Self-pairings on supersingular elliptic curves with embedding degree three

Self-pairings are a special subclass of pairings and have interesting applications in cryptographic schemes and protocols. In this paper, we explore the computation of the self-pairings on supersingular elliptic curves with embedding degree k = 3. We construct a novel self-pairing which has the same Miller loop as the Eta/Ate pairing. However, the proposed self-pairing has a simple final exponentiation. Our results suggest that the proposed self-pairings are more efficient than the other ones on the corresponding curves. We compare the efficiency of self-pairing computations on different curves over large characteristic and estimate that the proposed self-pairings on curves with k = 3 require 44% less field multiplications than the fastest ones on curves with k = 2 at AES 80-bit security level.