Faster maptopoint on supersingular elliptic curves in characteristic 3

Pairing-based cryptosystems are generally constructed us ing many functions such as pairing computation, arithmetic in finite fields, and arithmetic on elliptic curves. MapToPoint, which is a hashing algo rithm onto an elliptic curve point, is one of the functions for constructing pairing-based cryptosystems. There are two MapToPoint algorithms on supersingular elliptic curves in characteristic three, which is used by qy pairing. The first is computed by using a square root computation in IF and the computational cost of this algorithm is 0(log m) multiplications in F The second is computed by using an (m-1) × (m - 1) matrix over F It can be computed by O(1) multiplications in F However, this algorithm needs the off-line memory to store about m F₃^m -elements. In this paper, we propose an efficient MapToPoint algorithm on the supersingular ellip tic curves in characteristic three by using 1/3-trace over F₃^m We propose 1/3-trace over F₃^m which can compute solution x of ^x - x = c by using no multiplication in F₃^m The proposed algorithm is computed by O(1) multi plications in F ₃^m and it requires less than rn F₃^m -elements to be stored in the off-line memory to efficiently compute trace over F₃^m Moreover, in our software implementation of F ₃^m the proposed MapToPoint algorithm is approximately 35% faster than the conventional MapToPoint algorithm us ing the square root computation on an AMD Opteron processor (2.2 GHz).