Reduced Pairing Results with MNT Curves

[sshravan]

Why is e(0, any value) !=1 for MNT curves? Where 0 is the identity element of G1 and 1 is the identity element of GT. Or is there a change of notation between curves?

template<typename ppT>
void simple_test(){

    libff::G1<ppT> g1;
    libff::G2<ppT> g2;
    libff::GT<ppT> gt, a;

    g1 = libff::G1<ppT>::zero();
    g2 = libff::G2<ppT>::random_element();
    gt = libff::GT<ppT>::one();

    a = ppT::reduced_pairing(g1, g2);
    cout << "e(0, any) == 1 " << (a == gt ? "TRUE" : "FALSE") << endl;

    g1 = libff::G1<ppT>::random_element();
    g2 = libff::G2<ppT>::zero();
    gt = libff::GT<ppT>::one();

    a = ppT::reduced_pairing(g1, g2);
    cout << "e(any, 0) == 1 " << (a == gt ? "TRUE" : "FALSE") << endl;
    // return 0;

}

int main () {

    libff::inhibit_profiling_info = true;
    libff::inhibit_profiling_counters = true;

#ifdef CURVE_BN128
    libff::bn128_pp::init_public_params();
    cout << "BN128" << endl;
    simple_test<libff::bn128_pp>();
#endif

#ifdef CURVE_MNT4
    libff::mnt4_pp::init_public_params();
    cout << "MNT4" << endl;
    simple_test<libff::mnt4_pp>();
#endif

#ifdef CURVE_MNT6
    libff::mnt6_pp::init_public_params();
    cout << "MNT6" << endl;
    simple_test<libff::mnt6_pp>();
#endif

    return 0;
}

[sshravan]

Here is the link to the code.

[Pratyush]

GT::one() is the multiplicative identity element in GT, while e(0, random_element) is a random element in GT.

[ValarDragon]

GT::one() is the multiplicative identity element in GT, while e(0, random_element) is a random element in GT.

For any bilinear map e, it must be the case that e(0, y) = 0. Since the target group is Z_q*, the zero element is 1. So shouldn't this test hold, and this implies there is a bug in libff? (Or alternatively GT::one() returns g, not 1)

[Pratyush]

Hmm I must have misunderstood and thought 0 = generator, but that’s not the case.

[alinush]

I ran into the same bug here, for what it's worth.