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In the elliptic curve cryptosystem, we mainly calculate the scalar
multiplication. Since only the value of x-coordinates is used at some
elliptic curve cryptosystem algorithm (ECDH, ECDSA-S, etc.), the
algorithm which calculate scalar multiplication at high speed without
using y-coordinates is proposed. In Montgomery form elliptic curve,
there is the operation method using Lucas chain as a technique
equivalent to addition chain. In this research, some improvements are
added to scalar multiplication algorithm with Lucas chain and the effect
of each improvement in numerical experiments is shown. We proposed more
efficient scalar multiplication algorithm which combined those
improvements.
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