Exponentiation inversion problem reduced from fixed argument pairing inversion on twistable ate pairing and its difficulty

As one of problems that guarantee the security of pairing-based cryptography, pairing inversion problem is studied. Some recent works have reduced fixed argument pairing inversion (FAPI) problem to exponentiation inversion (EI) problem. According to the results, FAPI problem is solved if EI problem of exponent (q^k - 1)/Φ_k (q) is solved, where q, k, and r are the characteristic, embedding degree, and order of pairing group, respectively. Φ_k(x) is the cyclotomic polynomial of order k. This paper shows an approach for reducing the exponent of EI problem to q - 1 especially on Ate pairing. For many embedding degrees, it is considerably reduced from the previous result (q^k - 1)/Φ_k(q). After that, the difficulty of the reduced EI problem is discussed based on the distribution of correct (q - 1)-th roots on a small example.