This paper presents two main results relating to elliptic curve cryptography. First, a double point compression scheme is proposed which allows a compact representation of elliptic curve points without the computational cost associated with ordinary single point compression. A triple point compression scheme is also proposed which can result in more savings in memory and/or bandwidth. Second, a new approach to speeding up random point multiplication is given for the case where the base point is variable but available in a certificate. In this approach, some redundant information (a few multiples of the base point) is added to the certificate. It is shown that a significant speed up can be obtained by optimizing the Möller's algorithm for the case where only a portion of the lookup table is available. It is also shown how to use redundant information to compute random point multiplication using parallel processors. The proposed point compression schemes can be employed to reduce the required bandwidth when single point compression is computationally expensive.