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Because of their shorter key sizes, cryptosystems based on elliptic curves are being increasingly used in practical applications. A special class of elliptic curves, namely, Koblitz curves, offers an additional but crucial advantage of considerably reduced processing time. In this article, power analysis attacks are applied to cryptosystems that use scalar multiplication on Koblitz curves. Both the simple and the differential power analysis attacks are considered and a number of countermeasures are suggested. While the proposed countermeasures against the simple power analysis attacks rely on making the power consumption for the elliptic curve scalar multiplication independent of the secret key, those for the differential power analysis attacks depend on randomizing the secret key prior to each execution of the scalar multiplication.