# Counting the Number of Points on Elliptic Curve Over Finite Field of Characteristic Greater than Three.

December 2008

## Date of Award

Winter 12-12-2009

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

## Supervisor

Barua, Rana (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

The security of discrete logarithm based cryptosystem relies mainly on the order of the underlying group, unless special structures allow more efficient algorithms for breaking the system. If the group order is large enough, then square root attacks like Shankâ€™s baby-step giant-step or pollardâ€™s pmethods are not applicable. Also it is a good strategy to make sure that the group order contains a large prime factor, to prevent the Pohlig-Hellman attack. There are many way to choose an elliptic curves so that above attacks are not possible. The most secure way of selecting a curve is to fix an underlying field, randomly choose a curve and compute the group order until it is divisible by large prime and an Elliptic Curve Cryptosystem is designed using that elliptic curve. There are many algorithm to count such number. First Hasse gave a bound for that count. After that Baby step, Giant step method used Hasse bound to find the count. But most popular algorithm that use Hasse bound is Schoofâ€™s algorithm. Here we basically have designed an algorithm that can test irreducibility of any polynomial(Weierstrass form) of degree 3 without using gcd method. We have used our algorithm to find whether cardinality is even or odd where Schoof used to use gcd method. We have studied the problem to find degree of Frobenius Endomorphism directly. Also we have got some result related to count if we know the type of elliptic curve. Throughout this thesis p(prime) stands for characteristic of the underlying field and q(some power of p) stands for cardinality of the underlying field. Here we are considering only the fields with characteristic p > 3.Definition An elliptic curve E over the field F is of the formy 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6, where ai âˆˆ FThis equation also called generalized Weierstrass equation.We let E(F) denotes the set points (x, y) âˆˆ F 2 that satisfy this equation, along with a â€point at infinityâ€ denoted by âˆž

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843057

## Control Number

ISI-DISS-2008-219