In this paper, we particularly deal with no Fp-rational two-torsion elliptic curves, where Fp is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E ? 1)/2 and (#E?? 1)/2 are reciprocal to each other, where #E and #E? are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.
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