c = addii(subii(c, mulii(b, t)), mulii(at, t));
690 960317 : b = subii(b, shifti(at,1));
691 960317 : gcoeff(U,1,2) = subii( gcoeff(U,1,2), mulii(gcoeff(U,1,1), t));
692 960317 : gcoeff(U,2,2) = subii( gcoeff(U,2,2), mulii(gcoeff(U,2,1), t));
693 : } else
694 : {
695 : GEN t, ct, r;
696 960868 : GEN r2 = addii(shifti(c, m + 2), d);
697 960868 : long e2 = expi(r2);
698 960868 : r = int2n(signe(r2) < 0 || e2 < 2*m+2 ? m+1 : e2>>1);
699 960868 : t = truedivii(subii(b, r), shifti(c,1));
700 960868 : if (signe(t)==0) pari_err_BUG("pqfbred_1");
701 960868 : ct = mulii(c, t);
702 960868 : a = addii(subii(a, mulii(b, t)), mulii(ct, t));
703 960868 : b = subii(b, shifti(ct, 1));
704 960868 : gcoeff(U,1,1) = subii(gcoeff(U,1,1), mulii(gcoeff(U,1,2), t));
705 960868 : gcoeff(U,2,1) = subii(gcoeff(U,2,1), mulii(gcoeff(U,2,2), t));
706 : }
707 1921185 : return mkqfb(a,b,c,d);
708 : }
709 :
710 : static int
711 2046352 : is_minimal(GEN Q, long m)
712 : {
713 2046352 : pari_sp av = avma;
714 2046352 : GEN a = gel(Q,1), b = gel(Q,2), c = gel(Q,3);
715 4940497 : return gc_bool(av, lti2n(addii(subii(a,b), c), m)
716 1927231 : || (lti2n(subii(b, shifti(a,1)), m+1)
717 966914 : && lti2n(subii(b, shifti(c,1)), m+1)));
718 : }
719 :
720 : static GEN
721 124020 : pqfbred_iter_1(GEN Q, ulong m, GEN U)
722 : {
723 124020 : pari_sp av = avma;
724 1923484 : while (!is_minimal(Q,m))
725 : {
726 1799464 : Q = pqfbred_1(Q, m, U);
727 1799464 : if (gc_needed(av, 1))
728 : {
729 0 : if (DEBUGMEM>1) pari_warn(warnmem,"pqfbred_iter_1, lc = %ld", expi(gel(Q,3)));
730 0 : (void)gc_all(av, 3, &Q, &gel(U,1), &gel(U,2));
731 : }
732 : }
733 124020 : return Q;
734 : }
735 :
736 : static GEN
737 61492 : pqfbred_basecase(GEN Q, ulong m, GEN *pt_U)
738 : {
739 61492 : pari_sp av = avma;
740 61492 : GEN U = matid(2);
741 61492 : Q = pqfbred_iter_1(Q, m, U);
742 61492 : *pt_U = U;
743 61492 : return gc_all(av, 2, &Q, pt_U);
744 : }
745 :
746 : static long
747 101215826 : qfb_maxexpi(GEN Q)
748 101215826 : { return 1+maxss(expi(gel(Q,1)), maxss(expi(gel(Q,2)), expi(gel(Q,3)))); }
749 :
750 : static long
751 125056 : qfb_minexpi(GEN Q)
752 : {
753 125056 : long m = minss(expi(gel(Q,1)), minss(expi(gel(Q,2)), expi(gel(Q,3))));
754 125056 : return m < 0 ? 0: m;
755 : }
756 :
757 : static GEN
758 124020 : pqfbred_rec(GEN Q, long m, GEN *pt_U)
759 : {
760 124020 : pari_sp av = avma;
761 : GEN Q0, Q1, QR;
762 124020 : GEN d = qfb_disc(Q);
763 124020 : long n = qfb_maxexpi(Q) - m;
764 : long h;
765 124020 : int going_to_r8 = 0;
766 : GEN U;
767 :
768 124020 : if (n < 170)
769 61492 : return pqfbred_basecase(Q, m, pt_U);
770 :
771 62528 : if (qfb_minexpi(Q) <= m + 2)
772 : {
773 0 : U = matid(2);
774 0 : QR = Q;
775 : }
776 : else
777 : {
778 : long p, mm;
779 62528 : if (m <= n)
780 : {
781 650 : mm = m;
782 650 : p = 0;
783 650 : Q1 = Q;
784 : } else
785 : {
786 61878 : mm = n;
787 61878 : p = m + 1 - n;
788 61878 : Q0 = mkvec3(remi2n(gel(Q,1), p), remi2n(gel(Q,2), p), remi2n(gel(Q,3), p));
789 61878 : Q1 = qfb3(shifti(gel(Q,1),-p), shifti(gel(Q,2),-p), shifti(gel(Q,3),-p));
790 : }
791 62528 : h = mm + (n>>1);
792 62528 : if (qfb_minexpi(Q1) <= h)
793 : {
794 207 : U = matid(2);
795 207 : QR = Q1;
796 : }
797 : else
798 62321 : QR = pqfbred_rec(Q1, h, &U);
799 184249 : while (qfb_maxexpi(QR) > h)
800 : {
801 122868 : if (is_minimal(QR, mm))
802 : {
803 1147 : going_to_r8 = 1;
804 1147 : break;
805 : }
806 121721 : QR = pqfbred_1(QR, mm, U);
807 : }
808 62528 : if (!going_to_r8)
809 : {
810 : GEN V;
811 61381 : QR = pqfbred_rec(QR, mm, &V);
812 61381 : U = ZM2_mul(U,V);
813 : }
814 62528 : if (p > 0)
815 : {
816 61878 : GEN Q0U = qfb_ZM_apply(Q0,U);
817 123756 : QR = mkqfb(addii(shifti(gel(QR,1), p), gel(Q0U,1)),
818 61878 : addii(shifti(gel(QR,2), p), gel(Q0U,2)),
819 61878 : addii(shifti(gel(QR,3), p), gel(Q0U,3)), d);
820 : }
821 : }
822 62528 : QR = pqfbred_iter_1(QR, m, U);
823 62528 : *pt_U = U;
824 62528 : return gc_all(av, 2, &QR, pt_U);
825 : }
826 :
827 : static GEN
828 209575 : qfbredsl2_real(GEN Q, GEN isqrtD)
829 : {
830 209575 : pari_sp av = avma;
831 209575 : if (2*qfb_maxexpi(Q)-expi(gel(Q,4)) <= QFBRED_LIMIT)
832 209575 : return qfbredsl2_real_basecase(Q, isqrtD);
833 : else
834 : {
835 0 : GEN a = gel(Q,1), b = gel(Q,2), c = gel(Q,3), d = gel(Q,4);
836 0 : GEN Qf, Qr, W, U, t = NULL;
837 0 : long sa = signe(a), sb;
838 0 : if (sa < 0) { a = negi(a); b = negi(b); c = negi(c); }
839 0 : if (signe(c) < 0)
840 : {
841 : GEN at;
842 0 : t = addiu(truedivii(subii(isqrtD,b),shifti(a,1)),1);
843 0 : at = mulii(a,t);
844 0 : c = addii(subii(c, mulii(b, t)), mulii(at, t));
845 0 : b = subii(b, shifti(at,1));
846 : }
847 0 : sb = signe(b);
848 0 : Qr = pqfbred_rec(mkqfb(a, sb < 0 ? negi(b): b, c, d), 0, &U);
849 0 : if (sa < 0)
850 0 : Qr = mkqfb(negi(gel(Qr,1)), negi(gel(Qr,2)), negi(gel(Qr,3)), gel(Qr,4));
851 0 : if (sb < 0)
852 : {
853 0 : gcoeff(U,2,1) = negi(gcoeff(U,2,1));
854 0 : gcoeff(U,2,2) = negi(gcoeff(U,2,2));
855 : }
856 0 : if (t)
857 : {
858 0 : gcoeff(U,1,1) = subii( gcoeff(U,1,1), mulii(gcoeff(U,2,1), t));
859 0 : gcoeff(U,1,2) = subii( gcoeff(U,1,2), mulii(gcoeff(U,2,2), t));
860 : }
861 0 : W = qfbredsl2_real_basecase(Qr, isqrtD);
862 0 : Qf = gel(W,1);
863 0 : U = ZM2_mul(U,gel(W,2));
864 0 : return gc_GEN(av, mkvec2(Qf,U));
865 : }
866 : }
867 :
868 : static GEN
869 5044040 : qfbredsl2_imag(GEN Q)
870 : {
871 5044040 : pari_sp av = avma;
872 : GEN Qt, U;
873 5044040 : if (2*qfb_maxexpi(Q)-expi(gel(Q,4)) <= QFBRED_LIMIT)
874 5043734 : Qt = qfbredsl2_imag_basecase(Q, &U);
875 : else
876 : {
877 311 : long sb = signe(gel(Q,2));
878 : GEN Qr, W;
879 311 : Qr = pqfbred_rec(sb < 0 ? mkqfb(gel(Q,1), negi(gel(Q,2)), gel(Q,3), gel(Q,4)): Q, 0, &U);
880 311 : Qt = qfbredsl2_imag_basecase(Qr,&W);
881 311 : U = ZM2_mul(U,W);
882 311 : if (sb < 0)
883 : {
884 173 : gcoeff(U,2,1) = negi(gcoeff(U,2,1));
885 173 : gcoeff(U,2,2) = negi(gcoeff(U,2,2));
886 : }
887 : }
888 5044081 : return gc_GEN(av, mkvec2(Qt,U));
889 : }
890 :
891 : GEN
892 4733723 : redimagsl2(GEN Q, GEN *U)
893 : {
894 4733723 : GEN q = qfbredsl2_imag(Q);
895 4733774 : *U = gel(q,2); return gel(q,1);
896 : }
897 :
898 : GEN
899 519893 : qfbredsl2(GEN q, GEN isD)
900 : {
901 : pari_sp av;
902 519893 : if (typ(q) != t_QFB) pari_err_TYPE("qfbredsl2",q);
903 519893 : if (qfb_is_qfi(q))
904 : {
905 310311 : if (isD) pari_err_TYPE("qfbredsl2", isD);
906 310311 : return qfbredsl2_imag(q);
907 : }
908 209582 : av = avma;
909 209582 : if (!isD) isD = sqrti(qfb_disc(q));
910 208068 : else if (typ(isD) != t_INT) pari_err_TYPE("qfbredsl2",isD);
911 209575 : return gc_upto(av, qfbredsl2_real(q, isD));
912 : }
913 :
914 : static GEN
915 427 : qfbred_real_i(GEN Q, long flag, GEN isqrtD, GEN sqrtD)
916 : {
917 427 : if (typ(Q)!=t_QFB || 2*qfb_maxexpi(Q)-expi(gel(Q,4)) <= QFBRED_LIMIT)
918 427 : return qfbred_real_basecase_i(Q, flag, isqrtD, sqrtD);
919 : else
920 : {
921 0 : GEN a = gel(Q,1), b = gel(Q,2), c = gel(Q,3), d = gel(Q,4);
922 0 : GEN Qr, W, U, t = NULL;
923 0 : long sa = signe(a), sb;
924 0 : if (sa < 0) { a = negi(a); b = negi(b); c = negi(c); }
925 0 : if (signe(c) < 0)
926 : {
927 : GEN at;
928 0 : t = addiu(truedivii(subii(isqrtD,b),shifti(a,1)),1);
929 0 : at = mulii(a,t);
930 0 : c = addii(subii(c, mulii(b, t)), mulii(at, t));
931 0 : b = subii(b, shifti(at,1));
932 : }
933 0 : sb = signe(b);
934 0 : Qr = pqfbred_rec(mkqfb(a, sb < 0 ? negi(b): b, c, d), 0, &U);
935 0 : if (sa < 0)
936 0 : Qr = mkqfb(negi(gel(Qr,1)), negi(gel(Qr,2)), negi(gel(Qr,3)), gel(Qr,4));
937 0 : W = qfbred_real_basecase_i(Qr, flag, isqrtD, sqrtD);
938 0 : return gel(W,1);
939 : }
940 : }
941 :
942 : static GEN
943 63 : qfbred_real(GEN x) { return qfbred_real_i(x,0,NULL,NULL); }
944 :
945 : static GEN
946 95799727 : qfbred_imag_basecase_av(pari_sp av, GEN q)
947 : {
948 95799727 : GEN a = gel(q,1), b = gel(q,2), c = gel(q,3), D = gel(q,4);
949 95799727 : long cmp, lc = lgefint(c);
950 :
951 95799727 : if (lgefint(a) == 3 && lc == 3) return qfbred_imag_1(av, a, b, c, D);
952 911730 : cmp = abscmpii(a, b);
953 919092 : if (cmp < 0)
954 436231 : REDB(a,&b,&c);
955 482861 : else if (cmp == 0 && signe(b) < 0)
956 27 : b = negi(b);
957 : for(;;)
958 : {
959 3126345 : cmp = abscmpii(a, c); if (cmp <= 0) break;
960 2932679 : lc = lgefint(a); /* lg(future c): we swap a & c next */
961 2932679 : if (lc == 3) return qfbred_imag_1(av, a, b, c, D);
962 2207253 : swap(a,c); b = negi(b); /* apply rho */
963 2207253 : REDB(a,&b,&c);
964 2207253 : if (gc_needed(av, 2))
965 : {
966 0 : if (DEBUGMEM>1) pari_warn(warnmem,"qfbred_imag, lc = %ld", lc);
967 0 : (void)gc_all(av, 3, &a,&b,&c);
968 : }
969 : }
970 193666 : if (cmp == 0 && signe(b) < 0) b = negi(b);
971 193666 : return gc_GEN(av, mkqfb(a, b, c, D));
972 : }
973 : static GEN
974 95655226 : qfbred_imag_av(pari_sp av, GEN Q)
975 : {
976 95655226 : if (2*qfb_maxexpi(Q)-expi(gel(Q,4)) <= QFBRED_LIMIT)
977 95798320 : return qfbred_imag_basecase_av(av, Q);
978 : else
979 : {
980 7 : long sb = signe(gel(Q,2));
981 7 : GEN U, Qr = pqfbred_rec(sb < 0 ? mkqfb(gel(Q,1), negi(gel(Q,2)), gel(Q,3), gel(Q,4)): Q, 0, &U);
982 7 : return qfbred_imag_basecase_av(av, Qr);
983 : }
984 : }
985 :
986 : static GEN
987 22335163 : qfbred_imag(GEN q) { return qfbred_imag_av(avma, q); }
988 :
989 : GEN
990 54819 : qfbred0(GEN x, long flag, GEN isqrtD, GEN sqrtD)
991 : {
992 : pari_sp av;
993 54819 : GEN q = check_qfbext("qfbred",x);
994 54819 : if (qfb_is_qfi(q)) return (flag & qf_STEP)? rhoimag(x): qfbred_imag(x);
995 364 : if (typ(x)==t_QFB) flag |= qf_NOD;
996 49 : else flag &= ~qf_NOD;
997 364 : av = avma;
998 364 : return gc_GEN(av, qfbred_real_i(x,flag,isqrtD,sqrtD));
999 : }
1000 : /* t_QFB */
1001 : GEN
1002 22280718 : qfbred_i(GEN x) { return qfb_is_qfi(x)? qfbred_imag(x): qfbred_real(x); }
1003 : GEN
1004 53006 : qfbred(GEN x) { return qfbred0(x, 0, NULL, NULL); }
1005 : /***********************************************************************/
1006 : /** **/
1007 : /** Composition **/
1008 : /** **/
1009 : /***********************************************************************/
1010 :
1011 : static void
1012 33214850 : qfb_sqr(GEN z, GEN x)
1013 : {
1014 : GEN c, d1, x2, v1, v2, c3, m, p1, r;
1015 :
1016 33214850 : d1 = bezout(gel(x,2),gel(x,1),&x2, NULL); /* usually 1 */
1017 33214713 : c = gel(x,3);
1018 33214713 : m = mulii(c,x2);
1019 33214480 : if (equali1(d1))
1020 24737165 : v1 = v2 = gel(x,1);
1021 : else
1022 : {
1023 8477375 : v1 = diviiexact(gel(x,1),d1);
1024 8477377 : v2 = mulii(v1, gcdii(d1,c)); /* = v1 iff x primitive */
1025 8477379 : c = mulii(c, d1);
1026 : }
1027 33214542 : togglesign(m);
1028 33214470 : r = modii(m,v2);
1029 33214373 : p1 = mulii(r, v1);
1030 33214450 : c3 = addii(c, mulii(r,addii(gel(x,2),p1)));
1031 33214509 : gel(z,1) = mulii(v1,v2);
1032 33214466 : gel(z,2) = addii(gel(x,2), shifti(p1,1));
1033 33214458 : gel(z,3) = diviiexact(c3,v2);
1034 33214341 : }
1035 : /* z <- x * y */
1036 : static void
1037 75074406 : qfb_comp(GEN z, GEN x, GEN y)
1038 : {
1039 : GEN n, c, d, y1, v1, v2, c3, m, p1, r;
1040 :
1041 75074406 : if (x == y) { qfb_sqr(z,x); return; }
1042 42650434 : n = shifti(subii(gel(y,2),gel(x,2)), -1);
1043 42483617 : v1 = gel(x,1);
1044 42483617 : v2 = gel(y,1);
1045 42483617 : c = gel(y,3);
1046 42483617 : d = bezout(v2,v1,&y1,NULL);
1047 42579186 : if (equali1(d))
1048 22863875 : m = mulii(y1,n);
1049 : else
1050 : {
1051 19749021 : GEN s = subii(gel(y,2), n);
1052 19746850 : GEN x2, y2, d1 = bezout(s,d,&x2,&y2); /* x2 s + y2 (x1 v1 + y1 v2) = d1 */
1053 19750355 : if (!equali1(d1))
1054 : {
1055 10953798 : v1 = diviiexact(v1,d1);
1056 10952560 : v2 = diviiexact(v2,d1); /* gcd = 1 iff x or y primitive */
1057 10952294 : v1 = mulii(v1, gcdii(c,gcdii(gel(x,3),gcdii(d1,n))));
1058 10952695 : c = mulii(c, d1);
1059 : }
1060 19748928 : m = addii(mulii(mulii(y1,y2),n), mulii(gel(y,3),x2));
1061 : }
1062 42557570 : togglesign(m);
1063 42492554 : r = modii(m, v1);
1064 42455954 : p1 = mulii(r, v2);
1065 42493512 : c3 = addii(c, mulii(r,addii(gel(y,2),p1)));
1066 42492478 : gel(z,1) = mulii(v1,v2);
1067 42488298 : gel(z,2) = addii(gel(y,2), shifti(p1,1));
1068 42489059 : gel(z,3) = diviiexact(c3,v1);
1069 : }
1070 :
1071 : /* not meant to be efficient */
1072 : static GEN
1073 84 : qfb_comp_gen(GEN x, GEN y)
1074 : {
1075 84 : GEN d1 = qfb_disc(x), d2 = qfb_disc(y);
1076 84 : GEN a1 = gel(x,1), b1 = gel(x,2), c1 = gel(x,3), n1;
1077 84 : GEN a2 = gel(y,1), b2 = gel(y,2), c2 = gel(y,3), n2;
1078 84 : GEN cx = content(x), cy = content(y), A, B, C, D, U, m, m2;
1079 :
1080 84 : if (!is_pm1(cx))
1081 : {
1082 14 : a1 = diviiexact(a1, cx); b1 = diviiexact(b1, cx);
1083 14 : c1 = diviiexact(c1, cx); d1 = diviiexact(d1, sqri(cx));
1084 : }
1085 84 : if (!is_pm1(cy))
1086 : {
1087 28 : a2 = diviiexact(a2, cy); c2 = diviiexact(c2, cy);
1088 28 : b2 = diviiexact(b2, cy); d2 = diviiexact(d2, sqri(cy));
1089 : }
1090 84 : D = gcdii(d1, d2); if (signe(d1) < 0) setsigne(D, -1);
1091 133 : if (!Z_issquareall(diviiexact(d1, D), &n1) ||
1092 84 : !Z_issquareall(diviiexact(d2, D), &n2)) return NULL;
1093 49 : A = mulii(a1, n2);
1094 49 : B = mulii(a2, n1);
1095 49 : C = shifti(addii(mulii(b1, n2), mulii(b2, n1)), -1);
1096 49 : U = ZV_extgcd(mkvec3(A, B, C));
1097 49 : m = gel(U,1); U = gmael(U,2,3);
1098 49 : A = mulii(diviiexact(mulii(a1,b2),m), gel(U,1));
1099 49 : B = mulii(diviiexact(mulii(a2,b1),m), gel(U,2));
1100 49 : C = addii(mulii(b1,b2), mulii(D, mulii(n1,n2)));
1101 49 : C = mulii(diviiexact(shifti(C,-1), m), gel(U,3));
1102 49 : B = addii(A, addii(B, C));
1103 49 : m2 = sqri(m);
1104 49 : A = diviiexact(mulii(a1, a2), m2);
1105 49 : C = diviiexact(shifti(subii(sqri(B),D), -2), A);
1106 49 : cx = mulii(cx, cy);
1107 49 : if (!is_pm1(cx))
1108 : {
1109 14 : A = mulii(A, cx); B = mulii(B, cx);
1110 14 : C = mulii(C, cx); D = mulii(D, sqri(cx));
1111 : }
1112 49 : return mkqfb(A, B, C, D);
1113 : }
1114 :
1115 : static GEN
1116 72335979 : qficomp0(GEN x, GEN y, int raw)
1117 : {
1118 72335979 : pari_sp av = avma;
1119 72335979 : GEN z = cgetg(5,t_QFB);
1120 72331511 : gel(z,4) = gel(x,4);
1121 72331511 : qfb_comp(z, x,y);
1122 72111509 : if (raw) return gc_GEN(av,z);
1123 72109731 : return qfbred_imag_av(av, z);
1124 : }
1125 : static GEN
1126 441 : qfrcomp0(GEN x, GEN y, int raw)
1127 : {
1128 441 : pari_sp av = avma;
1129 441 : GEN dx = NULL, dy = NULL;
1130 441 : GEN z = cgetg(5,t_QFB);
1131 441 : if (typ(x)==t_VEC) { dx = gel(x,2); x = gel(x,1); }
1132 441 : if (typ(y)==t_VEC) { dy = gel(y,2); y = gel(y,1); }
1133 441 : gel(z,4) = gel(x,4);
1134 441 : qfb_comp(z, x,y);
1135 441 : if (dx) z = mkvec2(z, dy? addrr(dx, dy): dx); else if (dy) z = mkvec2(z, dy);
1136 441 : if (!raw) z = qfbred_real(z);
1137 441 : return gc_GEN(av, z);
1138 : }
1139 : /* same discriminant, no distance, no checks */
1140 : GEN
1141 27341630 : qfbcomp_i(GEN x, GEN y)
1142 27341630 : { return qfb_is_qfi(x)? qficomp0(x,y,0): qfrcomp0(x,y,0); }
1143 : GEN
1144 130651 : qfbcomp(GEN x, GEN y)
1145 : {
1146 130651 : GEN qx = check_qfbext("qfbcomp", x);
1147 130651 : GEN qy = check_qfbext("qfbcomp", y);
1148 130651 : if (!equalii(gel(qx,4),gel(qy,4)))
1149 : {
1150 63 : pari_sp av = avma;
1151 63 : GEN z = qfb_comp_gen(qx, qy);
1152 63 : if (typ(x) == t_VEC || typ(y) == t_VEC)
1153 7 : pari_err_IMPL("Shanks's distance in general composition");
1154 56 : if (!z) pari_err_OP("*",x,y);
1155 21 : return gc_upto(av, qfbred(z));
1156 : }
1157 130588 : return qfb_is_qfi(qx)? qficomp0(x,y,0): qfrcomp0(x,y,0);
1158 : }
1159 : /* same discriminant, no distance, no checks */
1160 : GEN
1161 0 : qfbcompraw_i(GEN x, GEN y)
1162 0 : { return qfb_is_qfi(x)? qficomp0(x,y,1): qfrcomp0(x,y,1); }
1163 : GEN
1164 2198 : qfbcompraw(GEN x, GEN y)
1165 : {
1166 2198 : GEN qx = check_qfbext("qfbcompraw", x);
1167 2198 : GEN qy = check_qfbext("qfbcompraw", y);
1168 2198 : if (!equalii(gel(qx,4),gel(qy,4)))
1169 : {
1170 21 : pari_sp av = avma;
1171 21 : GEN z = qfb_comp_gen(qx, qy);
1172 21 : if (typ(x) == t_VEC || typ(y) == t_VEC)
1173 0 : pari_err_IMPL("Shanks's distance in general composition");
1174 21 : if (!z) pari_err_OP("qfbcompraw",x,y);
1175 21 : return gc_GEN(av, z);
1176 : }
1177 2177 : if (!equalii(gel(qx,4),gel(qy,4))) pari_err_OP("qfbcompraw",x,y);
1178 2177 : return qfb_is_qfi(qx)? qficomp0(x,y,1): qfrcomp0(x,y,1);
1179 : }
1180 :
1181 : static GEN
1182 790845 : qfisqr0(GEN x, long raw)
1183 : {
1184 790845 : pari_sp av = avma;
1185 790845 : GEN z = cgetg(5,t_QFB);
1186 790845 : gel(z,4) = gel(x,4);
1187 790845 : qfb_sqr(z,x);
1188 790845 : if (raw) return gc_GEN(av,z);
1189 790845 : return qfbred_imag_av(av, z);
1190 : }
1191 : static GEN
1192 35 : qfrsqr0(GEN x, long raw)
1193 : {
1194 35 : pari_sp av = avma;
1195 35 : GEN dx = NULL, z = cgetg(5,t_QFB);
1196 35 : if (typ(x) == t_VEC) { dx = gel(x,2); x = gel(x,1); }
1197 35 : gel(z,4) = gel(x,4); qfb_sqr(z,x);
1198 35 : if (dx) z = mkvec2(z, shiftr(dx,1));
1199 35 : if (!raw) z = qfbred_real(z);
1200 35 : return gc_GEN(av, z);
1201 : }
1202 : /* same discriminant, no distance, no checks */
1203 : GEN
1204 697195 : qfbsqr_i(GEN x)
1205 697195 : { return qfb_is_qfi(x)? qfisqr0(x,0): qfrsqr0(x,0); }
1206 : GEN
1207 93685 : qfbsqr(GEN x)
1208 : {
1209 93685 : GEN qx = check_qfbext("qfbsqr", x);
1210 93685 : return qfb_is_qfi(qx)? qfisqr0(x,0): qfrsqr0(x,0);
1211 : }
1212 :
1213 : static GEN
1214 6867 : qfr_1_by_disc(GEN D)
1215 : {
1216 : GEN y, r, s;
1217 6867 : check_quaddisc_real(D, NULL, "qfr_1_by_disc");
1218 6867 : y = cgetg(5,t_QFB);
1219 6867 : s = sqrtremi(D, &r); togglesign(r); /* s^2 - r = D */
1220 6867 : if (mpodd(r))
1221 : {
1222 3535 : s = subiu(s,1);
1223 3535 : r = subii(r, addiu(shifti(s, 1), 1));
1224 3535 : r = shifti(r, -2); set_avma((pari_sp)y); s = icopy(s);
1225 : }
1226 : else
1227 3332 : { r = shifti(r, -2); set_avma((pari_sp)s); }
1228 6867 : gel(y,1) = gen_1;
1229 6867 : gel(y,2) = s;
1230 6867 : gel(y,3) = icopy(r);
1231 6867 : gel(y,4) = icopy(D); return y;
1232 : }
1233 :
1234 : static GEN
1235 35 : qfr_disc(GEN x)
1236 35 : { return qfb_disc(typ(x)==t_VEC ? gel(x,1): x); }
1237 :
1238 : static GEN
1239 35 : qfr_1(GEN x)
1240 35 : { return qfr_1_by_disc(qfr_disc(x)); }
1241 :
1242 : static void
1243 0 : qfr_1_fill(GEN y, struct qfr_data *S)
1244 : {
1245 0 : pari_sp av = avma;
1246 0 : GEN y2 = S->isqrtD;
1247 0 : gel(y,1) = gen_1;
1248 0 : if (mod2(S->D) != mod2(y2)) y2 = subiu(y,1);
1249 0 : gel(y,2) = y2; av = avma;
1250 0 : gel(y,3) = gc_INT(av, shifti(subii(sqri(y2), S->D),-2));
1251 0 : }
1252 : static GEN
1253 0 : qfr5_1(struct qfr_data *S, long prec)
1254 : {
1255 0 : GEN y = cgetg(6, t_VEC);
1256 0 : qfr_1_fill(y, S);
1257 0 : gel(y,4) = gen_0;
1258 0 : gel(y,5) = real_1(prec); return y;
1259 : }
1260 : static GEN
1261 0 : qfr3_1(struct qfr_data *S)
1262 : {
1263 0 : GEN y = cgetg(4, t_VEC);
1264 0 : qfr_1_fill(y, S); return y;
1265 : }
1266 :
1267 : /* Assume D < 0 is the discriminant of a t_QFB */
1268 : static GEN
1269 730958 : qfi_1_by_disc(GEN D)
1270 : {
1271 730958 : GEN b,c, y = cgetg(5,t_QFB);
1272 730958 : quadpoly_bc(D, mod2(D), &b,&c);
1273 730958 : if (b == gen_m1) b = gen_1;
1274 730958 : gel(y,1) = gen_1;
1275 730958 : gel(y,2) = b;
1276 730958 : gel(y,3) = c;
1277 730958 : gel(y,4) = icopy(D); return y;
1278 : }
1279 : static GEN
1280 719014 : qfi_1(GEN x)
1281 : {
1282 719014 : if (typ(x) != t_QFB) pari_err_TYPE("qfi_1",x);
1283 719014 : return qfi_1_by_disc(qfb_disc(x));
1284 : }
1285 :
1286 : GEN
1287 0 : qfb_1(GEN x) { return qfb_is_qfi(x) ? qfi_1(x): qfr_1(x); }
1288 :
1289 : static GEN
1290 13525489 : _qfimul(void *E, GEN x, GEN y) { (void) E; return qficomp0(x,y,0); }
1291 : static GEN
1292 31340722 : _qfisqr(void *E, GEN x) { (void) E; return qficomp0(x,x,0); }
1293 : static GEN
1294 7 : _qfimulraw(void *E, GEN x, GEN y) { (void) E; return qficomp0(x,y,1); }
1295 : static GEN
1296 7 : _qfisqrraw(void *E, GEN x) { (void) E; return qficomp0(x,x,1); }
1297 :
1298 : static GEN
1299 7 : qfipowraw(GEN x, long n)
1300 : {
1301 7 : pari_sp av = avma;
1302 : GEN y;
1303 7 : if (!n) return qfi_1(x);
1304 7 : if (n== 1) return gcopy(x);
1305 7 : if (n==-1) { x = gcopy(x); togglesign(gel(x,2)); return x; }
1306 7 : if (n < 0) x = qfb_inv(x);
1307 7 : y = gen_powu(x, labs(n), NULL, &_qfisqrraw, &_qfimulraw);
1308 7 : return gc_GEN(av,y);
1309 : }
1310 :
1311 : static GEN
1312 15830483 : qfipow(GEN x, GEN n)
1313 : {
1314 15830483 : pari_sp av = avma;
1315 : GEN y;
1316 15830483 : long s = signe(n);
1317 15830483 : if (!s) return qfi_1(x);
1318 15111469 : if (s < 0) x = qfb_inv(x);
1319 15111465 : y = gen_pow(qfbred_i(x), n, NULL, &_qfisqr, &_qfimul);
1320 15111475 : return gc_GEN(av,y);
1321 : }
1322 :
1323 : static long
1324 412328 : parteucl(GEN L, GEN *d, GEN *v3, GEN *v, GEN *v2)
1325 : {
1326 : long z;
1327 412328 : *v = gen_0; *v2 = gen_1;
1328 4351417 : for (z=0; abscmpii(*v3,L) > 0; z++)
1329 : {
1330 3939089 : GEN t3, t2 = subii(*v, mulii(truedvmdii(*d,*v3,&t3),*v2));
1331 3939089 : *v = *v2; *d = *v3; *v2 = t2; *v3 = t3;
1332 : }
1333 412328 : return z;
1334 : }
1335 :
1336 : /* composition: Shanks' NUCOMP & NUDUPL */
1337 : /* L = floor((|d|/4)^(1/4)) */
1338 : GEN
1339 400722 : nucomp(GEN x, GEN y, GEN L)
1340 : {
1341 400722 : pari_sp av = avma;
1342 : long z;
1343 : GEN a, a1, a2, b2, b, d, d1, g, n, p1, q1, q2, s, u, u1, v, v2, v3, Q;
1344 :
1345 400722 : if (x==y) return nudupl(x,L);
1346 400680 : if (!is_qfi(x)) pari_err_TYPE("nucomp",x);
1347 400680 : if (!is_qfi(y)) pari_err_TYPE("nucomp",y);
1348 :
1349 400680 : if (abscmpii(gel(x,1),gel(y,1)) < 0) swap(x, y);
1350 400680 : s = shifti(addii(gel(x,2),gel(y,2)), -1);
1351 400680 : n = subii(gel(y,2), s);
1352 400680 : a1 = gel(x,1);
1353 400680 : a2 = gel(y,1); d = bezout(a2,a1,&u,&v);
1354 400680 : if (equali1(d)) { a = negi(mulii(u,n)); d1 = d; }
1355 163576 : else if (dvdii(s,d)) /* d | s */
1356 : {
1357 83503 : a = negi(mulii(u,n)); d1 = d;
1358 83503 : a1 = diviiexact(a1, d1);
1359 83503 : a2 = diviiexact(a2, d1);
1360 83503 : s = diviiexact(s, d1);
1361 : }
1362 : else
1363 : {
1364 : GEN p2, l;
1365 80073 : d1 = bezout(s,d,&u1,NULL);
1366 80073 : if (!equali1(d1))
1367 : {
1368 2044 : a1 = diviiexact(a1,d1);
1369 2044 : a2 = diviiexact(a2,d1);
1370 2044 : s = diviiexact(s,d1);
1371 2044 : d = diviiexact(d,d1);
1372 : }
1373 80073 : p1 = remii(gel(x,3),d);
1374 80073 : p2 = remii(gel(y,3),d);
1375 80073 : l = modii(mulii(negi(u1), addii(mulii(u,p1),mulii(v,p2))), d);
1376 80073 : a = subii(mulii(l,diviiexact(a1,d)), mulii(u,diviiexact(n,d)));
1377 : }
1378 400680 : a = modii(a,a1); p1 = subii(a,a1); if (abscmpii(a,p1) > 0) a = p1;
1379 400680 : d = a1; v3 = a; z = parteucl(L, &d,&v3, &v,&v2);
1380 400680 : Q = cgetg(5,t_QFB);
1381 400680 : if (!z) {
1382 37632 : g = diviiexact(addii(mulii(v3,s),gel(y,3)), d);
1383 37632 : b = a2;
1384 37632 : b2 = gel(y,2);
1385 37632 : v2 = d1;
1386 37632 : gel(Q,1) = mulii(d,b);
1387 : } else {
1388 : GEN e, q3, q4;
1389 363048 : if (z&1) { v3 = negi(v3); v2 = negi(v2); }
1390 363048 : b = diviiexact(addii(mulii(a2,d), mulii(n,v)), a1);
1391 363048 : e = diviiexact(addii(mulii(s,d),mulii(gel(y,3),v)), a1);
1392 363048 : q3 = mulii(e,v2);
1393 363048 : q4 = subii(q3,s);
1394 363048 : b2 = addii(q3,q4);
1395 363048 : g = diviiexact(q4,v);
1396 363048 : if (!equali1(d1)) { v2 = mulii(d1,v2); v = mulii(d1,v); b2 = mulii(d1,b2); }
1397 363048 : gel(Q,1) = addii(mulii(d,b), mulii(e,v));
1398 : }
1399 400680 : q1 = mulii(b, v3);
1400 400680 : q2 = addii(q1,n);
1401 400680 : gel(Q,2) = addii(b2, z? addii(q1,q2): shifti(q1, 1));
1402 400680 : gel(Q,3) = addii(mulii(v3,diviiexact(q2,d)), mulii(g,v2));
1403 400680 : gel(Q,4) = gel(x,4);
1404 400680 : return qfbred_imag_av(av, Q);
1405 : }
1406 :
1407 : GEN
1408 11648 : nudupl(GEN x, GEN L)
1409 : {
1410 11648 : pari_sp av = avma;
1411 : long z;
1412 : GEN u, v, d, d1, p1, a, b, c, a2, b2, c2, Q, v2, v3, g;
1413 :
1414 11648 : if (!is_qfi(x)) pari_err_TYPE("nudupl",x);
1415 11648 : a = gel(x,1);
1416 11648 : b = gel(x,2);
1417 11648 : d1 = bezout(b,a, &u,NULL);
1418 11648 : if (!equali1(d1))
1419 : {
1420 4620 : a = diviiexact(a, d1);
1421 4620 : b = diviiexact(b, d1);
1422 : }
1423 11648 : c = modii(negi(mulii(u,gel(x,3))), a);
1424 11648 : p1 = subii(c,a); if (abscmpii(c,p1) > 0) c = p1;
1425 11648 : d = a; v3 = c; z = parteucl(L, &d,&v3, &v,&v2);
1426 11648 : a2 = sqri(d);
1427 11648 : c2 = sqri(v3);
1428 11648 : Q = cgetg(5,t_QFB);
1429 11648 : if (!z) {
1430 1281 : g = diviiexact(addii(mulii(v3,b),gel(x,3)), d);
1431 1281 : b2 = gel(x,2);
1432 1281 : v2 = d1;
1433 1281 : gel(Q,1) = a2;
1434 : } else {
1435 : GEN e;
1436 10367 : if (z&1) { v = negi(v); d = negi(d); }
1437 10367 : e = diviiexact(addii(mulii(gel(x,3),v), mulii(b,d)), a);
1438 10367 : g = diviiexact(subii(mulii(e,v2), b), v);
1439 10367 : b2 = addii(mulii(e,v2), mulii(v,g));
1440 10367 : if (!equali1(d1)) { b2 = mulii(d1,b2); v = mulii(d1,v); v2 = mulii(d1,v2); }
1441 10367 : gel(Q,1) = addii(a2, mulii(e,v));
1442 : }
1443 11648 : gel(Q,2) = addii(b2, subii(sqri(addii(d,v3)), addii(a2,c2)));
1444 11648 : gel(Q,3) = addii(c2, mulii(g,v2));
1445 11648 : gel(Q,4) = gel(x,4);
1446 11648 : return qfbred_imag_av(av, Q);
1447 : }
1448 :
1449 : static GEN
1450 4739 : mul_nucomp(void *l, GEN x, GEN y) { return nucomp(x, y, (GEN)l); }
1451 : static GEN
1452 11606 : mul_nudupl(void *l, GEN x) { return nudupl(x, (GEN)l); }
1453 : GEN
1454 1008 : nupow(GEN x, GEN n, GEN L)
1455 : {
1456 : pari_sp av;
1457 : GEN y, D;
1458 :
1459 1008 : if (typ(n) != t_INT) pari_err_TYPE("nupow",n);
1460 1008 : if (!is_qfi(x)) pari_err_TYPE("nupow",x);
1461 1008 : if (gequal1(n)) return gcopy(x);
1462 1008 : av = avma;
1463 1008 : D = qfb_disc(x);
1464 1008 : y = qfi_1_by_disc(D);
1465 1008 : if (!signe(n)) return y;
1466 959 : if (!L) L = sqrtnint(absi_shallow(D), 4);
1467 959 : y = gen_pow_i(x, n, (void*)L, &mul_nudupl, &mul_nucomp);
1468 959 : if (signe(n) < 0
1469 35 : && !absequalii(gel(y,1),gel(y,2))
1470 35 : && !absequalii(gel(y,1),gel(y,3))) togglesign(gel(y,2));
1471 959 : return gc_GEN(av, y);
1472 : }
1473 :
1474 : /* Not stack-clean */
1475 : GEN
1476 1735230 : qfr5_compraw(GEN x, GEN y)
1477 : {
1478 1735230 : GEN z = cgetg(6,t_VEC); qfb_comp(z,x,y);
1479 1735230 : if (x == y)
1480 : {
1481 34552 : gel(z,4) = shifti(gel(x,4),1);
1482 34552 : gel(z,5) = sqrr(gel(x,5));
1483 : }
1484 : else
1485 : {
1486 1700678 : gel(z,4) = addii(gel(x,4),gel(y,4));
1487 1700678 : gel(z,5) = mulrr(gel(x,5),gel(y,5));
1488 : }
1489 1735230 : fix_expo(z); return z;
1490 : }
1491 : GEN
1492 1735216 : qfr5_comp(GEN x, GEN y, struct qfr_data *S)
1493 1735216 : { return qfr5_red(qfr5_compraw(x, y), S); }
1494 : /* Not stack-clean */
1495 : GEN
1496 1009397 : qfr3_compraw(GEN x, GEN y)
1497 : {
1498 1009397 : GEN z = cgetg(4,t_VEC); qfb_comp(z,x,y);
1499 1009397 : return z;
1500 : }
1501 : GEN
1502 1009397 : qfr3_comp(GEN x, GEN y, struct qfr_data *S)
1503 1009397 : { return qfr3_red(qfr3_compraw(x,y), S); }
1504 :
1505 : /* m > 0. Not stack-clean */
1506 : static GEN
1507 7 : qfr5_powraw(GEN x, long m)
1508 : {
1509 7 : GEN y = NULL;
1510 14 : for (; m; m >>= 1)
1511 : {
1512 14 : if (m&1) y = y? qfr5_compraw(y,x): x;
1513 14 : if (m == 1) break;
1514 7 : x = qfr5_compraw(x,x);
1515 : }
1516 7 : return y;
1517 : }
1518 :
1519 : /* return x^n. Not stack-clean */
1520 : GEN
1521 21 : qfr5_pow(GEN x, GEN n, struct qfr_data *S)
1522 : {
1523 21 : GEN y = NULL;
1524 21 : long i, m, s = signe(n);
1525 21 : if (!s) return qfr5_1(S, lg(gel(x,5)));
1526 21 : if (s < 0) x = qfb_inv(x);
1527 42 : for (i=lgefint(n)-1; i>1; i--)
1528 : {
1529 21 : m = n[i];
1530 56 : for (; m; m>>=1)
1531 : {
1532 56 : if (m&1) y = y? qfr5_comp(y,x,S): x;
1533 56 : if (m == 1 && i == 2) break;
1534 35 : x = qfr5_comp(x,x,S);
1535 : }
1536 : }
1537 21 : return y;
1538 : }
1539 : /* m > 0; return x^m. Not stack-clean */
1540 : static GEN
1541 0 : qfr3_powraw(GEN x, long m)
1542 : {
1543 0 : GEN y = NULL;
1544 0 : for (; m; m>>=1)
1545 : {
1546 0 : if (m&1) y = y? qfr3_compraw(y,x): x;
1547 0 : if (m == 1) break;
1548 0 : x = qfr3_compraw(x,x);
1549 : }
1550 0 : return y;
1551 : }
1552 : /* return x^n. Not stack-clean */
1553 : GEN
1554 4557 : qfr3_pow(GEN x, GEN n, struct qfr_data *S)
1555 : {
1556 4557 : GEN y = NULL;
1557 4557 : long i, m, s = signe(n);
1558 4557 : if (!s) return qfr3_1(S);
1559 4557 : if (s < 0) x = qfb_inv(x);
1560 9130 : for (i=lgefint(n)-1; i>1; i--)
1561 : {
1562 4573 : m = n[i];
1563 5312 : for (; m; m>>=1)
1564 : {
1565 5292 : if (m&1) y = y? qfr3_comp(y,x,S): x;
1566 5292 : if (m == 1 && i == 2) break;
1567 739 : x = qfr3_comp(x,x,S);
1568 : }
1569 : }
1570 4557 : return y;
1571 : }
1572 :
1573 : static GEN
1574 7 : qfrinvraw(GEN x)
1575 : {
1576 7 : if (typ(x) == t_VEC) retmkvec2(qfbinv(gel(x,1)), negr(gel(x,2)));
1577 7 : return qfbinv(x);
1578 : }
1579 : static GEN
1580 14 : qfrpowraw(GEN x, long n)
1581 : {
1582 14 : struct qfr_data S = { NULL, NULL, NULL };
1583 14 : pari_sp av = avma;
1584 14 : if (n==1) return gcopy(x);
1585 14 : if (n==-1) return qfrinvraw(x);
1586 7 : if (typ(x)==t_QFB)
1587 : {
1588 0 : GEN D = qfb_disc(x);
1589 0 : if (!n) return qfr_1(x);
1590 0 : if (n < 0) { x = qfb_inv(x); n = -n; }
1591 0 : x = qfr3_powraw(x, n);
1592 0 : x = qfr3_to_qfr(x, D);
1593 : }
1594 : else
1595 : {
1596 7 : GEN d0 = gel(x,2);
1597 7 : x = gel(x,1);
1598 7 : if (!n) retmkvec2(qfr_1(x), real_0(precision(d0)));
1599 7 : if (n < 0) { x = qfb_inv(x); n = -n; }
1600 7 : x = qfr5_init(x, d0, &S);
1601 7 : if (labs(n) != 1) x = qfr5_powraw(x, n);
1602 7 : x = qfr5_to_qfr(x, S.D, mulrs(d0,n));
1603 : }
1604 7 : return gc_GEN(av, x);
1605 : }
1606 : static GEN
1607 112 : qfrpow(GEN x, GEN n)
1608 : {
1609 112 : struct qfr_data S = { NULL, NULL, NULL };
1610 112 : long s = signe(n);
1611 112 : pari_sp av = avma;
1612 112 : if (typ(x)==t_QFB)
1613 : {
1614 42 : if (!s) return qfr_1(x);
1615 28 : if (s < 0) x = qfb_inv(x);
1616 28 : x = qfr3_init(x, &S);
1617 28 : x = is_pm1(n)? qfr3_red(x, &S): qfr3_pow(x, n, &S);
1618 28 : x = qfr3_to_qfr(x, S.D);
1619 : }
1620 : else
1621 : {
1622 70 : GEN d0 = gel(x,2);
1623 70 : x = gel(x,1);
1624 70 : if (!s) retmkvec2(qfr_1(x), real_0(precision(d0)));
1625 49 : if (s < 0) x = qfb_inv(x);
1626 49 : x = qfr5_init(x, d0, &S);
1627 49 : x = is_pm1(n)? qfr5_red(x, &S): qfr5_pow(x, n, &S);
1628 49 : x = qfr5_to_qfr(x, S.D, mulri(d0,n));
1629 : }
1630 77 : return gc_GEN(av, x);
1631 : }
1632 : GEN
1633 21 : qfbpowraw(GEN x, long n)
1634 : {
1635 21 : GEN q = check_qfbext("qfbpowraw",x);
1636 21 : return qfb_is_qfi(q)? qfipowraw(x,n): qfrpowraw(x,n);
1637 : }
1638 : /* same discriminant, no distance, no checks */
1639 : GEN
1640 14418813 : qfbpow_i(GEN x, GEN n) { return qfb_is_qfi(x)? qfipow(x,n): qfrpow(x,n); }
1641 : GEN
1642 1411784 : qfbpow(GEN x, GEN n)
1643 : {
1644 1411784 : GEN q = check_qfbext("qfbpow",x);
1645 1411783 : return qfb_is_qfi(q)? qfipow(x,n): qfrpow(x,n);
1646 : }
1647 : GEN
1648 1313390 : qfbpows(GEN x, long n)
1649 : {
1650 1313390 : long N[] = { evaltyp(t_INT) | _evallg(3), 0, 0};
1651 1313390 : affsi(n, N); return qfbpow(x, N);
1652 : }
1653 :
1654 : /* Prime forms attached to prime ideals of degree 1 */
1655 :
1656 : /* assume x != 0 a t_INT, p > 0
1657 : * Return a t_QFB, but discriminant sign is not checked: can be used for
1658 : * real forms as well */
1659 : GEN
1660 12282095 : primeform_u(GEN x, ulong p)
1661 : {
1662 12282095 : GEN c, y = cgetg(5, t_QFB);
1663 12281945 : pari_sp av = avma;
1664 : ulong b;
1665 : long s;
1666 :
1667 12281945 : s = mod8(x); if (signe(x) < 0 && s) s = 8-s;
1668 : /* 2 or 3 mod 4 */
1669 12281949 : if (s & 2) pari_err_DOMAIN("primeform", "disc % 4", ">",gen_1, x);
1670 12282213 : if (p == 2) {
1671 3944237 : switch(s) {
1672 589486 : case 0: b = 0; break;
1673 3003209 : case 1: b = 1; break;
1674 351542 : case 4: b = 2; break;
1675 0 : default: pari_err_SQRTN("primeform", mkintmod(x,utoi(p)) );
1676 0 : b = 0; /* -Wall */
1677 : }
1678 3944237 : c = shifti(subsi(s,x), -3);
1679 : } else {
1680 8337976 : b = Fl_sqrt(umodiu(x,p), p);
1681 8339669 : if (b == ~0UL) pari_err_SQRTN("primeform", mkintmod(x,utoi(p)) );
1682 : /* mod(b) != mod2(x) ? */
1683 8340514 : if ((b ^ s) & 1) b = p - b;
1684 8340514 : c = diviuexact(shifti(subii(sqru(b), x), -2), p);
1685 : }
1686 12270809 : gel(y,3) = gc_INT(av, c);
1687 12277550 : gel(y,4) = icopy(x);
1688 12280680 : gel(y,2) = utoi(b);
1689 12280992 : gel(y,1) = utoipos(p); return y;
1690 : }
1691 :
1692 : /* special case: p = 1 return unit form */
1693 : GEN
1694 135445 : primeform(GEN x, GEN p)
1695 : {
1696 135445 : const char *f = "primeform";
1697 : pari_sp av;
1698 135445 : long s, sx = signe(x), sp = signe(p);
1699 : GEN y, b, absp;
1700 :
1701 135445 : if (typ(x) != t_INT) pari_err_TYPE(f,x);
1702 135445 : if (typ(p) != t_INT) pari_err_TYPE(f,p);
1703 135445 : if (!sp) pari_err_DOMAIN(f,"p","=",gen_0,p);
1704 135445 : if (!sx) pari_err_DOMAIN(f,"D","=",gen_0,x);
1705 135445 : if (lgefint(p) == 3)
1706 : {
1707 135431 : ulong pp = p[2];
1708 135431 : if (pp == 1) {
1709 17768 : if (sx < 0) {
1710 : long r;
1711 10936 : if (sp < 0) pari_err_IMPL("negative definite t_QFB");
1712 10936 : r = mod4(x);
1713 10936 : if (r && r != 3) pari_err_DOMAIN(f,"disc % 4",">", gen_1,x);
1714 10936 : return qfi_1_by_disc(x);
1715 : }
1716 6832 : y = qfr_1_by_disc(x);
1717 6832 : if (sp < 0) { gel(y,1) = gen_m1; togglesign(gel(y,3)); }
1718 6832 : return y;
1719 : }
1720 117663 : y = primeform_u(x, pp);
1721 117656 : if (sx < 0) {
1722 89957 : if (sp < 0) pari_err_IMPL("negative definite t_QFB");
1723 89957 : return y;
1724 : }
1725 27699 : if (sp < 0) { togglesign(gel(y,1)); togglesign(gel(y,3)); }
1726 27699 : return gcopy( qfr3_to_qfr(y, x) );
1727 : }
1728 14 : s = mod8(x);
1729 14 : if (sx < 0)
1730 : {
1731 7 : if (sp < 0) pari_err_IMPL("negative definite t_QFB");
1732 7 : if (s) s = 8-s;
1733 : }
1734 14 : y = cgetg(5, t_QFB);
1735 : /* 2 or 3 mod 4 */
1736 14 : if (s & 2) pari_err_DOMAIN(f, "disc % 4", ">",gen_1, x);
1737 14 : absp = absi_shallow(p); av = avma;
1738 14 : b = Fp_sqrt(x, absp); if (!b) pari_err_SQRTN(f, mkintmod(x,absp));
1739 14 : s &= 1; /* s = x mod 2 */
1740 : /* mod(b) != mod2(x) ? [Warning: we may have b == 0] */
1741 14 : if ((!signe(b) && s) || mod2(b) != s) b = gc_INT(av, subii(absp,b));
1742 :
1743 14 : av = avma;
1744 14 : gel(y,3) = gc_INT(av, diviiexact(shifti(subii(sqri(b), x), -2), p));
1745 14 : gel(y,4) = icopy(x);
1746 14 : gel(y,2) = b;
1747 14 : gel(y,1) = icopy(p);
1748 14 : return y;
1749 : }
1750 :
1751 : static GEN
1752 2620772 : normforms(GEN D, GEN fa)
1753 : {
1754 : long i, j, k, lB, aN, sa;
1755 : GEN a, L, V, B, N, N2;
1756 2620772 : int D_odd = mpodd(D);
1757 2620772 : a = typ(fa) == t_INT ? fa: typ(fa) == t_VEC? gel(fa,1): factorback(fa);
1758 2620772 : sa = signe(a);
1759 2620772 : if (sa==0 || (signe(D)<0 && sa<0)) return NULL;
1760 1203972 : V = D_odd? Zn_quad_roots(fa, gen_1, shifti(subsi(1, D), -2))
1761 2551766 : : Zn_quad_roots(fa, gen_0, negi(shifti(D, -2)));
1762 2551766 : if (!V) return NULL;
1763 511966 : N = gel(V,1); B = gel(V,2); lB = lg(B);
1764 511966 : N2 = shifti(N,1);
1765 511966 : aN = itou(diviiexact(a, N)); /* |a|/N */
1766 511966 : L = cgetg((lB-1)*aN+1, t_VEC);
1767 2360558 : for (k = 1, i = 1; i < lB; i++)
1768 : {
1769 1848592 : GEN b = shifti(gel(B,i), 1), c, C;
1770 1848589 : if (D_odd) b = addiu(b , 1);
1771 1848589 : c = diviiexact(shifti(subii(sqri(b), D), -2), a);
1772 1848585 : for (j = 0;; b = addii(b, N2))
1773 : {
1774 2216659 : gel(L, k++) = mkqfb(a, b, c, D);
1775 2216668 : if (++j == aN) break;
1776 368074 : C = addii(b, N); if (aN > 1) C = diviuexact(C, aN);
1777 368074 : c = sa > 0? addii(c, C): subii(c, C);
1778 : }
1779 : }
1780 511966 : return L;
1781 : }
1782 :
1783 : /* Let M and N in SL2(Z), return (N*M^-1)[,1] */
1784 : static GEN
1785 344318 : SL2_div_mul_e1(GEN N, GEN M)
1786 : {
1787 344318 : GEN b = gcoeff(M,2,1), d = gcoeff(M,2,2);
1788 344318 : GEN A = mulii(gcoeff(N,1,1), d), B = mulii(gcoeff(N,1,2), b);
1789 344309 : GEN C = mulii(gcoeff(N,2,1), d), D = mulii(gcoeff(N,2,2), b);
1790 344305 : retmkvec2(subii(A,B), subii(C,D));
1791 : }
1792 : static GEN
1793 1445669 : qfisolve_normform(GEN Q, GEN P)
1794 : {
1795 1445669 : GEN a = gel(Q,1), N = gel(Q,2);
1796 1445669 : GEN M, b = qfbredsl2_imag_basecase(P, &M);
1797 1445679 : if (!qfb_equal(a,b)) return NULL;
1798 102125 : return SL2_div_mul_e1(N,M);
1799 : }
1800 :
1801 : /* Test equality modulo GL2 of two reduced forms */
1802 : static int
1803 61068 : GL2_qfb_equal(GEN a, GEN b)
1804 : {
1805 61068 : return equalii(gel(a,1),gel(b,1))
1806 11361 : && absequalii(gel(a,2),gel(b,2))
1807 72429 : && equalii(gel(a,3),gel(b,3));
1808 : }
1809 :
1810 : /* Q(u,v) = p; if s < 0 return that solution; else the set of all solutions */
1811 : static GEN
1812 48027 : allsols(GEN Q, long s, GEN u, GEN v)
1813 : {
1814 48027 : GEN w = mkvec2(u, v), b;
1815 48019 : if (signe(v) < 0) { u = negi(u); v = negi(v); } /* normalize for v >= 0 */
1816 48019 : w = mkvec2(u, v); if (s < 0) return w;
1817 41373 : if (!s) return mkvec(w);
1818 38944 : b = gel(Q,2); /* sum of the 2 solutions (if they exist) is -bv / a */
1819 38944 : if (signe(b))
1820 : { /* something to check */
1821 : GEN r, t;
1822 13433 : t = dvmdii(mulii(b, v), gel(Q,1), &r);
1823 13433 : if (signe(r)) return mkvec(w);
1824 1820 : u = addii(u, t);
1825 : }
1826 27331 : return mkvec2(w, mkvec2(negi(u), v));
1827 : }
1828 : static GEN
1829 223039 : qfisolvep_all(GEN Q, GEN p, long all)
1830 : {
1831 223039 : GEN R, U, V, M, N, x, q, D = qfb_disc(Q);
1832 223038 : long s = kronecker(D, p);
1833 :
1834 223041 : if (s < 0) return NULL;
1835 126966 : if (!all) s = -1; /* to indicate we want a single solution */
1836 : /* Solutions iff a class of maximal ideal above p is the class of Q;
1837 : * Two solutions iff (s > 0 and the class has order > 2), else one */
1838 126966 : if (!signe(gel(Q,2)))
1839 : { /* if principal form, use faster cornacchia */
1840 43644 : GEN a = gel(Q,1), c = gel(Q,3);
1841 43644 : if (equali1(a))
1842 : {
1843 38171 : if (!cornacchia(c, p, &M,&N)) return NULL;
1844 33711 : return allsols(Q, s, M, N);
1845 : }
1846 5474 : if (equali1(c))
1847 : {
1848 5194 : if (!cornacchia(a, p, &M,&N)) return NULL;
1849 721 : return allsols(Q, s, N, M);
1850 : }
1851 : }
1852 83602 : R = qfbredsl2_imag_basecase(Q, &U);
1853 83601 : if (equali1(gel(R,1)))
1854 : { /* principal form */
1855 22533 : if (!signe(gel(R,2)))
1856 : {
1857 4396 : if (!cornacchia(gel(R,3), p, &M,&N)) return NULL;
1858 812 : x = mkvec2(M,N);
1859 : }
1860 : else
1861 : { /* x^2 + xy + ((1-D)/4)y^2 = p <==> (2x + y)^2 - D y^2 = 4p */
1862 18137 : if (!cornacchia2(negi(D), p, &M, &N)) return NULL;
1863 2331 : x = subii(M,N); if (mpodd(x)) return NULL;
1864 2331 : x = mkvec2(shifti(x,-1), N);
1865 : }
1866 3143 : x = ZM_ZC_mul(U, x); x[0] = evaltyp(t_VEC) | _evallg(3); /* transpose */
1867 3143 : return allsols(Q, s, gel(x,1), gel(x,2));
1868 : }
1869 61068 : q = qfbredsl2_imag_basecase(primeform(D, p), &V);
1870 61068 : if (!GL2_qfb_equal(R,q)) return NULL;
1871 10451 : if (signe(gel(R,2)) != signe(gel(q,2))) gcoeff(V,2,1) = negi(gcoeff(V,2,1));
1872 10451 : x = SL2_div_mul_e1(U,V); return allsols(Q, s, gel(x,1), gel(x,2));
1873 : }
1874 : GEN
1875 0 : qfisolvep(GEN Q, GEN p)
1876 : {
1877 0 : pari_sp av = avma;
1878 0 : GEN x = qfisolvep_all(Q, p, 0);
1879 0 : return x? gc_GEN(av, x): gc_const(av, gen_0);
1880 : }
1881 :
1882 : static GEN
1883 770126 : qfrsolve_normform(GEN N, GEN Ps, GEN rd)
1884 : {
1885 770126 : pari_sp av = avma, btop;
1886 770126 : GEN M = N, P = qfbredsl2_real_basecase(Ps, rd), Q = P;
1887 :
1888 770126 : btop = avma;
1889 : for(;;)
1890 : {
1891 5840681 : if (qfb_equal(gel(M,1), gel(P,1)))
1892 154084 : return gc_upto(av, SL2_div_mul_e1(gel(M,2),gel(P,2)));
1893 5686597 : if (qfb_equal(gel(N,1), gel(Q,1)))
1894 77658 : return gc_upto(av, SL2_div_mul_e1(gel(N,2),gel(Q,2)));
1895 5608939 : M = rhorealsl2(M, rd);
1896 5608939 : if (qfb_equal(gel(M,1), gel(N,1))) return gc_NULL(av);
1897 5201735 : Q = rhorealsl2(Q, rd);
1898 5201735 : if (qfb_equal(gel(P,1), gel(Q,1))) return gc_NULL(av);
1899 5070555 : if (gc_needed(btop, 1)) (void)gc_all(btop, 2, &M, &Q);
1900 : }
1901 : }
1902 :
1903 : GEN
1904 0 : qfrsolvep(GEN Q, GEN p)
1905 : {
1906 0 : pari_sp av = avma;
1907 0 : GEN N, x, rd, d = qfb_disc(Q);
1908 :
1909 0 : if (kronecker(d, p) < 0) return gc_const(av, gen_0);
1910 0 : rd = sqrti(d);
1911 0 : N = qfbredsl2_real(Q, rd);
1912 0 : x = qfrsolve_normform(N, primeform(d, p), rd);
1913 0 : return x? gc_upto(av, x): gc_const(av, gen_0);
1914 : }
1915 :
1916 : static GEN
1917 1862934 : known_prime(GEN v)
1918 : {
1919 1862934 : GEN p, e, fa = check_arith_all(v, "qfbsolve");
1920 1862935 : if (!fa) return BPSW_psp(v)? v: NULL;
1921 42075 : if (lg(gel(fa,1)) != 2) return NULL;
1922 29349 : p = gcoeff(fa,1,1);
1923 29349 : e = gcoeff(fa,1,2);
1924 29349 : return (equali1(e) && !is_pm1(p) && signe(p) > 0)? p: NULL;
1925 : }
1926 : static GEN
1927 2215795 : qfsolve_normform(GEN Q, GEN f, GEN rd)
1928 2215795 : { return rd? qfrsolve_normform(Q, f, rd): qfisolve_normform(Q, f); }
1929 : static GEN
1930 2843810 : qfbsolve_primitive_i(GEN Q, GEN rd, GEN *Qr, GEN fa, long all)
1931 : {
1932 : GEN x, W, F, p;
1933 : long i, j, l;
1934 2843810 : if (!rd && (p = known_prime(fa))) return qfisolvep_all(Q, p, all);
1935 2620760 : F = normforms(qfb_disc(Q), fa);
1936 2620772 : if (!F) return NULL;
1937 511966 : if (!*Qr) *Qr = qfbredsl2(Q, rd);
1938 511966 : l = lg(F); W = all? cgetg(l, t_VEC): NULL;
1939 2727251 : for (j = i = 1; i < l; i++)
1940 2215795 : if ((x = qfsolve_normform(*Qr, gel(F,i), rd)))
1941 : {
1942 333861 : if (!all) return x;
1943 333350 : gel(W,j++) = x;
1944 : }
1945 511456 : if (j == 1) return NULL;
1946 127457 : setlg(W,j); return lexsort(W);
1947 : }
1948 :
1949 : static GEN
1950 2838515 : qfb_initrd(GEN Q) { GEN d = qfb_disc(Q); return signe(d) > 0? sqrti(d): NULL; }
1951 : static GEN
1952 2828289 : qfbsolve_primitive(GEN Q, GEN fa, long all)
1953 : {
1954 2828289 : GEN x, Qr = NULL, rdQ = qfb_initrd(Q);
1955 2828282 : x = qfbsolve_primitive_i(Q, rdQ, &Qr, fa, all);
1956 2828289 : if (!x) return cgetg(1, t_VEC);
1957 174750 : return x;
1958 : }
1959 :
1960 : /* f / g^2 */
1961 : static GEN
1962 5299 : famat_divsqr(GEN f, GEN g)
1963 5299 : { return famat_reduce(famat_div_shallow(f, famat_pows_shallow(g,2))); }
1964 : static GEN
1965 10227 : qfbsolve_all(GEN Q, GEN n, long all)
1966 : {
1967 10227 : GEN W, Qr = NULL, fa = factorint(n, 0), rdQ = qfb_initrd(Q);
1968 10227 : GEN D = divisors_factored(mkmat2(gel(fa,1), gshift(gel(fa,2),-1)));
1969 10227 : long i, j, l = lg(D);
1970 10227 : W = all? cgetg(l, t_VEC): NULL;
1971 25151 : for (i = j = 1; i < l; i++)
1972 : {
1973 15526 : GEN w, d = gel(D,i), FA = i == 1? fa: famat_divsqr(fa, gel(d,2));
1974 15526 : if ((w = qfbsolve_primitive_i(Q, rdQ, &Qr, FA, all)))
1975 : {
1976 1218 : if (i != 1) w = RgV_Rg_mul(w, gel(d,1));
1977 1218 : if (!all) return w;
1978 616 : gel(W,j++) = w;
1979 : }
1980 : }
1981 9625 : if (j == 1) return cgetg(1, t_VEC);
1982 525 : setlg(W,j); return lexsort(shallowconcat1(W));
1983 : }
1984 :
1985 : GEN
1986 2838518 : qfbsolve(GEN Q, GEN n, long fl)
1987 : {
1988 2838518 : pari_sp av = avma;
1989 2838518 : if (typ(Q) != t_QFB) pari_err_TYPE("qfbsolve",Q);
1990 2838518 : if (fl < 0 || fl > 3) pari_err_FLAG("qfbsolve");
1991 5666797 : return gc_GEN(av, (fl & 2)? qfbsolve_all(Q, n, fl & 1)
1992 2828284 : : qfbsolve_primitive(Q, n, fl & 1));
1993 : }
1994 :
1995 : /* 1 if there exists x,y such that x^2 + dy^2 = p, 0 otherwise;
1996 : * Assume d > 0 and p is prime */
1997 : long
1998 55244 : cornacchia(GEN d, GEN p, GEN *px, GEN *py)
1999 : {
2000 55244 : pari_sp av = avma;
2001 : GEN b, c, r;
2002 :
2003 55244 : *px = *py = gen_0;
2004 55244 : b = subii(p, d);
2005 55205 : if (signe(b) < 0) return gc_long(av,0);
2006 54993 : if (signe(b) == 0) { *py = gen_1; return gc_long(av,1); }
2007 54986 : b = Fp_sqrt(b, p); /* sqrt(-d) */
2008 55049 : if (!b) return gc_long(av,0);
2009 51318 : b = gmael(halfgcdii(p, b), 2, 2);
2010 51361 : c = dvmdii(subii(p, sqri(b)), d, &r);
2011 51234 : if (r != gen_0 || !Z_issquareall(c, &c)) return gc_long(av,0);
2012 35504 : set_avma(av);
2013 35499 : *px = icopy(b);
2014 35479 : *py = icopy(c); return 1;
2015 : }
2016 :
2017 : static GEN
2018 1416256 : lastqi(GEN Q)
2019 : {
2020 1416256 : GEN s = gcoeff(Q,1,1), q = gcoeff(Q,1,2), p = absi_shallow(gcoeff(Q,2,2));
2021 1416250 : if (!signe(q)) return gen_0;
2022 1416061 : if (!signe(s)) return p;
2023 1409677 : if (is_pm1(q)) return subiu(p,1);
2024 1409679 : return divii(p, absi_shallow(q));
2025 : }
2026 :
2027 : static long
2028 1416254 : cornacchia2_i(long av, GEN d, GEN p, GEN b, GEN px4, GEN *px, GEN *py)
2029 : {
2030 : GEN M, Q, V, c, r, b2;
2031 1416254 : if (!signe(b)) { /* d = p,2p,3p,4p */
2032 14 : set_avma(av);
2033 14 : if (absequalii(d, px4)){ *py = gen_1; return 1; }
2034 14 : if (absequalii(d, p)) { *py = gen_2; return 1; }
2035 0 : return 0;
2036 : }
2037 1416240 : if (mod2(b) != mod2(d)) b = subii(p,b);
2038 1416227 : M = halfgcdii(shifti(p,1), b); Q = gel(M,1); V = gel(M, 2);
2039 1416256 : b = addii(mulii(gel(V,1), lastqi(Q)), gel(V,2));
2040 1416183 : b2 = sqri(b);
2041 1416168 : if (cmpii(b2,px4) > 0)
2042 : {
2043 1408002 : b = gel(V,1); b2 = sqri(b);
2044 1408013 : if (cmpii(b2,px4) > 0) { b = gel(V,2); b2 = sqri(b); }
2045 : }
2046 1416176 : c = dvmdii(subii(px4, b2), d, &r);
2047 1416179 : if (r != gen_0 || !Z_issquareall(c, &c)) return gc_long(av,0);
2048 1375656 : set_avma(av);
2049 1375651 : *px = icopy(b);
2050 1375642 : *py = icopy(c); return 1;
2051 : }
2052 :
2053 : /* 1 if there exists x,y such that x^2 + dy^2 = 4p, 0 otherwise;
2054 : * Assume d > 0 is congruent to 0 or 3 mod 4 and p is prime */
2055 : long
2056 1381726 : cornacchia2(GEN d, GEN p, GEN *px, GEN *py)
2057 : {
2058 1381726 : pari_sp av = avma;
2059 1381726 : GEN b, p4 = shifti(p,2);
2060 :
2061 1381697 : *px = *py = gen_0;
2062 1381697 : if (abscmpii(p4, d) < 0) return gc_long(av,0);
2063 1380874 : if (absequaliu(p, 2))
2064 : {
2065 7 : set_avma(av);
2066 7 : switch (itou_or_0(d)) {
2067 0 : case 4: *px = gen_2; break;
2068 0 : case 7: *px = gen_1; break;
2069 7 : default: return 0;
2070 : }
2071 0 : *py = gen_1; return 1;
2072 : }
2073 1380866 : b = Fp_sqrt(negi(d), p);
2074 1380925 : if (!b) return gc_long(av,0);
2075 1380841 : return cornacchia2_i(av, d, p, b, p4, px, py);
2076 : }
2077 :
2078 : /* 1 if there exists x,y such that x^2 + dy^2 = 4p [p prime], 0 otherwise */
2079 : long
2080 35413 : cornacchia2_sqrt(GEN d, GEN p, GEN b, GEN *px, GEN *py)
2081 : {
2082 35413 : pari_sp av = avma;
2083 35413 : GEN p4 = shifti(p,2);
2084 35413 : *px = *py = gen_0;
2085 35413 : if (abscmpii(p4, d) < 0) return gc_long(av,0);
2086 35413 : return cornacchia2_i(av, d, p, b, p4, px, py);
2087 : }
2088 :
2089 : GEN
2090 7630 : qfbcornacchia(GEN d, GEN p)
2091 : {
2092 7630 : pari_sp av = avma;
2093 : GEN x, y;
2094 7630 : if (typ(d) != t_INT || signe(d) <= 0) pari_err_TYPE("qfbcornacchia", d);
2095 7630 : if (typ(p) != t_INT || cmpiu(p, 2) < 0) pari_err_TYPE("qfbcornacchia", p);
2096 7630 : if (mod4(p)? cornacchia(d, p, &x, &y): cornacchia2(d, shifti(p, -2), &x, &y))
2097 287 : return gc_GEN(av, mkvec2(x, y));
2098 7343 : retgc_const(av, cgetg(1, t_VEC));
2099 : }
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