Efficient Squaring Algorithm in 2-nd Tower Field Available for Various Pairing-Based Cryptographies

Many public-key cryptographers have recently focused on cryptographic schemes based on pairing, which is a bilinear map from two elliptic curve groups to a group in an extension field. In order to provide efficient pairings, several kinds of pairing-friendly curves have been proposed. Since most of the pairing-friendly curves are defined over a certain extension field, arithmetic operations in extension field should be carried out efficiently. Especially for final exponentiation included in pairing calculation, squaring is more important than multiplication. This paper proposes an efficient squaring algorithm in 2-nd tower field available for various pairing-friendly curves.