In this thesis we present an improved algorithm for counting points on elliptic curves over finite fields. It is mainly based on Satoh-Skjernaa-Taguchi algorithm, and uses a Gaussian Normal Basis (GNB) of small type t≤4. In practice, about 42% (36% for prime N) of fields in cryptographic context (i.e., for p=2 and 160＜ N＜600) have such bases. They can be lifted from $\F_{p^N}$ to $\Z_{p^N}$ in a natural way. From the specific properties of GNBs, efficient multiplication and the Frobenius substitutions are available. Thus a fast norm computation algorithm is derived, which runs in $O(N^{2μ logN)$ with $O(N^2)$ space, where the time complexity of multiplying two n-bit objects is $O(n^μ)$. As a result, for all small characteristic p, we reduced the time complexity of the SST-algorithm from $O(N^{2μ+ 0.5})$ to $O(N^{2μ + \frac{1}{μ + 1}})$ and the space complexity still fits in $O(N^2)$.

- Advisors
- Hahn, Sang-Geun
*researcher*; 한상근*researcher*

- Description
- 한국과학기술원 : 수학전공,

- Publisher
- 한국과학기술원

- Issue Date
- 2002

- Identifier
- 177223/325007 / 000995108

- Language
- eng

- Description
학위논문(박사) - 한국과학기술원 : 수학전공, 2002.8, [ [ii], 38 p. ; ]

- Keywords
타원곡선; 위수 계산; Gaussian normal basis; elliptic curve; order counting

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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