A scalar multiplication algorithm with recovery of the y-coordinate on the montgomery form and analysis of efficiency for elliptic curve cryptosystems

We present a scalar multiplication algorithm with recovery of the y-coordinate on a Montgomery-form elliptic curve over any non-binary field. The previous algorithms for scalar multiplication on a Montgomery form do not consider how to recover the y-coordinate. So although they can be applicable to certain restricted schemes (e.g. ECDH and ECDSA-S), some schemes (e.g. ECDSA-V and MQV) require scalar multiplication with recovery of the y-coordinate. We compare our proposed scalar multiplication algorithm with the traditional scalar multiplication algorithms (including Window-methods on the Weierstrass form), and discuss the Montgomery form versus the Weierstrass form in the performance of implementation with several techniques of elliptic curve cryptosystems (including ECES, ECDSA, and ECMQV). Our results clarify the advantage of the cryptographic usage of Montgomery-form elliptic curve in constrained environments such as mobile devices and smart cards.