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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳君明 | |
| dc.contributor.author | Jeng-Rung Jiang | en |
| dc.contributor.author | 江政融 | zh_TW |
| dc.date.accessioned | 2021-06-15T05:53:08Z | - |
| dc.date.available | 2010-08-20 | |
| dc.date.copyright | 2010-08-20 | |
| dc.date.issued | 2010 | |
| dc.date.submitted | 2010-08-17 | |
| dc.identifier.citation | [1] W. Diffie and M. Hellman, New directions in cryptography, IEEE Transactions
on Information Theory IT-22, 644-654, 1976. [2] R.L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM 21, 120-126, 1978. [3] J. Hoffstein, J. Pipher and J. H. Silverman, NTRU, A Ring-Based Public Key Cryptosystem, Algorithmic Number Theory (ANTS III), Portland, OR, June 1998, J.P. Buhler (ed.), LNCS 1423, Springer-Verlag, Berlin, 267-288, 1998. [4] Craig Gentry, Fully Homomorphic Encryption Using Ideal Lattices, ACM 41, 169-178, 2009. [5] Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman, An Introduction to Mathematical Cryptography, Springer-Verlag, New York, Undergraduate Texts in Mathematics, 2008. [6] Jörn Steuding, Diophantine Analysis, Discrete Mathematics and Its Applications, Chapman & Hall/CRC, 2005. [7] O. N. Vasilenko, Number-theoretic Algorithms in Cryptography (Translations of Mathematical Monographs), American Mathematical Society, 2006. [8] J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, 65 Berlin, 1971. [9] Tommi Meskanen, On the NTRU Cryptosystem, TUCS Dissertations No 63, 2005. [10] Daniel Rosenberg, NTRUEncrypt and Lattice Attacks, KTH Department of Numerical Analysis and Computer Science, Royal Institute of Technology SE-100 44 Stockholm, Sweden, 2004. [11] C. Dwork, Lattices and Their Application to Cryptography, Lecture Notes, Stanford University, 1998. [12] János Pach, Pankaj K. Agarwal, Combinatorial Geometry, John Wiley & Sons Inc., 1995. [13] John Horton Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd ed., 1998. [14] C. D. Olds, Anneli Lax, Giuliana P. Davidoff, The Geometry of Numbers (New Mathematical Library), The Mathematical Association of America, 2001. [15] A. K. Lenstra, H. W. Lenstra, and L. Lovász, Factoring Polynomials with Rational Coefficients, Math. Ann. 261, 515-534, 1982. [16] H. Cohen, A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993. [17] B. de Weger, Algorithms for Diophantine equations, Dissertation, Centrum voor 66 Wiskunde en Informatica, Amsterdam, 1988. [18] C.P. Schnorr and M. Euchner, Lattice basis reduction: Improved practical algorithms and solving subset sum problems, Proc. of the FCT 1991, LN in Camp. ScL 529, Springer-Verlag, Berlin, Heidelberg, 68-85, 1991. [19] M. Pohst, A modification of the LLL-algorithm, J. Symb. Camp. 4, 123-128, 1987. [20] C.P. Schnorr, A hierarchy of polynomial time lattice basis reduction algorithms, Theoretical Computer Science 53, 201-224, North-Holland, 1987. [21] A. Korkine and G. Zolotareff, Sur les forms quadratiques, Math. Annafen 6, 366-389, 1873. [22] J.C. Lagarias, H.W. Lenstra, Jr. and C.P. Schnorr, Korkine-Zolotareff bases and successive minima of a lattice and its reciprocal lattice, Tech. Rept., MSRI 07718-86, Mathematical Sciences Research Institute, Berkeley, 1989. [23] Walter Rudin, Principles of Mathematical Analysis, 3rd edition, New York, McGraw-Hill, 1976. [24] NTRU Cryptosystems. Estimated breaking times for NTRU lattices, Technical report, Technical Report 012, 1999, updated 2003. http://www.securityinnovation.com/cryptolab/tech_notes.shtml. [25] Alexander May, Cryptanalysis of NTRU, 1999. 67 [26] J. H. Silverman, Dimension-Reduced Lattices, Zero-Forced Lattices, and the NTRU Public Key Cryptosystem, Technical Report 013, Version 1, NTRU Cryptosystems, 1999. http://www.securityinnovation.com/cryptolab/tech_notes.shtml. [27] R.C. Merkle and M.E. Hellman, Hiding information and signatures in trapdoor knapsacks, IEEE Transactions on Information Theory IT-24(5), 525–530, 1978. [28] A. Shamir, A polynomial-time algorithm for breaking the basic Merkle-Hellman cryptosystem, IEEE Trans. Inform. Theory, 30(5):699–704, 1984. [29] A. M. Odlyzko, The rise and fall of knapsack cryptosystems, In Cryptology and Computational Number Theory (Boulder, CO, 1989), volume 42 of Proc. Sympos. Appl. Math., pages 75–88. Amer. Math. Soc., Providence, RI, 1990. [30] Marten van Dijk, Craig Gentry, Shai Halevi, and Vinod Vaikuntanathan, Fully Homomorphic Encryption over the Integers, Eurocrypt 2010, LNCS 6110, 24–43, 2010. [31] Vadim Lyubashevsky, Chris Peikert, and Oded Regev, On Ideal Lattices and Learning with Errors over Rings, Eurocrypt 2010, LNCS 6110, 1–23, 2010. [32] Nigel P. Smart and Frederik Vercauteren, Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes, PKC 2010, LNCS 6056, 420–443, 2010. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/47273 | - |
| dc.description.abstract | 首先，本論文將介紹NTRU公鑰系統的基本運作，隨即以密碼分析的角度帶出lattice結構的一些相關知識，並且描述和證明LLL演算法，以說明現今攻擊NTRU系統的主要方法。最後將NTRU系統做更進一步的推廣，於參數上使用一些限制條件，賦予加密函數同態的特性。 | zh_TW |
| dc.description.abstract | This thesis introduces how the NTRU cryptosystem works and an elementary cryptanalysis about lattice. After ntroducing NTRU, we briefly describe the lattice structure and LLL, the lattice reduction algorithm from a cryptanalytic point of view, and then express the relations between NTRU and the lattice structure. Finally, we extend the system by adjusting the key space, parameters and message space with
appropriate restrictions to endow NTRU encryption with various properties of ring homomorphism. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T05:53:08Z (GMT). No. of bitstreams: 1 ntu-99-R97221001-1.pdf: 429582 bytes, checksum: 8316ad2110ce9910ffef1fc45990e325 (MD5) Previous issue date: 2010 | en |
| dc.description.tableofcontents | 口試委員會審定書I
中文摘要 II Abstract II 目錄 III 1 Introduction 1 2 NTRU Public-Key Cryptosystem 3 2.1 Polynomial rings 3 2.2 Encryption and decryption of NTRU 4 3 Lattice Reduction 9 3.1 An introduction to the lattice structure 9 3.2 Original LLL algorithm 20 3.3 Variants of LLL algorithm 29 4 Cryptanalysis by Lattice Reduction 34 4.1 Gaussian expected shortest length 34 4.2 NTRU as a lattice problem 38 4.3 Knapsack as a lattice problem 44 5 NTRU with Homomorphism 47 5.1 Homomorphic encryption 47 5.2 Construct a new NTRU with Homomorphism 48 5.3 An application 60 6 Discussion about the extended NTRU 62 參考文獻 64 | |
| dc.language.iso | en | |
| dc.title | NTRU 密碼系統之同態運算及其分析 | zh_TW |
| dc.title | Homomorphism and Cryptanalysis of NTRU | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 98-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 楊柏因,鄭振牟,黃柏嶧 | |
| dc.subject.keyword | 多項式環,NTRU,lattice,LLL演算法,apprSVP,homomorphic encryption, | zh_TW |
| dc.subject.keyword | Polynomial rings,NTRU,lattice,LLL algorithm,apprSVP,homomorphic encryption, | en |
| dc.relation.page | 67 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2010-08-18 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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