Related papers: Mathematics of Isogeny Based Cryptography

In 2022, Persianom, Phan and Yung outlined the creation of Anamorphic Cryptography. With this, we can create a public key to encrypt data, and then have two secret keys. These secret keys are used to decrypt the cipher into different…

Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of…

Historically, Elliptic Curve Cryptography (ECC) is an active field of applied cryptography where recent focus is on high speed, constant time, and formally verified implementations. While there are a handful of outliers where all these…

Cryptography is an art and science of secure communication. Here the sender and receiver are guaranteed the security through encryption of their data, with the help of a common key. Both the parties should agree on this key prior to…

An analysis is made of the properties and conditions for the existence of 3- and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of…

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic…

These lecture notes offer a pedagogical yet concise introduction to topological quantum computing. The material focuses on topological superconductors and Majorana qubits. It concludes with a discussion of more general braiding phenomena.…

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by…

This text arises from teaching advanced undergraduate courses in differential topology for the master curriculum in Mathematics at the University of Pisa. So it is mainly addressed to motivated and collaborative master undergraduate…

In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These…

These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced…

I present examples of mathematical objects that are of interest for public key cryptography. Text for the Journ\'ee Annuelle 2007 of the SMF.

We work out the complete descent via 4-isogeny for a family of rational elliptic curves with a rational point of order 4; such a family is of the form $y^2 + x y + a y = x^3 + a x^2$ where $\sqrt{-a} \in \mathbb Q^\times$. In the process we…

The fundamental principles of quantum mechanics are considered to be hard for understanding by unprepared listeners, many attempts of its popularization turned out to be either difficult to grasp or incorrect. We propose quantum…

These lecture notes are meant to serve as an introduction to some geometric constructions and techniques (in particular the ones of toric geometry) often employed by the physicist working on string theory compactifications. The emphasis is…

With the ever-growing concern for internet security, the field of quantum cryptography emerges as a promising solution for enhancing the security of networking systems. In this paper, 20 notable papers from leading conferences and journals…

Like all of quantum information theory, quantum cryptography is traditionally based on two level quantum systems. In this letter, a new protocol for quantum key distribution based on higher dimensional systems is presented. An experimental…

These notes are from a series of lectures given at the Universidad de Los Andes in Bogot\'a, Colombia on some topics of current interest in quantum information. While they aim to be self-contained, they are necessarily incomplete and…

These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…

This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties.…