In this paper we show how using a representation of an elliptic curve as the intersection of two quadrics in $\PP^3$ can provide a defence against Simple and Differental Power Analysis (SPA/DPA) style attacks. We combine this with a `random window' method of point multiplication and point blinding. The proposed method offers considerable advantages over standard algorithmic techniques of preventing SPA and DPA which usually require a significant increased computational cost, usually more than double. Our method requires roughly a seventy percent increase in computational cost of the basic cryptographic operation, although we give some indication as to how this can be reduced. In addition we show that the Jacobi form is also more efficient than the standard Weierstrass form for elliptic curves in the situation where SPA and DPA are not a concern.