Elliptical Curve Cryptography

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• Topic: Cryptography, Public-key cryptography, Key
• Pages : 27 (7500 words )
• Published : May 22, 2011

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CSE 450/598
Design and Analysis of Algorithms
Project ID: P113
Elliptic Curve Cryptography

Final Report
Abstract
The idea of information security lead to the evolution of Cryptography. In other words, Cryptography is the science of keeping information secure. It involves encryption and decryption of messages. Encryption is the process of converting a plain text into cipher text and decryption is the process of getting back the original message from the encrypted text. Cryptography, in addition to providing confidentiality, also provides Authentication, Integrity and Non-repudiation. The crux of cryptography lies in the key involved and the secrecy of the keys used to encrypt or decrypt. Another important factor is the key strength, i.e. the size of the key so that it is difficult to perform a brute force on the plain and cipher text and retrieve the key. There have been various cryptographic algorithms suggested. In this project we study and analyze the Elliptic Curve cryptosystems. This system has been proven to be stronger than known algorithms like RSA/DSA.

Keywords
Cryptography, Public Key Systems, Galois Fields, Elliptic Curve, Scalar Multiplication

Abstract1
Keywords1
Table of Figures3
Table of Algorithms3
1Introduction4
2Individual contributions of the team members5
3Cryptosystems and Public key cryptography6
3.1Brief Overview of some known algorithms7
3.1.1Diffie-Hellman (DH) public-key algorithm:7
3.1.2RSA8
3.1.2.1Working of RSA8
3.1.2.2Security of RSA8
3.1.2.3Difference between RSA and Diffie-Hellman9
3.1.3DSA9
4Mathematical Overview11
4.1Groups11
4.2Rings11
4.3Fields and Vector Spaces11
4.4Finite Fields13
4.4.1Prime Field Fp13
4.4.2Binary Finite Field F2m13
4.4.2.1Polynomial basis representation of F2m14
4.4.2.2Normal basis representation of F2m15
4.5Elliptic Curves16
4.5.1Elliptic Curves over Finite Fields16
4.5.1.1Elliptic Curves over Fp16
4.5.1.2Elliptic curves over F2m19
4.5.2Elliptic Curve: Some Definitions20
5Elliptical Curve Discrete Logarithm Problem21
6Application of Elliptical Curves in Key Exchange22
6.1Elliptic Curve Cryptography (ECC) domain parameters22 6.2Elliptic Curve protocols22
6.2.1Elliptic Curve Diffie-Helman protocol (ECDH)23
6.2.2Elliptic Curve Digital Signature Authentication (ECDSA)24 6.2.3Elliptic Curve Authentication Encryption Scheme (ECAES)26 7Algorithms for Elliptic Scalar Multiplication28
7.2Complexity analysis of the Elliptic Scalar Multiplication algorithms29 7.2.1Binary Method29