@inbook{1c80bf18b41b4ff5b242aba0e45befcf,

title = "A general expansion method using efficient endomorphisms",

abstract = "There are various expansion methods to accelerate scalar multiplication on special types of elliptic curves. In this paper we present a general expansion method that uses efficient endomorphisms. We first show that the set of all endomorphisms over a non-supersingular elliptic curve E is isomorphic to ℤ[ω] = {a + bω | a, b ∈ ℤ}, where ω is an algebraic integer with the smallest norm in an imaginary quadratic field, if ω is an endomorphism over E. Then we present a new division algorithm in ℤ[ω], by which an integer k can be expanded by the Frobenius endomorphism and ω. If ω is more efficient than a point doubling, we can use it to improve the performance of scalar multiplication by replacing some point doublings with the w maps. As an instance of this general method, we give a new expansion method using the efficiently computable endomorphisms used by Ciet et al. [1].",

keywords = "Efficient Endomorphism, Elliptic Curve, Frobenius Expansion, Scalar Multiplication",

author = "Park, {Tae Jun} and Lee, {Mun Kyu} and Kim, {E. Yong} and Kunsoo Park",

year = "2004",

doi = "10.1007/978-3-540-24691-6_10",

language = "English",

isbn = "3540213767",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "112--126",

editor = "Lim, {Jong In} and Lee, {Dong Hoon}",

booktitle = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

address = "Germany",

}