Abstract
Since Elliptic Curve Cryptosystems(ECCs) support the same security as RSA cryptosystem and ElGamal cryptosystem with 1/6 size key, ECCs are the most efficient to smart cards, cellular phone and small-size computers restricted by high memory capacity and power of process. In this paper, we explicitly explain methods for finite fields operations used in ECC, and then construct some composite fields over finite fields which are secure under Weil's decent attack and maximize the speed of operations. Lastly, we propose a fast multiplier over our composite fields.
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