Experimantal Analysis of Cheon's Algorithm Against Pairing-Friendly Curves

The discrete logarithm problem (DLP) is one of the familiar problem on which some cryptographic schemes rely. In 2006, Cheon proposed an algorithm for solving DLP with auxiliary input which works better than conventional algorithms. In this paper, we show our experimental results of Cheon's algorithm on a pairing-friendly elliptic curve defined over GF(3
¹²⁷
). It is shown that the algorithm combined with the kangaroo method has an advantage over that combined with the baby-step giant-step method in the sense that the required time and space are smaller. Then, for the algorithm combined with the kangaroo-method, speeding-up techniques are introduced. Based on our experimental results and the speeding-up techniques, we evaluate the required time and space for some pairing-friendly elliptic curves curves. As results, a portion of pairing-friendly elliptic curves can be analyzed by Cheon's algorithm at reasonable cost.