Eliptičke krivulje i kriptiranje

Musulin, Zdravko (2016) Eliptičke krivulje i kriptiranje. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

Elliptic curves have been intensively studied in theory of algebraic geometry for many years. However, by development of computers they had a big breakthrough and have been playing an increasingly important role both in number theory and in related fields such as cryptography. At that time, elliptic curve techniques for factorization and primality testing were developed and also hardness of the elliptic curve discrete logarithm problem was discovered, which led to its application in algorithms based on that problem. Cryptography is fascinating because of the close ties it forges between theory and practice. Because of that, today's practical applications of cryptography are pervasive and crucial components of our information-based society. The theoretical work refines and improves the practice, while the practice challenges and inspires the theoretical study. When some system is “broken", our knowledge expands and next, upgraded system repairs the previous defect. The importance of elliptic curves is best shown in 1995, when they figured prominently in the proof of Fermat's Last Theorem by A. Wiles. This thesis provides a general overview of elliptic curves and their properties over the field of rational numbers and also over finite fields. Public key cryptography is also described, focusing on both the discrete logarithm problem and the elliptic curve discrete logarithm problem. What makes ECC interesting is that, as of today, the discrete logarithm problem for elliptic curves seems to be “harder" if compared to other similar problems used in cryptography. This implies that we need keys with fewer bits in order to achieve the same level of security as with other cryptosystems. Furthermore, we explain Diffie-Hellman key exchange protocol and finally study some public key cryptosystems, especially ones using elliptic curves.

Item Type: Thesis (Diploma thesis)
Supervisor: Franušić, Zrinka
Date: 2016
Number of Pages: 58
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 26 Aug 2016 08:13
Last Modified: 26 Aug 2016 08:13
URI: http://digre.pmf.unizg.hr/id/eprint/4991

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