faculty: "FNWI" and publication year: "2010"
| Author||Tom Veerman|
|Title||Elliptic Curve Cryptography|
|Faculty||Faculty of Science|
|Institute/dept.||FNWI: Korteweg-De Vries Instituut|
|Programme||FNWI MSc Mathematics|
|Keywords||Cryptography; Elliptic curves; Discrete logarithm problem|
|Abstract||In this thesis, we examine the mathematics behind elliptic curve cryptography. We first discuss the notion of a public key cryptosystems (PKC), which are based on mathematical problems that take much time to solve. Elliptic curve cryptosystems form examples of PKC's, and are based on the discrete logarithm problem (DLP). This is the problem of finding a number k, such that kg = h for some elements g; h in an abelian group. We examine the difficulty of this problem by investigating some algorithms for solving it, namely
the baby-step giant-step algorithm, the Pollard p-algorithm, index calculus on F*p, and also the Pohlig-Hellman algorithm that speeds up finding discrete logarithms.
We also treat some theory about elliptic curves, especially the ones over finite fields. Important for cryptographic purposes, and therefore included in this thesis, are torsion points, the Weil-pairing from torsion points to roots of unity, the Schoof-algorithm for determining the number of points, and supersingular elliptic curves.
Finally we glue all the theory together, and define PKC's over elliptic curve groups. We also discuss the MOV-algorithm, which uses the Weil-pairing to translate a DLP on elliptic curves to a DLP on the multiplicative group of a finite field. When the elliptic curve in consideration is supersingular, this algorithm is relatively fast, so we should exclude this class from being used for cryptographic purposes.|
|Document type|| scriptie master|
Use this url to link to this page: http://dare.uva.nl/en/scriptie/365297
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