Efficient Construction of CGL Hash Function Using Legendre Curves

The CGL hash function is a provably secure hash function using walks on isogeny graphs of supersingular elliptic curves. A dominant cost of its computation comes from iterative computations of power roots over quadratic extension fields. In this paper, we reduce the necessary number of power root computations by almost half, by applying and also extending an existing method of efficient isogeny sequence computation on Legendre curves (Hashimoto and Nuida, CASC 2021). We also point out some relationship between 2-isogenies for Legendre curves and those for Edwards curves, which is of independent interests, and develop a method of efficient computation for 2^e-th roots in quadratic extension fields.