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The elliptic curve cryptosystem (ECC) is very attractive for the use in portable devices because of the small key size. The finite field multiplication over GF(2m) is the most important arithmetic for performing the ECC. Portable devices usually have restricted computation power and memory resources. This work will present a simple method for designing a Gaussian normal basis (GNB) multiplier over GF(2m) needing only fewer computation power whereas keeping lower cost. The proposed Gaussian NB multiplier saves ~57~ space complexity as compared with existing GNB multiplier.