Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - basemath - Fle.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.1 lcov report (development 28695-49bb1ac00f) Lines: 427 456 93.6 %
Date: 2023-09-24 07:47:42 Functions: 57 64 89.1 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2014  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             : 
      15             : #include "pari.h"
      16             : #include "paripriv.h"
      17             : 
      18             : /* Not so fast arithmetic with points over elliptic curves over Fl */
      19             : 
      20             : /***********************************************************************/
      21             : /**                              Flj                                  **/
      22             : /***********************************************************************/
      23             : /* Jacobian coordinates: we represent a projective point (x:y:z) on E by
      24             :  * [z*x, z^2*y, z]. Not the fastest representation available for the problem,
      25             :  * but easy to implement and up to 60% faster than the school-book method. */
      26             : 
      27             : /* Cost: 1M + 8S + 1*a + 10add + 1*8 + 2*2 + 1*3.
      28             :  * http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl */
      29             : INLINE void
      30    25092162 : Flj_dbl_indir_pre(GEN P, GEN Q, ulong a4, ulong p, ulong pi)
      31             : {
      32             :   ulong X1, Y1, Z1;
      33             :   ulong XX, YY, YYYY, ZZ, S, M, T;
      34             : 
      35    25092162 :   X1 = P[1]; Y1 = P[2]; Z1 = P[3];
      36             : 
      37    25092162 :   if (Z1 == 0) { Q[1] = X1; Q[2] = Y1; Q[3] = Z1; return; }
      38             : 
      39    24910445 :   XX = Fl_sqr_pre(X1, p, pi);
      40    24913178 :   YY = Fl_sqr_pre(Y1, p, pi);
      41    24904322 :   YYYY = Fl_sqr_pre(YY, p, pi);
      42    24903319 :   ZZ = Fl_sqr_pre(Z1, p, pi);
      43    24901086 :   S = Fl_double(Fl_sub(Fl_sqr_pre(Fl_add(X1, YY, p), p, pi),
      44             :                        Fl_add(XX, YYYY, p), p), p);
      45    24894779 :   M = Fl_addmul_pre(Fl_triple(XX, p), a4, Fl_sqr_pre(ZZ, p, pi), p, pi);
      46    24913624 :   T = Fl_sub(Fl_sqr_pre(M, p, pi), Fl_double(S, p), p);
      47    24905445 :   Q[1] = T;
      48    24905445 :   Q[2] = Fl_sub(Fl_mul_pre(M, Fl_sub(S, T, p), p, pi),
      49             :                 Fl_double(Fl_double(Fl_double(YYYY, p), p), p), p);
      50    24899877 :   Q[3] = Fl_sub(Fl_sqr_pre(Fl_add(Y1, Z1, p), p, pi),
      51             :                 Fl_add(YY, ZZ, p), p);
      52             : }
      53             : 
      54             : INLINE void
      55    21185362 : Flj_dbl_pre_inplace(GEN P, ulong a4, ulong p, ulong pi)
      56             : {
      57    21185362 :   Flj_dbl_indir_pre(P, P, a4, p, pi);
      58    21186712 : }
      59             : 
      60             : GEN
      61     3905475 : Flj_dbl_pre(GEN P, ulong a4, ulong p, ulong pi)
      62             : {
      63     3905475 :   GEN Q = cgetg(4, t_VECSMALL);
      64     3905196 :   Flj_dbl_indir_pre(P, Q, a4, p, pi);
      65     3905031 :   return Q;
      66             : }
      67             : 
      68             : /* Cost: 11M + 5S + 9add + 4*2.
      69             :  * http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl */
      70             : INLINE void
      71     7597743 : Flj_add_indir_pre(GEN P, GEN Q, GEN R, ulong a4, ulong p, ulong pi)
      72             : {
      73             :   ulong X1, Y1, Z1, X2, Y2, Z2;
      74             :   ulong Z1Z1, Z2Z2, U1, U2, S1, S2, H, I, J, r, V, W;
      75     7597743 :   X1 = P[1]; Y1 = P[2]; Z1 = P[3];
      76     7597743 :   X2 = Q[1]; Y2 = Q[2]; Z2 = Q[3];
      77             : 
      78     7597743 :   if (Z2 == 0) { R[1] = X1; R[2] = Y1; R[3] = Z1; return; }
      79     7597432 :   if (Z1 == 0) { R[1] = X2; R[2] = Y2; R[3] = Z2; return; }
      80             : 
      81     7581071 :   Z1Z1 = Fl_sqr_pre(Z1, p, pi);
      82     7581508 :   Z2Z2 = Fl_sqr_pre(Z2, p, pi);
      83     7580870 :   U1 = Fl_mul_pre(X1, Z2Z2, p, pi);
      84     7580989 :   U2 = Fl_mul_pre(X2, Z1Z1, p, pi);
      85     7581128 :   S1 = Fl_mul_pre(Y1, Fl_mul_pre(Z2, Z2Z2, p, pi), p, pi);
      86     7581199 :   S2 = Fl_mul_pre(Y2, Fl_mul_pre(Z1, Z1Z1, p, pi), p, pi);
      87     7581359 :   H = Fl_sub(U2, U1, p);
      88     7580867 :   r = Fl_double(Fl_sub(S2, S1, p), p);
      89             : 
      90     7580937 :   if (H == 0) {
      91      598493 :     if (r == 0) Flj_dbl_indir_pre(P, R, a4, p, pi); /* P = Q */
      92      593746 :     else { R[1] = R[2] = 1; R[3] = 0; } /* P = -Q */
      93      598492 :     return;
      94             :   }
      95     6982444 :   I = Fl_sqr_pre(Fl_double(H, p), p, pi);
      96     6983358 :   J = Fl_mul_pre(H, I, p, pi);
      97     6983282 :   V = Fl_mul_pre(U1, I, p, pi);
      98     6983198 :   W = Fl_sub(Fl_sqr_pre(r, p, pi), Fl_add(J, Fl_double(V, p), p), p);
      99     6983033 :   R[1] = W;
     100     6983033 :   R[2] = Fl_sub(Fl_mul_pre(r, Fl_sub(V, W, p), p, pi),
     101             :                 Fl_double(Fl_mul_pre(S1, J, p, pi), p), p);
     102     6983172 :   R[3] = Fl_mul_pre(Fl_sub(Fl_sqr_pre(Fl_add(Z1, Z2, p), p, pi),
     103             :                            Fl_add(Z1Z1, Z2Z2, p), p), H, p, pi);
     104             : }
     105             : 
     106             : INLINE void
     107     7506314 : Flj_add_pre_inplace(GEN P, GEN Q, ulong a4, ulong p, ulong pi)
     108     7506314 : { Flj_add_indir_pre(P, Q, P, a4, p, pi); }
     109             : 
     110             : GEN
     111       90573 : Flj_add_pre(GEN P, GEN Q, ulong a4, ulong p, ulong pi)
     112             : {
     113       90573 :   GEN R = cgetg(4, t_VECSMALL);
     114       90571 :   Flj_add_indir_pre(P, Q, R, a4, p, pi);
     115       90577 :   return R;
     116             : }
     117             : 
     118             : GEN
     119     2440139 : Flj_neg(GEN Q, ulong p)
     120     2440139 : { return mkvecsmall3(Q[1], Fl_neg(Q[2], p), Q[3]); }
     121             : 
     122             : typedef struct {
     123             :   ulong pbits, nbits;  /* Positive bits and negative bits */
     124             :   ulong lnzb; /* Leading nonzero bit */
     125             : } naf_t;
     126             : 
     127             : /* Return the signed binary representation (i.e. the Non-Adjacent Form
     128             :  * in base 2) of a; a = x.pbits - x.nbits (+ 2^BILif < 0; this
     129             :  * exceptional case can happen if a > 2^(BIL-1)) */
     130             : static void
     131     4438496 : naf_repr(naf_t *x, ulong a)
     132             : {
     133     4438496 :   ulong pbits = 0, nbits = 0, c0 = 0, c1, a0;
     134             :   long t, i;
     135             : 
     136    44674746 :   for (i = 0; a; a >>= 1, ++i) {
     137    40236250 :     a0 = a & 1;
     138    40236250 :     c1 = (c0 + a0 + ((a & 2) >> 1)) >> 1;
     139    40236250 :     t = c0 + a0 - (c1 << 1);
     140    40236250 :     if (t < 0)
     141     6249765 :       nbits |= (1UL << i);
     142    33986485 :     else if (t > 0)
     143     7100313 :       pbits |= (1UL << i);
     144    40236250 :     c0 = c1;
     145             :   }
     146     4438496 :   c1 = c0 >> 1;
     147     4438496 :   t = c0 - (c1 << 1);
     148             :   /* since a >= 0, we have t >= 0; if i = BIL, pbits (virtually) overflows;
     149             :    * that leading overflowed bit is implied and not recorded in pbits */
     150     4438496 :   if (t > 0 && i != BITS_IN_LONG) pbits |= (1UL << i);
     151     4438496 :   x->pbits = pbits;
     152     4438496 :   x->nbits = nbits;
     153     4438496 :   x->lnzb = t? i-2: i-3;
     154     4438496 : }
     155             : 
     156             : /* Standard left-to-right signed double-and-add to compute [n]P. */
     157             : static GEN
     158     3970125 : Flj_mulu_pre_naf(GEN P, ulong n, ulong a4, ulong p, ulong pi, const naf_t *x)
     159             : {
     160             :   GEN R, Pi;
     161             :   ulong pbits, nbits;
     162             :   ulong m;
     163             : 
     164     3970125 :   if (n == 0) return mkvecsmall3(1, 1, 0);
     165     3970125 :   if (n == 1) return Flv_copy(P);
     166             : 
     167     3905639 :   R = Flj_dbl_pre(P, a4, p, pi);
     168     3904934 :   if (n == 2) return R;
     169             : 
     170     3064662 :   pbits = x->pbits;
     171     3064662 :   nbits = x->nbits;
     172     3064662 :   Pi = nbits? Flj_neg(P, p): NULL;
     173     3065196 :   m = (1UL << x->lnzb);
     174    24251435 :   for ( ; m; m >>= 1) {
     175    21186543 :     Flj_dbl_pre_inplace(R, a4, p, pi);
     176    21185810 :     if (m & pbits)
     177     3292255 :       Flj_add_pre_inplace(R, P, a4, p, pi);
     178    17893555 :     else if (m & nbits)
     179     4216714 :       Flj_add_pre_inplace(R, Pi, a4, p, pi);
     180             :   }
     181     3064892 :   return gc_const((pari_sp)R, R);
     182             : }
     183             : 
     184             : GEN
     185     2736527 : Flj_mulu_pre(GEN P, ulong n, ulong a4, ulong p, ulong pi)
     186             : {
     187     2736527 :   naf_t x; naf_repr(&x, n);
     188     2736919 :   return Flj_mulu_pre_naf(P, n, a4, p, pi, &x);
     189             : }
     190             : 
     191             : struct _Flj { ulong a4, p, pi; };
     192             : 
     193             : static GEN
     194       90573 : _Flj_add(void *E, GEN P, GEN Q)
     195             : {
     196       90573 :   struct _Flj *ell=(struct _Flj *) E;
     197       90573 :   return Flj_add_pre(P, Q, ell->a4, ell->p, ell->pi);
     198             : }
     199             : 
     200             : static GEN
     201      106191 : _Flj_mul(void *E, GEN P, GEN n)
     202             : {
     203      106191 :   struct _Flj *ell = (struct _Flj *) E;
     204      106191 :   long s = signe(n);
     205             :   GEN Q;
     206      106191 :   if (s==0) return mkvecsmall3(1, 1, 0);
     207      106191 :   Q = Flj_mulu_pre(P, itou(n), ell->a4, ell->p, ell->pi);
     208      106191 :   return s>0 ? Q : Flj_neg(Q, ell->p);
     209             : }
     210             : static GEN
     211           0 : _Flj_one(void *E)
     212           0 : { (void) E; return mkvecsmall3(1, 1, 0); }
     213             : 
     214             : GEN
     215       15617 : FljV_factorback_pre(GEN P, GEN L, ulong a4, ulong p, ulong pi)
     216             : {
     217             :   struct _Flj E;
     218       15617 :   E.a4 = a4; E.p = p; E.pi = pi;
     219       15617 :   return gen_factorback(P, L, (void*)&E, &_Flj_add, &_Flj_mul, &_Flj_one);
     220             : }
     221             : 
     222             : ulong
     223      291573 : Flj_order_ufact(GEN P, ulong n, GEN fa, ulong a4, ulong p, ulong pi)
     224             : {
     225      291573 :   pari_sp av = avma;
     226      291573 :   GEN T = gel(fa,1), E = gel(fa,2);
     227      291573 :   long i, l = lg(T);
     228      291573 :   ulong res = 1;
     229             : 
     230      822275 :   for (i = 1; i < l; i++, set_avma(av))
     231             :   {
     232      657330 :     ulong j, t = T[i], e = E[i];
     233      657330 :     GEN b = P;
     234      657330 :     naf_t x; naf_repr(&x, t);
     235      657396 :     if (l != 2) b = Flj_mulu_pre(b, n / upowuu(t,e), a4, p, pi);
     236     1890716 :     for (j = 0; j < e && b[3]; j++) b = Flj_mulu_pre_naf(b, t, a4, p, pi, &x);
     237      657274 :     if (b[3]) return 0;
     238      530635 :     res *= upowuu(t, j);
     239             :   }
     240      164945 :   return res;
     241             : }
     242             : 
     243             : GEN
     244     1564179 : Fle_to_Flj(GEN P)
     245     3128203 : { return ell_is_inf(P) ? mkvecsmall3(1UL, 1UL, 0UL):
     246     1564102 :                          mkvecsmall3(P[1], P[2], 1UL); }
     247             : 
     248             : GEN
     249       68257 : Flj_to_Fle(GEN P, ulong p)
     250             : {
     251       68257 :   if (P[3] == 0) return ellinf();
     252             :   else
     253             :   {
     254       67501 :     ulong Z = Fl_inv(P[3], p);
     255       67501 :     ulong Z2 = Fl_sqr(Z, p);
     256       67501 :     ulong X3 = Fl_mul(P[1], Z2, p);
     257       67501 :     ulong Y3 = Fl_mul(P[2], Fl_mul(Z, Z2, p), p);
     258       67501 :     return mkvecsmall2(X3, Y3);
     259             :   }
     260             : }
     261             : 
     262             : GEN
     263     1579470 : Flj_to_Fle_pre(GEN P, ulong p, ulong pi)
     264             : {
     265     1579470 :   if (P[3] == 0) return ellinf();
     266             :   else
     267             :   {
     268     1393323 :     ulong Z = Fl_inv(P[3], p);
     269     1393746 :     ulong Z2 = Fl_sqr_pre(Z, p, pi);
     270     1393580 :     ulong X3 = Fl_mul_pre(P[1], Z2, p, pi);
     271     1393602 :     ulong Y3 = Fl_mul_pre(P[2], Fl_mul_pre(Z, Z2, p, pi), p, pi);
     272     1393626 :     return mkvecsmall2(X3, Y3);
     273             :   }
     274             : }
     275             : 
     276             : INLINE void
     277     8270609 : random_Fle_pre_indir(ulong a4, ulong a6, ulong p, ulong pi,
     278             :                      ulong *pt_x, ulong *pt_y)
     279             : {
     280             :   ulong x, x2, y, rhs;
     281             :   do
     282             :   {
     283     8270609 :     x   = random_Fl(p); /*  x^3+a4*x+a6 = x*(x^2+a4)+a6  */
     284     8279642 :     x2  = Fl_sqr_pre(x, p, pi);
     285     8257800 :     rhs = Fl_addmul_pre(a6, x, Fl_add(x2, a4, p), p, pi);
     286     8257936 :   } while ((!rhs && !Fl_add(Fl_triple(x2,p),a4,p)) || krouu(rhs, p) < 0);
     287     4139756 :   y = Fl_sqrt_pre(rhs, p, pi);
     288     4139373 :   *pt_x = x; *pt_y = y;
     289     4139373 : }
     290             : 
     291             : GEN
     292      490932 : random_Flj_pre(ulong a4, ulong a6, ulong p, ulong pi)
     293             : {
     294             :   ulong x, y;
     295      490932 :   random_Fle_pre_indir(a4, a6, p, pi, &x, &y);
     296      490907 :   return mkvecsmall3(x, y, 1);
     297             : }
     298             : 
     299             : GEN
     300      157800 : Flj_changepointinv_pre(GEN P, GEN ch, ulong p, ulong pi)
     301             : {
     302             :   ulong c, u, r, s, t, u2, u3, z2;
     303      157800 :   ulong x  = uel(P,1), y = uel(P,2), z = uel(P,3);
     304             :   GEN w;
     305      157800 :   if (z == 0) return Flv_copy(P);
     306      157772 :   u = ch[1]; r = ch[2];
     307      157772 :   s = ch[3]; t = ch[4];
     308      157772 :   u2 = Fl_sqr_pre(u, p, pi); u3 = Fl_mul_pre(u, u2, p, pi);
     309      157769 :   c = Fl_mul_pre(u2, x, p, pi);
     310      157770 :   z2 = Fl_sqr_pre(z, p, pi);
     311      157769 :   w = cgetg(4, t_VECSMALL);
     312      157767 :   uel(w,1) = Fl_add(c, Fl_mul_pre(r, z2, p, pi), p);
     313      157772 :   uel(w,2) = Fl_add(Fl_mul_pre(u3 ,y, p, pi),
     314             :                     Fl_mul_pre(z, Fl_add(Fl_mul_pre(s,c,p,pi),
     315             :                                          Fl_mul_pre(z2,t,p,pi), p), p, pi), p);
     316      157774 :   uel(w,3) = z;
     317      157774 :   return w;
     318             : }
     319             : 
     320             : /***********************************************************************/
     321             : /**                              Fle                                  **/
     322             : /***********************************************************************/
     323             : GEN
     324       16336 : Fle_changepoint(GEN P, GEN ch, ulong p)
     325             : {
     326             :   ulong c, u, r, s, t, v, v2, v3;
     327             :   GEN z;
     328       16336 :   if (ell_is_inf(P)) return ellinf();
     329       16336 :   u = ch[1]; r = ch[2];
     330       16336 :   s = ch[3]; t = ch[4];
     331       16336 :   v = Fl_inv(u, p); v2 = Fl_sqr(v,p); v3 = Fl_mul(v,v2,p);
     332       16336 :   c = Fl_sub(uel(P,1),r,p);
     333       16336 :   z = cgetg(3,t_VECSMALL);
     334       16336 :   z[1] = Fl_mul(v2, c, p);
     335       16336 :   z[2] = Fl_mul(v3, Fl_sub(uel(P,2), Fl_add(Fl_mul(s,c, p),t, p),p),p);
     336       16336 :   return z;
     337             : }
     338             : 
     339             : GEN
     340       77361 : Fle_changepointinv(GEN P, GEN ch, ulong p)
     341             : {
     342             :   ulong c, u, r, s, t, u2, u3;
     343             :   GEN z;
     344       77361 :   if (ell_is_inf(P)) return ellinf();
     345       76605 :   u = ch[1]; r = ch[2];
     346       76605 :   s = ch[3]; t = ch[4];
     347       76605 :   u2 = Fl_sqr(u, p); u3 = Fl_mul(u,u2,p);
     348       76605 :   c = Fl_mul(u2,uel(P,1), p);
     349       76605 :   z = cgetg(3, t_VECSMALL);
     350       76605 :   z[1] = Fl_add(c,r,p);
     351       76605 :   z[2] = Fl_add(Fl_mul(u3,uel(P,2),p), Fl_add(Fl_mul(s,c,p), t, p), p);
     352       76605 :   return z;
     353             : }
     354             : static GEN
     355      589933 : Fle_dbl_slope(GEN P, ulong a4, ulong p, ulong *slope)
     356             : {
     357             :   ulong x, y, Qx, Qy;
     358      589933 :   if (ell_is_inf(P) || !P[2]) return ellinf();
     359      519722 :   x = P[1]; y = P[2];
     360      519722 :   *slope = Fl_div(Fl_add(Fl_triple(Fl_sqr(x,p), p), a4, p),
     361             :                   Fl_double(y, p), p);
     362      519737 :   Qx = Fl_sub(Fl_sqr(*slope, p), Fl_double(x, p), p);
     363      519725 :   Qy = Fl_sub(Fl_mul(*slope, Fl_sub(x, Qx, p), p), y, p);
     364      519723 :   return mkvecsmall2(Qx, Qy);
     365             : }
     366             : 
     367             : GEN
     368      431294 : Fle_dbl(GEN P, ulong a4, ulong p)
     369             : {
     370             :   ulong slope;
     371      431294 :   return Fle_dbl_slope(P,a4,p,&slope);
     372             : }
     373             : 
     374             : static GEN
     375     1195522 : Fle_add_slope(GEN P, GEN Q, ulong a4, ulong p, ulong *slope)
     376             : {
     377             :   ulong Px, Py, Qx, Qy, Rx, Ry;
     378     1195522 :   if (ell_is_inf(P)) return Q;
     379     1195523 :   if (ell_is_inf(Q)) return P;
     380     1195519 :   Px = P[1]; Py = P[2];
     381     1195519 :   Qx = Q[1]; Qy = Q[2];
     382     1195519 :   if (Px==Qx) return Py==Qy ? Fle_dbl_slope(P, a4, p, slope): ellinf();
     383     1064239 :   *slope = Fl_div(Fl_sub(Py, Qy, p), Fl_sub(Px, Qx, p), p);
     384     1064353 :   Rx = Fl_sub(Fl_sub(Fl_sqr(*slope, p), Px, p), Qx, p);
     385     1064318 :   Ry = Fl_sub(Fl_mul(*slope, Fl_sub(Px, Rx, p), p), Py, p);
     386     1064319 :   return mkvecsmall2(Rx, Ry);
     387             : }
     388             : 
     389             : GEN
     390     1190596 : Fle_add(GEN P, GEN Q, ulong a4, ulong p)
     391             : {
     392             :   ulong slope;
     393     1190596 :   return Fle_add_slope(P,Q,a4,p,&slope);
     394             : }
     395             : 
     396             : static GEN
     397      130886 : Fle_neg(GEN P, ulong p)
     398             : {
     399      130886 :   if (ell_is_inf(P)) return P;
     400      130886 :   return mkvecsmall2(P[1], Fl_neg(P[2], p));
     401             : }
     402             : 
     403             : GEN
     404           0 : Fle_sub(GEN P, GEN Q, ulong a4, ulong p)
     405             : {
     406           0 :   pari_sp av = avma;
     407             :   ulong slope;
     408           0 :   return gerepileupto(av, Fle_add_slope(P, Fle_neg(Q, p), a4, p, &slope));
     409             : }
     410             : 
     411             : struct _Fle { ulong a4, a6, p; };
     412             : 
     413             : static GEN
     414           0 : _Fle_dbl(void *E, GEN P)
     415             : {
     416           0 :   struct _Fle *ell = (struct _Fle *) E;
     417           0 :   return Fle_dbl(P, ell->a4, ell->p);
     418             : }
     419             : 
     420             : static GEN
     421      366870 : _Fle_add(void *E, GEN P, GEN Q)
     422             : {
     423      366870 :   struct _Fle *ell=(struct _Fle *) E;
     424      366870 :   return Fle_add(P, Q, ell->a4, ell->p);
     425             : }
     426             : 
     427             : GEN
     428     2001024 : Fle_mulu(GEN P, ulong n, ulong a4, ulong p)
     429             : {
     430             :   ulong pi;
     431     2001024 :   if (!n || ell_is_inf(P)) return ellinf();
     432     2000938 :   if (n==1) return zv_copy(P);
     433     1988212 :   if (n==2) return Fle_dbl(P, a4, p);
     434     1564043 :   pi = get_Fl_red(p);
     435     1564195 :   return Flj_to_Fle_pre(Flj_mulu_pre(Fle_to_Flj(P), n, a4, p, pi), p, pi);
     436             : }
     437             : 
     438             : static GEN
     439     1481641 : _Fle_mul(void *E, GEN P, GEN n)
     440             : {
     441     1481641 :   pari_sp av = avma;
     442     1481641 :   struct _Fle *e=(struct _Fle *) E;
     443     1481641 :   long s = signe(n);
     444             :   GEN Q;
     445     1481641 :   if (!s || ell_is_inf(P)) return ellinf();
     446     1463888 :   if (s < 0) P = Fle_neg(P, e->p);
     447     1463888 :   if (is_pm1(n)) return s > 0? zv_copy(P): P;
     448     1146606 :   Q = (lgefint(n)==3) ? Fle_mulu(P, uel(n,2), e->a4, e->p):
     449           0 :                         gen_pow(P, n, (void*)e, &_Fle_dbl, &_Fle_add);
     450     1146606 :   return s > 0? Q: gerepileuptoleaf(av, Q);
     451             : }
     452             : 
     453             : GEN
     454         238 : Fle_mul(GEN P, GEN n, ulong a4, ulong p)
     455             : {
     456             :   struct _Fle E;
     457         238 :   E.a4 = a4; E.p = p;
     458         238 :   return _Fle_mul(&E, P, n);
     459             : }
     460             : 
     461             : /* Finds a random nonsingular point on E */
     462             : GEN
     463     3647655 : random_Fle_pre(ulong a4, ulong a6, ulong p, ulong pi)
     464             : {
     465             :   ulong x, y;
     466     3647655 :   random_Fle_pre_indir(a4, a6, p, pi, &x, &y);
     467     3648646 :   return mkvecsmall2(x, y);
     468             : }
     469             : 
     470             : GEN
     471       26040 : random_Fle(ulong a4, ulong a6, ulong p)
     472       26040 : { return random_Fle_pre(a4, a6, p, get_Fl_red(p)); }
     473             : 
     474             : static GEN
     475           0 : _Fle_rand(void *E)
     476             : {
     477           0 :   struct _Fle *e=(struct _Fle *) E;
     478           0 :   return random_Fle(e->a4, e->a6, e->p);
     479             : }
     480             : 
     481             : static const struct bb_group Fle_group={_Fle_add,_Fle_mul,_Fle_rand,hash_GEN,zv_equal,ell_is_inf,NULL};
     482             : 
     483             : GEN
     484      288787 : Fle_order(GEN z, GEN o, ulong a4, ulong p)
     485             : {
     486      288787 :   pari_sp av = avma;
     487             :   struct _Fle e;
     488      288787 :   e.a4=a4;
     489      288787 :   e.p=p;
     490      288787 :   return gerepileuptoint(av, gen_order(z, o, (void*)&e, &Fle_group));
     491             : }
     492             : 
     493             : GEN
     494       54278 : Fle_log(GEN a, GEN b, GEN o, ulong a4, ulong p)
     495             : {
     496       54278 :   pari_sp av = avma;
     497             :   struct _Fle e;
     498       54278 :   e.a4=a4;
     499       54278 :   e.p=p;
     500       54278 :   return gerepileuptoint(av, gen_PH_log(a, b, o, (void*)&e, &Fle_group));
     501             : }
     502             : 
     503             : ulong
     504           0 : Fl_ellj(ulong a4, ulong a6, ulong p)
     505             : {
     506           0 :   if (SMALL_ULONG(p))
     507             :   { /* a43 = 4 a4^3 */
     508           0 :     ulong a43 = Fl_double(Fl_double(Fl_mul(a4, Fl_sqr(a4, p), p), p), p);
     509             :     /* a62 = 27 a6^2 */
     510           0 :     ulong a62 = Fl_mul(Fl_sqr(a6, p), 27 % p, p);
     511           0 :     ulong z1 = Fl_mul(a43, 1728 % p, p);
     512           0 :     ulong z2 = Fl_add(a43, a62, p);
     513           0 :     return Fl_div(z1, z2, p);
     514             :   }
     515           0 :   return Fl_ellj_pre(a4, a6, p, get_Fl_red(p));
     516             : }
     517             : 
     518             : void
     519      174715 : Fl_ellj_to_a4a6(ulong j, ulong p, ulong *pt_a4, ulong *pt_a6)
     520             : {
     521      174715 :   ulong zagier = 1728 % p;
     522      174715 :   if (j == 0)           { *pt_a4 = 0; *pt_a6 =1; }
     523      174701 :   else if (j == zagier) { *pt_a4 = 1; *pt_a6 =0; }
     524             :   else
     525             :   {
     526      174687 :     ulong k = Fl_sub(zagier, j, p);
     527      174668 :     ulong kj = Fl_mul(k, j, p);
     528      174658 :     ulong k2j = Fl_mul(kj, k, p);
     529      174654 :     *pt_a4 = Fl_triple(kj, p);
     530      174643 :     *pt_a6 = Fl_double(k2j, p);
     531             :   }
     532      174666 : }
     533             : 
     534             : ulong
     535     2120115 : Fl_elldisc_pre(ulong a4, ulong a6, ulong p, ulong pi)
     536             : { /* D = -(4A^3 + 27B^2) */
     537             :   ulong t1, t2;
     538     2120115 :   t1 = Fl_mul_pre(a4, Fl_sqr_pre(a4, p, pi), p, pi);
     539     2119247 :   t1 = Fl_double(Fl_double(t1, p), p);
     540     2118389 :   t2 = Fl_mul_pre(27 % p, Fl_sqr_pre(a6, p, pi), p, pi);
     541     2119274 :   return Fl_neg(Fl_add(t1, t2, p), p);
     542             : }
     543             : 
     544             : ulong
     545           0 : Fl_elldisc(ulong a4, ulong a6, ulong p)
     546             : {
     547           0 :   if (SMALL_ULONG(p))
     548             :   { /* D = -(4A^3 + 27B^2) */
     549             :     ulong t1, t2;
     550           0 :     t1 = Fl_mul(a4, Fl_sqr(a4, p), p);
     551           0 :     t1 = Fl_double(Fl_double(t1, p), p);
     552           0 :     t2 = Fl_mul(27 % p, Fl_sqr(a6, p), p);
     553           0 :     return Fl_neg(Fl_add(t1, t2, p), p);
     554             :   }
     555           0 :   return Fl_elldisc_pre(a4, a6, p, get_Fl_red(p));
     556             : }
     557             : 
     558             : void
     559      105157 : Fl_elltwist_disc(ulong a4, ulong a6, ulong D, ulong p, ulong *pa4, ulong *pa6)
     560             : {
     561      105157 :   ulong D2 = Fl_sqr(D, p);
     562      105157 :   *pa4 = Fl_mul(a4, D2, p);
     563      105157 :   *pa6 = Fl_mul(a6, Fl_mul(D, D2, p), p);
     564      105156 : }
     565             : 
     566             : void
     567           0 : Fl_elltwist(ulong a4, ulong a6, ulong p, ulong *pt_a4, ulong *pt_a6)
     568           0 : { Fl_elltwist_disc(a4, a6, nonsquare_Fl(p), p, pt_a4, pt_a6); }
     569             : 
     570             : static void
     571    52556365 : Fle_dbl_sinv_pre_inplace(GEN P, ulong a4, ulong sinv, ulong p, ulong pi)
     572             : {
     573             :   ulong x, y, slope;
     574    52556365 :   if (uel(P,1)==p) return;
     575    52313448 :   if (!P[2]) { P[1] = p; return; }
     576    52165581 :   x = P[1]; y = P[2];
     577    52165581 :   slope = Fl_mul_pre(Fl_add(Fl_triple(Fl_sqr_pre(x, p, pi), p), a4, p),
     578             :                 sinv, p, pi);
     579    52326138 :   P[1] = Fl_sub(Fl_sqr_pre(slope, p, pi), Fl_double(x, p), p);
     580    52138972 :   P[2] = Fl_sub(Fl_mul_pre(slope, Fl_sub(x, P[1], p), p, pi), y, p);
     581             : }
     582             : 
     583             : static void
     584     7216420 : Fle_add_sinv_pre_inplace(GEN P, GEN Q, ulong a4, ulong sinv, ulong p, ulong pi)
     585             : {
     586             :   ulong Px, Py, Qx, Qy, slope;
     587     7216420 :   if (uel(P,1)==p) { P[1] = Q[1]; P[2] = Q[2]; }
     588     7216420 :   if (ell_is_inf(Q)) return;
     589     7214952 :   Px = P[1]; Py = P[2];
     590     7214952 :   Qx = Q[1]; Qy = Q[2];
     591     7214952 :   if (Px==Qx)
     592             :   {
     593       29376 :     if (Py==Qy) Fle_dbl_sinv_pre_inplace(P, a4, sinv, p, pi);
     594       13944 :     else P[1] = p;
     595       29376 :     return;
     596             :   }
     597     7185576 :   slope = Fl_mul_pre(Fl_sub(Py, Qy, p), sinv, p, pi);
     598     7187600 :   P[1] = Fl_sub(Fl_sub(Fl_sqr_pre(slope, p, pi), Px, p), Qx, p);
     599     7175406 :   P[2] = Fl_sub(Fl_mul_pre(slope, Fl_sub(Px, P[1], p), p, pi), Py, p);
     600             : }
     601             : 
     602             : static void
     603     8050032 : Fle_sub_sinv_pre_inplace(GEN P, GEN Q, ulong a4, ulong sinv, ulong p, ulong pi)
     604             : {
     605             :   ulong Px, Py, Qx, Qy, slope;
     606     8050032 :   if (uel(P,1)==p) { P[1] = Q[1]; P[2] = Fl_neg(Q[2], p); }
     607     8050032 :   if (ell_is_inf(Q)) return;
     608     8046771 :   Px = P[1]; Py = P[2];
     609     8046771 :   Qx = Q[1]; Qy = Q[2];
     610     8046771 :   if (Px==Qx)
     611             :   {
     612       34985 :     if (Py==Qy) P[1] = p;
     613             :     else
     614       13608 :       Fle_dbl_sinv_pre_inplace(P, a4, sinv, p, pi);
     615       34986 :     return;
     616             :   }
     617     8011786 :   slope = Fl_mul_pre(Fl_add(Py, Qy, p), sinv, p, pi);
     618     8020949 :   P[1] = Fl_sub(Fl_sub(Fl_sqr_pre(slope, p, pi), Px, p), Qx, p);
     619     8004691 :   P[2] = Fl_sub(Fl_mul_pre(slope, Fl_sub(Px, P[1], p), p, pi), Py, p);
     620             : }
     621             : 
     622             : static long
     623    67967608 : skipzero(long n) { return n ? n:1; }
     624             : 
     625             : void
     626     1781647 : FleV_add_pre_inplace(GEN P, GEN Q, GEN a4, ulong p, ulong pi)
     627             : {
     628     1781647 :   long i, l=lg(a4);
     629     1781647 :   GEN sinv = cgetg(l, t_VECSMALL);
     630     9013755 :   for(i=1; i<l; i++)
     631     7230517 :     uel(sinv,i) = umael(P,i,1)==p? 1: skipzero(Fl_sub(mael(P,i,1), mael(Q,i,1), p));
     632     1783238 :   Flv_inv_pre_inplace(sinv, p, pi);
     633     8997467 :   for (i=1; i<l; i++)
     634     7216978 :     Fle_add_sinv_pre_inplace(gel(P,i), gel(Q,i), uel(a4,i), uel(sinv,i), p, pi);
     635     1780489 : }
     636             : 
     637             : void
     638     2056864 : FleV_sub_pre_inplace(GEN P, GEN Q, GEN a4, ulong p, ulong pi)
     639             : {
     640     2056864 :   long i, l=lg(a4);
     641     2056864 :   GEN sinv = cgetg(l, t_VECSMALL);
     642    10123757 :   for(i=1; i<l; i++)
     643     8068676 :     uel(sinv,i) = umael(P,i,1)==p? 1: skipzero(Fl_sub(mael(P,i,1), mael(Q,i,1), p));
     644     2055081 :   Flv_inv_pre_inplace(sinv, p, pi);
     645    10111621 :   for (i=1; i<l; i++)
     646     8052379 :     Fle_sub_sinv_pre_inplace(gel(P,i), gel(Q,i), uel(a4,i), uel(sinv,i), p, pi);
     647     2059242 : }
     648             : 
     649             : void
     650    13508994 : FleV_dbl_pre_inplace(GEN P, GEN a4, ulong p, ulong pi)
     651             : {
     652    13508994 :   long i, l=lg(a4);
     653    13508994 :   GEN sinv = cgetg(l, t_VECSMALL);
     654    66938780 :   for(i=1; i<l; i++)
     655    53448920 :     uel(sinv,i) = umael(P,i,1)==p? 1: skipzero(Fl_double(umael(P,i,2), p));
     656    13489860 :   Flv_inv_pre_inplace(sinv, p, pi);
     657    66084788 :   for(i=1; i<l; i++)
     658    52648181 :     Fle_dbl_sinv_pre_inplace(gel(P,i), uel(a4,i), uel(sinv,i), p, pi);
     659    13436607 : }
     660             : 
     661             : static void
     662     1045326 : FleV_mulu_pre_naf_inplace(GEN P, ulong n, GEN a4, ulong p, ulong pi, const naf_t *x)
     663             : {
     664     1045326 :   pari_sp av = avma;
     665             :   ulong pbits, nbits, m;
     666             :   GEN R;
     667     1045326 :   if (n == 1) return;
     668             : 
     669     1045326 :   R = P; P = gcopy(P);
     670     1045197 :   FleV_dbl_pre_inplace(R, a4, p, pi);
     671     1044232 :   if (n == 2) return;
     672             : 
     673     1044138 :   pbits = x->pbits;
     674     1044138 :   nbits = x->nbits;
     675     1044138 :   m = (1UL << x->lnzb);
     676    13520406 :   for ( ; m; m >>= 1) {
     677    12476349 :     FleV_dbl_pre_inplace(R, a4, p, pi);
     678    12471255 :     if (m & pbits)
     679     1781944 :       FleV_add_pre_inplace(R, P, a4, p, pi);
     680    10689311 :     else if (m & nbits)
     681     2057352 :       FleV_sub_pre_inplace(R, P, a4, p, pi);
     682             :   }
     683     1044057 :   set_avma(av);
     684             : }
     685             : 
     686             : void
     687     1045019 : FleV_mulu_pre_inplace(GEN P, ulong n, GEN a4, ulong p, ulong pi)
     688             : {
     689     1045019 :   naf_t x; naf_repr(&x, n);
     690     1045418 :   FleV_mulu_pre_naf_inplace(P, n, a4, p, pi, &x);
     691     1044058 : }
     692             : 
     693             : /***********************************************************************/
     694             : /**                            Pairings                               **/
     695             : /**               Derived from APIP by Jerome Milan, 2012             **/
     696             : /***********************************************************************/
     697             : static ulong
     698      111116 : Fle_vert(GEN P, GEN Q, ulong a4, ulong p)
     699             : {
     700      111116 :   if (ell_is_inf(P))
     701       38479 :     return 1;
     702       72637 :   if (uel(Q, 1) != uel(P, 1))
     703       68857 :     return Fl_sub(uel(Q, 1), uel(P, 1), p);
     704        3780 :   if (uel(P,2) != 0 ) return 1;
     705        2692 :   return Fl_inv(Fl_add(Fl_triple(Fl_sqr(uel(P,1),p), p), a4, p), p);
     706             : }
     707             : 
     708             : static ulong
     709       38028 : Fle_Miller_line(GEN R, GEN Q, ulong slope, ulong a4, ulong p)
     710             : {
     711       38028 :   ulong x = uel(Q, 1), y = uel(Q, 2);
     712       38028 :   ulong tmp1 = Fl_sub(x, uel(R, 1), p);
     713       38028 :   ulong tmp2 = Fl_add(Fl_mul(tmp1, slope, p), uel(R,2), p);
     714       38028 :   if (y != tmp2)
     715       36378 :     return Fl_sub(y, tmp2, p);
     716        1650 :   if (y == 0)
     717        1080 :     return 1;
     718             :   else
     719             :   {
     720             :     ulong s1, s2;
     721         570 :     ulong y2i = Fl_inv(Fl_double(y, p), p);
     722         570 :     s1 = Fl_mul(Fl_add(Fl_triple(Fl_sqr(x, p), p), a4, p), y2i, p);
     723         570 :     if (s1 != slope)
     724         278 :       return Fl_sub(s1, slope, p);
     725         292 :     s2 = Fl_mul(Fl_sub(Fl_triple(x, p), Fl_sqr(s1, p), p), y2i, p);
     726         292 :     return s2 ? s2: y2i;
     727             :   }
     728             : }
     729             : 
     730             : /* Computes the equation of the line tangent to R and returns its
     731             :  * evaluation at the point Q. Also doubles the point R. */
     732             : static ulong
     733       59030 : Fle_tangent_update(GEN R, GEN Q, ulong a4, ulong p, GEN *pt_R)
     734             : {
     735       59030 :   if (ell_is_inf(R)) { *pt_R = ellinf(); return 1; }
     736       54684 :   else if (uel(R,2) == 0) { *pt_R = ellinf(); return Fle_vert(R, Q, a4, p); }
     737             :   else
     738             :   {
     739             :     ulong slope;
     740       33109 :     *pt_R = Fle_dbl_slope(R, a4, p, &slope);
     741       33109 :     return Fle_Miller_line(R, Q, slope, a4, p);
     742             :   }
     743             : }
     744             : 
     745             : /* Computes the equation of the line through R and P, and returns its
     746             :  * evaluation at the point Q. Also adds P to the point R. */
     747             : static ulong
     748       17715 : Fle_chord_update(GEN R, GEN P, GEN Q, ulong a4, ulong p, GEN *pt_R)
     749             : {
     750       17715 :   if (ell_is_inf(R)) { *pt_R = P; return Fle_vert(P, Q, a4, p); }
     751       17477 :   else if (ell_is_inf(P)) { *pt_R = R; return Fle_vert(R, Q, a4, p); }
     752       17477 :   else if (uel(P, 1) == uel(R, 1))
     753             :   {
     754       12558 :     if (uel(P, 2) == uel(R, 2)) return Fle_tangent_update(R, Q, a4, p, pt_R);
     755       12558 :     else { *pt_R = ellinf(); return Fle_vert(R, Q, a4, p); }
     756             :   }
     757             :   else
     758             :   {
     759             :     ulong slope;
     760        4919 :     *pt_R = Fle_add_slope(P, R, a4, p, &slope);
     761        4919 :     return Fle_Miller_line(R, Q, slope, a4, p);
     762             :   }
     763             : }
     764             : 
     765             : struct _Fle_miller { ulong p, a4; GEN P; };
     766             : static GEN
     767       59030 : Fle_Miller_dbl(void* E, GEN d)
     768             : {
     769       59030 :   struct _Fle_miller *m = (struct _Fle_miller *)E;
     770       59030 :   ulong p = m->p, a4 = m->a4;
     771       59030 :   GEN P = m->P;
     772             :   ulong v, line;
     773       59030 :   ulong N = Fl_sqr(umael(d,1,1), p);
     774       59030 :   ulong D = Fl_sqr(umael(d,1,2), p);
     775       59030 :   GEN point = gel(d,2);
     776       59030 :   line = Fle_tangent_update(point, P, a4, p, &point);
     777       59030 :   N  = Fl_mul(N, line, p);
     778       59030 :   v = Fle_vert(point, P, a4, p);
     779       59030 :   D = Fl_mul(D, v, p); return mkvec2(mkvecsmall2(N, D), point);
     780             : }
     781             : static GEN
     782       17715 : Fle_Miller_add(void* E, GEN va, GEN vb)
     783             : {
     784       17715 :   struct _Fle_miller *m = (struct _Fle_miller *)E;
     785       17715 :   ulong p = m->p, a4= m->a4;
     786       17715 :   GEN P = m->P, point;
     787             :   ulong v, line;
     788       17715 :   ulong na = umael(va,1,1), da = umael(va,1,2);
     789       17715 :   ulong nb = umael(vb,1,1), db = umael(vb,1,2);
     790       17715 :   GEN pa = gel(va,2), pb = gel(vb,2);
     791       17715 :   ulong N = Fl_mul(na, nb, p);
     792       17715 :   ulong D = Fl_mul(da, db, p);
     793       17715 :   line = Fle_chord_update(pa, pb, P, a4, p, &point);
     794       17715 :   N = Fl_mul(N, line, p);
     795       17715 :   v = Fle_vert(point, P, a4, p);
     796       17715 :   D = Fl_mul(D, v, p); return mkvec2(mkvecsmall2(N, D), point);
     797             : }
     798             : 
     799             : /* Returns the Miller function f_{m, Q} evaluated at the point P using
     800             :  * the standard Miller algorithm. */
     801             : static ulong
     802       33895 : Fle_Miller(GEN Q, GEN P, ulong m, ulong a4, ulong p)
     803             : {
     804       33895 :   pari_sp av = avma;
     805             :   struct _Fle_miller d;
     806             :   GEN v;
     807             :   ulong N, D;
     808             : 
     809       33895 :   d.a4 = a4; d.p = p; d.P = P;
     810       33895 :   v = gen_powu_i(mkvec2(mkvecsmall2(1,1), Q), m, (void*)&d,
     811             :                 Fle_Miller_dbl, Fle_Miller_add);
     812       33895 :   N = umael(v,1,1); D = umael(v,1,2);
     813       33895 :   return gc_ulong(av, Fl_div(N, D, p));
     814             : }
     815             : 
     816             : ulong
     817        9636 : Fle_weilpairing(GEN P, GEN Q, ulong m, ulong a4, ulong p)
     818             : {
     819        9636 :   pari_sp ltop = avma;
     820             :   ulong N, D, w;
     821        9636 :   if (ell_is_inf(P) || ell_is_inf(Q) || zv_equal(P,Q)) return 1;
     822        8509 :   N = Fle_Miller(P, Q, m, a4, p);
     823        8509 :   D = Fle_Miller(Q, P, m, a4, p);
     824        8509 :   w = Fl_div(N, D, p);
     825        8509 :   if (odd(m)) w  = Fl_neg(w, p);
     826        8509 :   return gc_ulong(ltop, w);
     827             : }
     828             : 
     829             : ulong
     830       16877 : Fle_tatepairing(GEN P, GEN Q, ulong m, ulong a4, ulong p)
     831             : {
     832       16877 :   if (ell_is_inf(P) || ell_is_inf(Q)) return 1;
     833       16877 :   return Fle_Miller(P, Q, m, a4, p);
     834             : }
     835             : 
     836             : GEN
     837       11438 : Fl_ellptors(ulong l, ulong N, ulong a4, ulong a6, ulong p)
     838             : {
     839       11438 :   long v = z_lval(N, l);
     840             :   ulong N0, N1;
     841             :   GEN F;
     842       11438 :   if (v==0) return cgetg(1,t_VEC);
     843       11438 :   N0 = upowuu(l, v); N1 = N/N0;
     844       11438 :   F = mkmat2(mkcols(l), mkcols(v));
     845             :   while(1)
     846        1582 :   {
     847       13020 :     pari_sp av2 = avma;
     848             :     GEN P, Q;
     849             :     ulong d, s, t;
     850             : 
     851       13020 :     P = Fle_mulu(random_Fle(a4, a6, p), N1, a4, p);
     852       13020 :     Q = Fle_mulu(random_Fle(a4, a6, p), N1, a4, p);
     853       13020 :     s = itou(Fle_order(P, F, a4, p));
     854       13020 :     t = itou(Fle_order(Q, F, a4, p));
     855       13020 :     if (s < t) { swap(P,Q); lswap(s,t); }
     856       13020 :     if (s == N0) retmkvec(Fle_mulu(P, s/l, a4, p));
     857        2394 :     d = Fl_order(Fle_weilpairing(P, Q, s, a4, p), s, p);
     858        2394 :     if (s*d == N0)
     859         812 :       retmkvec2(Fle_mulu(P, s/l, a4, p), Fle_mulu(Q, t/l, a4, p));
     860        1582 :     set_avma(av2);
     861             :   }
     862             : }

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