Elliptical Curve Cryptography

Topics: Cryptography, Public-key cryptography, Key Pages: 27 (7500 words) Published: May 22, 2011
CSE 450/598
Design and Analysis of Algorithms
Project ID: P113
Elliptic Curve Cryptography

|Vikram V Kumar (vikramv@asu.edu) |[Grad.] | |Satish Doraiswamy (satish.d@asu.edu) |[Grad.] | |Zabeer Jainullabudeen (zabeer@asu.edu) |[Grad.] |

Final Report
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.

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

Table of Contents
Table of Contents2
Table of Figures3
Table of Algorithms3
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 of RSA8 of RSA8 between RSA and Diffie-Hellman9
4Mathematical Overview11
4.3Fields and Vector Spaces11
4.4Finite Fields13
4.4.1Prime Field Fp13
4.4.2Binary Finite Field F2m13 basis representation of F2m14 basis representation of F2m15
4.5Elliptic Curves16
4.5.1Elliptic Curves over Finite Fields16 Curves over Fp16 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
7.1Non adjacent form (NAF)28
7.2Complexity analysis of the Elliptic Scalar Multiplication algorithms29 7.2.1Binary Method29
7.2.2Addition-Subtraction method30
7.2.3Repeated doubling method31

Table of Figures
Figure 1:Elliptic curve over R2: y2 = x3 – 3x + 316
Figure 2:Addition of 2 points P and Q on the curve y2 = x3 – 3x + 317 Figure 3:Doubling of a point P, R = 2P on the curve y2 = x3 – 3x + 318 Figure 4:Illustration of Elliptic Curve Diffie-Hellman Protocol24 Figure 5:Illustration of Elliptic Curve Digital Signature Algorithm25 Figure 6:Illustration of Elliptic Curve Authentication Encryption Scheme27 Figure 7:Illustration of computation of NAF(7)29

Figure 8:Comparison of the key strengths of RSA/DSA and ECC32

Table of Algorithms
Algorithm 1:Computation of the NAF of a scalar28
Algorithm 2:Scalar Multiplication using the Addition-Subtraction method30 Algorithm 3:Scalar Multiplication using Repeated Additions31...
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