Computing pairings using <Emphasis Type="Italic">x</Emphasis>-coordinates only

To reduce bandwidth in elliptic curve cryptography one can transmit only x-coordinates of points (or x-coordinates together with an extra bit). This is called point compression. For further computation using the points one can either recover the y-coordinates by taking square roots or one can use point multiplication formulae which use x-coordinates only. We consider how to efficiently use point compression in pairing-based cryptography when the embedding degree is even. We give a method to compute compressed pairings using x-coordinates only. We also show how to compute the compressed pairings using two x-coordinates and one y-coordinate. Our methods are more efficient than taking square roots when the embedding degree is small. We implemented the algorithms in the case of embedding degree 2 curves over where (mod 4) and found that our methods can be 10–15% faster than the analogous methods using square roots.