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This paper presents a new coordinate system for elliptic curves that accelerates the elliptic curve addition and doubling over an optimal extension field (OEF). Many coordinate systems for elliptic curves have been proposed to accelerate elliptic curve cryptosystems. This paper is a natural extension of these papers and the new coordinates are much faster when the elliptic curve is defined over an OEF. This paper also shows that the total computational cost is reduced by 28% when the elliptic curve is defined over ${\mathbb F}_{q^m}$, q = 261−1 for m = 5 and the speed of a scalar multiplication on an elliptic curve becomes 41.9 μsec per operation on a 2.82-GHz Athlon 64 FX PC.