International Association
for Cryptologic Research

A recent area of interest in cryptography is recursive composition of proof systems. One of the approaches to make recursive composition efficient involves cycles of pairing-friendly elliptic curves of prime order. However, known constructions have very low embedding degrees. This entails large parameter sizes, which makes the overall system inefficient. In this paper, we explore 2-cycles composed of curves from families parameterized by polynomials, and show that such cycles do not exist unless a strong condition holds. As a consequence, we prove that no 2-cycles can arise from the known families, except for those cycles already known. Additionally, we show some general properties about cycles, and provide a detailed computation on the density of pairing-friendly cycles among all cycles.