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-based cryptography and its security analysis 47 4.1 Pairing In this paper we deal with Tate pairing using the supersingular curves over finite fields of characteristic three, which have been used for efficient implementation of pairing [4, 15, 24]. Let GF.3n/ be the finite field of characteristic three and extension degree n. For b D ˙1 the supersingular elliptic curve Eb3 .n/ over GF.3n/ is defined by Eb3 .n/ D ¹.x; y/ 2 GF.3n/ GF.3n/ jy2 D x3 x C bº [ ¹1º: We usually choose n as a prime number in cryptography. The order of Eb3 .n/ is #Eb3 .n/ D 3n C 1C b 3 n 3.nC1/=2; where