Rho Method with Period k on Non -- symmetric Ordinary Pairings of Embedding Degree k -- Especially for Barreto -- Naehrig Curves

Pollard´s ρ method is well known as an efficient method for solving elliptic curve discrete logarithm problem (ECDLP). This paper applies it to non-super singular pairing-friendly curves, especially to Barreto-Naehrig curves, together with Frobenius endomorphism. Then, it is shown that the list of rational points can be reduced with norms of their x and y coordinates without loss of theoretic uniqueness.