### Abstract

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 F_{3}^{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 F_{3}^{m} We propose 1/3-trace over F_{3}^{m} which can compute solution x of ^{x} - x = c by using no multiplication in F_{3}^{m} The proposed algorithm is computed by O(1) multi plications in F _{3}^{m} and it requires less than rn F_{3}^{m} -elements to be stored in the off-line memory to efficiently compute trace over F_{3}^{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).

Original language | English |
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Pages (from-to) | 150-155 |

Number of pages | 6 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E94-A |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2011 |

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### All Science Journal Classification (ASJC) codes

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E94-A*(1), 150-155. https://doi.org/10.1587/transfun.E94.A.150