Essays about: "Elliptic Curves"

Showing result 1 - 5 of 19 essays containing the words Elliptic Curves.

  1. 1. The Elliptic Curve Method : A Modern Approach to Integer Factorization

    University essay from KTH/Skolan för teknikvetenskap (SCI)

    Author : Felix Cao; [2023]
    Keywords : Pollard s p-1 method; Factorization; Elliptic curves; Lentra s; Elliptic curve method;

    Abstract : In this paper, we present a study of elliptic curves, focusing on theirunderlying mathematical concepts, properties and applications in numbertheory. We begin by introducing elliptic curves and their unique features,discussing their algebraic structure, and exploring their group law, pro-viding examples and geometric interpretations. READ MORE

  3. 2. Elliptic Curves and Cryptography

    University essay from Uppsala universitet/Sannolikhetsteori och kombinatorik

    Author : David Ahlqvist; [2022]
    Keywords : ;

    Abstract : .... READ MORE

  4. 3. Effective quasiparallelogram laws on elliptic curves over number fields

    University essay from Göteborgs universitet/Institutionen för matematiska vetenskaper

    Author : Douglas Molin; [2021-05-21]
    Keywords : height; elliptic curve; quasiparallelogram law; canonical height; difference bounds;

    Abstract : We introduce the classical theory of heights on projective space and prove explicit quasiparallelogram laws for the ordinary height and the naive height on elliptic curves over number fields with shortWeierstrass equations. As corollaries, we obtain bounds for the differences between the classical heights and the canonical height, similar to the well-known Silverman bounds. READ MORE

  5. 4. Pairing-Based Cryptography in Theory and Practice

    University essay from Umeå universitet/Institutionen för matematik och matematisk statistik

    Author : Hannes Salin; [2021]
    Keywords : cryptography; pairings; elliptic curves; algebra; provable security; algebraic geometry;

    Abstract : In this thesis we review bilinear maps and their usage in modern cryptography, i.e. the theoretical framework of pairing-based cryptography including the underlying mathematical hardness assumptions. The theory is based on algebraic structures, elliptic curves and divisor theory from which explicit constructions of pairings can be defined. READ MORE

  6. 5. Scheme Theory & Weak Mordell-Weil for Elliptic Curves Over Number Fields

    University essay from Lunds universitet/Matematik LTH; Lunds universitet/Matematik (naturvetenskapliga fakulteten)

    Author : Carl-Fredrik Lidgren; [2021]
    Keywords : algebraic geometry; schemes; scheme theory; elliptic curves; Mordell-Weil; weak Mordell-Weil; algebraic number theory; arithmetic geometry; number theory; geometry; Mathematics and Statistics;

    Abstract : We provide an introduction to scheme-theoretic algebraic geometry, which studies spaces that are in essence locally solutions to systems of polynomial equations, and prove the weak Mordell-Weil theorem. The weak Mordell-Weil theorem states that for an elliptic curve $E$ over a number field $K$, the quotient $E(K)/mE(K)$ is finite for all $m\geq 2$. READ MORE