Pairing Inversion via Non-degenerate Auxiliary Pairings

The security of pairing-based cryptosystems is closely related to the difficulty of the pairing inversion problem(PI). In this paper, we discuss the difficulty of pairing inversion on the generalized ate pairings of Vercauteren. First, we provide a simpler approach for PI by generalizing and simplifying Kanayama-Okamotos approach; our approach involves modifications of exponentiation inversion(EI) and Miller inversion(MI), via an auxiliary pairing. Then we provide a complexity of the modified MI, showing that the complexity depends on the sum-norm of the integer vector defining the auxiliary pairing. Next, we observe that degenerate auxiliary pairings expect to make modified EI harder. We provide a sufficient condition on the integer vector, in terms of its max norm, so that the corresponding auxiliary paring is non-degenerate. Finally, we define an infinite set of curve parameters, which includes those of typical pairing friendly curves, and we show that, within those parameters, PI of arbitrarily given generalized ate pairing can be reduced to modified EI in polynomial time.