Efficient Computations of the Tate Pairing for the Large MOV Degrees

Abstract

The Tate pairing hasp lenty of attractive applications, e.g., ID-based cryptosystems, short signatures, etc. Recently several fast implementationsof the Tate pairing hasb een reported, which make it appear that the Tate pairing is capable to be used in practical applications. The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields. However these fast implementation are restricted to supersingular curves with small MOV degrees. In this paper we propose several improvements of computing the Tate pairing over general elliptic curves over finite fields IF q (q = pm, p > 3) - some of them can be efficiently applied to supersingular curves. The proposed methods can be combined with previous techniques. The proposed algorithm iss pecially effective upon the curvest hat hasa large MOV degree k. We develop several formulas that compute the Tate pairing using the small number of multiplications over IFq k. For k = 6, the proposed algorithm is about 20% faster than previously fastest algorithm.KeywordsElliptic curve cryptosystemTate pairingJacobian coordinateMOV degreeefficient computation