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Let q be a power of a prime and φ be the Frobenius endomorphism on E(Fqλ), then q = tφ- O2. Applying this equation, a new algorithm to compute rational point scalar multiplications on elliptic curves by finding a suitable small positive integer s such that qs can be represented as some very sparse φ-polynomial is proposed. If a Normal Basis (NB) or Optimal Normal Basis (ONB) is applied and the precomputations are considered free, our algorithm will cost, on average, about 55% to 80% less than binary method, and about 42% to 74% less than q-ary method. For some elliptic curves, our algorithm is also faster than Muller’s algorithm. In addition, an effective algorithm is provided for finding such integer s.
Let q be a power of a prime and φ be the Frobenius endomorphism on E (Fqλ), then q = tφ-O2. Applying this equation, a new algorithm to compute rational point scalar multiplications on elliptic curves by finding a suitable small positive integer If such a normal Basis (NB) or Optimal Normal Basis (ONB) is applied and the precomputations are considered free, our algorithm will cost, on average, about 55 % to 80% less than binary method, and about 42% to 74% less than q-ary method. Our algorithm is also faster than Muller’s algorithm. In addition, an effective algorithm is provided for finding such integer s .