On the Computation of the Tate Pairing for Elliptic Curves over Fields of Large Characteristic

A few years ago a new kind of public key cryptosystems was proposed, called Identity Based Cryptography (IBC), which uses each user identity as his public key. IBC has had practical applications in e-mail communications and instant messengers. One of its most important goals is the efficient evaluation of cryptographically secure bilinear maps, as the Tate or Weil pairings. Most people in pairing research have focused on fields of characteristic 2 and 3, and they put away the fields of characteristic p > 3. In this paper we summarize the techniques to calculate the Weil and Tate pairings using elliptic curves over fields of characteristic a large prime p. We analyze the Miller’s algorithm for pairing calculation in fields of characteristic a large prime number and we analyze the efficiency in finding prime numbers that guarantee the correctness of the proposed algorithm modifications.