Reduced Tate pairing in supersingular elliptic curves

[Aleksandr Lenin <aleksandr.lenin@cyber.ee>]

Hi all,

I have stepped across the case when I get a trivial value 1 for the
reduced Tate pairing for seemingly legit inputs. Both inputs belong to
the r-torsion, point 1 is defined over the base-field subgroup, point 2
does not belong to the base-field subgroup (and it shouldn't, as the
reduced Tate pairing is supposed to return 1 if two points belong to the
same subgroup). But the result is nevertheless 1, although in theory it
looks like it shouldn't.

I attach the C++ code for libPARI with inputs and the code I was using
to produce this result.

I would be grateful for any advice on why could this happen, possible
fixes, and possible ways to optimize this code a bit so that it would
work faster (I know the code is ugly, I am a beginner PARI user).

Thanks in advance,

-- 
Aleksandr

    pari_init(1024*1024, 2);

    GEN r = gp_read_str("115792089237316195423570985008687907853269984665640564039457584007913130327851");
    GEN gen = ffgen(gp_read_str("Mod(1, 729995325678669359612263452605506580733004427132673582314497567805014278041635994590090082180502116143263532359048738406597239443318640672711245526346238092246683746134657288054890921912991126239130697620428047828998444390759826609224613037843714024073696709708567493436988013034917053907792300971937911565720840872897751218935438210856450629634873555502858190089546937212523125070225588545551391707034179535007705558338179039614684525084857725397040972387874272971448091159792452744469591)*x^2 + Mod(1, 729995325678669359612263452605506580733004427132673582314497567805014278041635994590090082180502116143263532359048738406597239443318640672711245526346238092246683746134657288054890921912991126239130697620428047828998444390759826609224613037843714024073696709708567493436988013034917053907792300971937911565720840872897751218935438210856450629634873555502858190089546937212523125070225588545551391707034179535007705558338179039614684525084857725397040972387874272971448091159792452744469591)*x + Mod(1, 729995325678669359612263452605506580733004427132673582314497567805014278041635994590090082180502116143263532359048738406597239443318640672711245526346238092246683746134657288054890921912991126239130697620428047828998444390759826609224613037843714024073696709708567493436988013034917053907792300971937911565720840872897751218935438210856450629634873555502858190089546937212523125070225588545551391707034179535007705558338179039614684525084857725397040972387874272971448091159792452744469591)"),0);
    GEN curve = ellinit(mkvec2(gen_0, gp_read_str("124077955980")), gen, DEFAULTPREC);
    GEN p1 = gp_read_str("[416866340151231144999071547104363552860995592342107617914079000324381628161252265155645646854917176101609645586710923664886999924176044846510900094872811821512999385612906577590788149280356319811591000347648102887159624329635123286323730839020460834613075504904339611008560087599883844273800883881168320111158488361278933397816853937934926779232014231202042266788881007785375396656800844388723109494478629332994684564657346416564422982075930877830457781120143476348463212228485913738818378, 332617282081743765958457595331209402426543969823409409920211593987172038898954201278875792928497905587381429484928180618013398755967181957508337582717533897724873379440281463870584527916994925025070333137982197591362084522892577435310917767393712302706039550334126677727240346450052996401195274561866971151644056816349514618325673949647197767156490279473407609178487581825885626398304873791614461472577390168158829970346191915053707382080614765449833113137839636705734751352621516743726381]");
    GEN p = gp_read_str("[323835096672091128109519914566785095547699666695964334770724989155913496620293120337354058374930769954669940899969219387554719328257359458694798778402657621168324676375937875283168007232503735955552269877916079662046089679497781438420526238043055627372609457292555529361293689880121259497378515059917784373024675650018604892937621023989959813453649239377792409026313473481431097062900458413710095621691098110439012225036374236170080882525581730564537966637105841683438706003865158334361879, 96995708969082174029386372192146598856582423767391084007802154833605167125565120086193266518810107953218158243787512290935368962360366938408600464516374396487143688894851855808206836903352887290635476739165504596690182400273250031906505481976985386408091703095585683044627711755143903867142203095362609317948954191252558670513895497284388066199500635347784010694984498336131022426955975086103409966851132198916955647772119315622637951174020723482166760611028951847777953307999952433119298]");
    GEN p2 = mkvec2(FF_mul(Fq_to_FF(gp_read_str("x"), gen),FF_Z_mul(FF_1(gen),gel(p,1))),FF_Z_mul(FF_1(gen),gel(p,2)));
    GEN pairing = FF_pow(FF_elltatepairing(curve,p1,p2,r),gdiv(gsub(FF_q(gen),gen_1),r));

    if ( gequal(ellisoncurve(curve,p1),gen_1) &&
         gequal(ellisoncurve(curve,p2),gen_1) ) {
        std::cout << "p1 and p2 are elliptic curve points" << std::endl;
    } else {
        std::cout << "p1 or p2 are not elliptic curve points" << std::endl;
    }

    pari_printf("Point 1 %Ps\n", p1);
    pari_printf("Point 2 %Ps\n", p2);
    pari_printf("Reduced pairing %Ps\n", pairing);

    GEN random_pairing = FF_pow(FF_elltatepairing(curve,p1,ellrandom(curve),r), gdiv(gsub(FF_q(gen),gen_1),r));
    pari_printf("Random pairing %Ps\n", random_pairing);

    pari_close();