有限體上超橢圓曲線之數目

Yan-Chen Liu; 劉彥辰

請用此 Handle URI 來引用此文件： http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/29395

完整後設資料紀錄

DC 欄位	值	語言
dc.contributor.advisor	朱樺(Huah Chu)
dc.contributor.author	Yan-Chen Liu	en
dc.contributor.author	劉彥辰	zh_TW
dc.date.accessioned	2021-06-13T01:06:02Z	-
dc.date.available	2007-07-26
dc.date.copyright	2007-07-26
dc.date.issued	2007
dc.date.submitted	2007-07-20
dc.identifier.citation	[1] Bertrand Byramjee and Sylvain Duquesne. Classification of genus 2 curves over F_{2^n} and optimization of their arithmetic, 2004. Cryptology ePrint Archive 2004/107. [2] L. Carlitz. The arithmetic of polynomials in a galois field. Amer. J. Math., 54:39–50, 1932. [3] Youngju Choie and Eunkyung Jeong. Isomorphism classes of hyperelliptic curves of genus 2 over F2n, 2003. Cryptology ePrint Archive 2003/213. [4] Youngju Choie and D. Yun. Isomorphism classes of hyperelliptic curves of genus 2 over F_q, pages 190–202. LNCS 2384. Springer, 2002. [5] Huah Chu, Yingpu Deng, and Tse-Chung Yang. Isomorphism classes of hyperelliptic curves of genus 4 over finite fields with even characteristic. preprint. [6] Henri Cohen and Gerhard Frey. Handbook of elliptic and hyperelliptic curve cryptography. Chapman & Hall/CRC, 2006.Homepage : http://www.hyperelliptic.org/HEHCC/. [7] Yingpu Deng. Isomorphism classes of hyperelliptic curves of genus 4 over finite fields with odd characteristic. preprint. [8] Yingpu Deng. Isomorphism classes of hyperelliptic curves of genus 3 over finite fields. Finite Fields and Their Applications, 12:248–282, 2006. [9] Yingpu Deng and M. Liu. Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2. Sci. China Ser. A, 49(2):173–184, 2005. [10] L. Hern´andez Encinasa and J. Mu˜noz Masqu´e. Isomorphism classes of hyperelliptic curves of genus 2 in characteristic 5. Technical Report CORR2002-07, Centre For Applied Cryptographic Research, University of Waterloo, Canada, 2002. Available at http://www.cacr.math.uwaterloo.ca/techreports/2002/corr2002-07.ps. [11] L. Hern´andez Encinasa, Alfred J. Menezes, and J. Mu˜noz Masqu´e. Isomorphism classes of genus−2 hyperelliptic curves over finite fields. Appl. Algebra Engrg. Comm. Comput., 13(1):57–65, Apr 2002. [12] N. J. Fine. Binomial coefficients modulo a prime. Am. Math. Mon.,54(10):589–592, Dec 1947. [13] J. Espinosa Garcia, L. Hern´andez Encinas, and J. Mu˜noz Masqu´e. A review on the isomorphism classes of hyperelliptic curves of genus 2 over finite fields admitting a weierstrass point. Acta. Appl. Math., 93(1-3):299–318, 2006. [14] Pierrick Gaudry. An algorithm for solving the discrete log problem on hyperelliptic curves, pages 19–34. LNCS 1807. Springer, 2000. [15] Nathan Jacobson. Basic Algebra, volume 1. W. H. Freeman and Company. In page 145, chapter 2, exercise 20. [16] Eunkyung Jeong. Isomorphism classes of hyperelliptic curves of genus 3 over finite fields, 2003. Cryptology ePrint Archive 2003/251. [17] N. Koblitz. Hyperelliptic cryptosystems. J. Crypto., 1:139–150, 203–209,1989. [18] Neal Koblitz, A. Menezes, Yi-Hong Wu, and Robert J. Zuccherato. Algebraic aspects of cryptography, pages 155–178. Algorithms and computation in mathematics, vol. 3. Springer, 2nd edition, 1998. In Appendix:“An elementary introduction to hyperelliptic curves”. [19] P. Lockhart. On the discriminant of a hyperelliptic curve. Trans. Amer. Math. Soc., 342(2):729–752, Apr 1994. [20] A. Menezes, Yi-Hong Wu, and Robert J. Zuccherato. An elementary introduction to hyperelliptic curves. Technical Report CORR96-19, Department of C&O, University of Waterloo, Canada, 1996. Available at http://www.cacr.math.uwaterloo.ca/techreports/1997/corr96-19.ps. [21] Michael Rosen. Number theory in function fields. GTM 210. Springer, 2002. Chapter 2 :“Primes, Arithmetic Functions, and the Zeta Function”. [22] Josuph J. Rotman. The Theory of Groups, An Introduction. 2nd edition. In page 45, Theorem 3.23. [23] H. Wilf. Generatingfunctionology. Academic Press, first edition. Available at http://www.math.upenn.edu/˜wilf/DownldGF.html. [24] Tse-Chung Yang. The isomorphism classes of hyperelliptic curves over finite fields eith characteristic 2. Master’s thesis, Nation Taiwan University, Department of Mathematics, Jun 2005.
dc.identifier.uri	http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/29395	-
dc.description.abstract	這篇論文中, 我們利用有別於前人的方法估計了特徵數為 $2$ 的有限體上具有Weierstrass點的超橢圓曲線的同構類數目. 其結果為 $2q^{2g−1}+q^{g−1}+O(q^{g−2})$ 若 $g$ 為奇數; $2q^{2g−1}+q^g+O(q^{g−1})$ 若 $g$ 為偶數. 其中 $g$ 為虧格數, $q$ 為有限體的元素個數.	zh_TW
dc.description.abstract	In this thesis, we will give an asymptotic behavior of the number of hyperelliptic curves with Weierstrass points of arbitrary genus $g$ over $F_q$ when $q$ is even. Our result is $2q^{2g−1}+q^{g−1}+O(q^{g−2})$ if $g$ is odd; $2q^{2g−1}+q^g+O(q^{g−1})$ if $g$is even.	en
dc.description.provenance	Made available in DSpace on 2021-06-13T01:06:02Z (GMT). No. of bitstreams: 1 ntu-96-R94221030-1.pdf: 476801 bytes, checksum: d1dd7613da548bdc78aaf138a06bb454 (MD5) Previous issue date: 2007	en
dc.description.tableofcontents	1.Introduction..............................1 2.Preliminaries..............................3 3.(Rg)s..............................5 3.1.Sing(h)..............................6 3.2.Lemma and Corollary..............................8 3.3.Weight..............................10 3.4.Zeta function..............................11 3.5.Prove Lemma 3.6..............................12 4.Technique Lemmas..............................14 5.FixTg(σ) for a eq1..............................16 5.1.T0..............................19 5.2.T3,T5,...,T2g+1..............................19 5.2.1.g ≤ 4..............................20 5.2.2.g ≥ 5,T3..............................20 5.2.3.g ≥ 5,T5..............................26 6.FixTg(σ) for a=1..............................30 6.1.b = 0..............................30 6.2.b eq0,FixRg (σ)..............................31 6.3.b eq0,Fix(Rg)s(σ)..............................36 Reference..............................38
dc.language.iso	en
dc.subject	超橢圓曲線	zh_TW
dc.subject	同構類	zh_TW
dc.subject	hyperelliptic curve	en
dc.title	有限體上超橢圓曲線之數目	zh_TW
dc.title	The Number of Hyperelliptic Curves over Finite Fields with Even Characteristic	en
dc.type	Thesis
dc.date.schoolyear	95-2
dc.description.degree	碩士
dc.contributor.oralexamcommittee	康明昌(Ming-Chang Kang),陳永秋(Eng-Tjioe Tan),胡守仁(Shou-jen Hu),陳榮凱(Jungkai Alfred Chen)
dc.subject.keyword	超橢圓曲線,同構類,	zh_TW
dc.subject.keyword	hyperelliptic curve,	en
dc.relation.page	40
dc.rights.note	有償授權
dc.date.accepted	2007-07-24
dc.contributor.author-college	理學院	zh_TW
dc.contributor.author-dept	數學研究所	zh_TW
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