Curves, Jacobians, and Cryptography

The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms. Several open problems and new directions are suggested.Comment: 66 page

Towards a deeper understanding of APN functions and related longstanding problems

This dissertation is dedicated to the properties, construction and analysis of APN and AB functions. Being cryptographically optimal, these functions lack any general structure or patterns, which makes their study very challenging. Despite intense work since at least the early 90's, many important questions and conjectures in the area remain open. We present several new results, many of which are directly related to important longstanding open problems; we resolve some of these problems, and make significant progress towards the resolution of others. More concretely, our research concerns the following open problems: i) the maximum algebraic degree of an APN function, and the Hamming distance between APN functions (open since 1998); ii) the classification of APN and AB functions up to CCZ-equivalence (an ongoing problem since the introduction of APN functions, and one of the main directions of research in the area); iii) the extension of the APN binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$ into an infinite family (open since 2006); iv) the Walsh spectrum of the Dobbertin function (open since 2001); v) the existence of monomial APN functions CCZ-inequivalent to ones from the known families (open since 2001); vi) the problem of efficiently and reliably testing EA- and CCZ-equivalence (ongoing, and open since the introduction of APN functions). In the course of investigating these problems, we obtain i.a. the following results: 1) a new infinite family of APN quadrinomials (which includes the binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$); 2) two new invariants, one under EA-equivalence, and one under CCZ-equivalence; 3) an efficient and easily parallelizable algorithm for computationally testing EA-equivalence; 4) an efficiently computable lower bound on the Hamming distance between a given APN function and any other APN function; 5) a classification of all quadratic APN polynomials with binary coefficients over $F_{2^n}$ for $n \le 9$; 6) a construction allowing the CCZ-equivalence class of one monomial APN function to be obtained from that of another; 7) a conjecture giving the exact form of the Walsh spectrum of the Dobbertin power functions; 8) a generalization of an infinite family of APN functions to a family of functions with a two-valued differential spectrum, and an example showing that this Gold-like behavior does not occur for infinite families of quadratic APN functions in general; 9) a new class of functions (the so-called partially APN functions) defined by relaxing the definition of the APN property, and several constructions and non-existence results related to them.Doktorgradsavhandlin

Advances of Machine Learning in Materials Science: Ideas and Techniques

In this big data era, the use of large dataset in conjunction with machine learning (ML) has been increasingly popular in both industry and academia. In recent times, the field of materials science is also undergoing a big data revolution, with large database and repositories appearing everywhere. Traditionally, materials science is a trial-and-error field, in both the computational and experimental departments. With the advent of machine learning-based techniques, there has been a paradigm shift: materials can now be screened quickly using ML models and even generated based on materials with similar properties; ML has also quietly infiltrated many sub-disciplinary under materials science. However, ML remains relatively new to the field and is expanding its wing quickly. There are a plethora of readily-available big data architectures and abundance of ML models and software; The call to integrate all these elements in a comprehensive research procedure is becoming an important direction of material science research. In this review, we attempt to provide an introduction and reference of ML to materials scientists, covering as much as possible the commonly used methods and applications, and discussing the future possibilities.Comment: 80 pages; 22 figures. To be published in Frontiers of Physics, 18, xxxxx, (2023

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Line Primitives and Their Applications in Geometric Computer Vision

Line primitives are widely found in structured scenes which provide a higher level of structure information about the scenes than point primitives. Furthermore, line primitives in space are closely related to Euclidean transformations, because the dual vector (also known as Pluecker coordinates) representation of 3D lines is the counterpart of the dual quaternion which depicts an Euclidean transformation. These geometric properties of line primitives motivate the work in this thesis with the following contributions: Firstly, by combining local appearances of lines and geometric constraints between line pairs in images, a line segment matching algorithm is developed which constructs a novel line band descriptor to depict the local appearance of a line and builds a relational graph to measure the pair-wise consistency between line correspondences. Experiments show that the matching algorithm is robust to various image transformations and more efficient than conventional graph based line matching algorithms. Secondly, by investigating the symmetric property of line directions in space, this thesis presents a complete analysis about the solutions of the Perspective-3-Line (P3L) problem which estimates the camera pose from three reference lines in space and their 2D projections. For three spatial lines in general configurations, a P3L polynomial is derived which is employed to develop a solution of the Perspective-n-Line problem. The proposed robust PnL algorithm can efficiently and accurately estimate the camera pose for both small numbers and large numbers of line correspondences. For three spatial lines in special configurations (e.g., in a Manhattan world which consists of three mutually orthogonal dominant directions), the solution of the P3L problem is employed to solve the vanishing point estimation and line classification problem. The proposed vanishing point estimation algorithm achieves high accuracy and efficiency by thoroughly utilizing the Manhattan world characteristic. Another advantage of the proposed framework is that it can be easily generalized to images taken by central catadioptric cameras or uncalibrated cameras. The third major contribution of this thesis is about structure-from-motion using line primitives. To circumvent the Pluecker constraints on the Pluecker coordinates of lines, the Cayley representation of lines is developed which is inspired by the geometric property of the Pluecker coordinates of lines. To build the line observation model, two derivations of line projection functions are presented: one is based on the dual relationship between points and lines; and the other is based on the relationship between Pluecker coordinates and the Pluecker matrix. Then the motion and structure parameters are initialized by an incremental approach and optimized by sparse bundle adjustment. Quantitative validations show the increase in performance when compared to conventional line reconstruction algorithms

Hadrons Under Extreme Conditions

At high temperature and density, strongly interacting matter experiences a phase transition from a hadronic phase to a quark-gluon plasma. Heavier hadrons are able to survive longer inside the quark-gluon plasma, but their properties change as they approach deconfinement. This thesis aims to investigate the onset of deconfinement and the properties of heavy hadrons as they approach this state, with a particular focus on baryons under extreme conditions. The research will utilise non-relativistic potential models and will build upon the method of Silvestre et al. (2020) [1], which utilises a variational approach to solve three-body potential models through expansion of the wave function in a simple harmonic oscillator basis. The project will also extend this method to solve spin-dependent baryon models. The main focus of this thesisis the application of this method to temperature-dependent baryon potential models. To this end, we will solve two such models that include a spin-spin interaction term. We find that the masses of heavy baryons decrease as temperature is increased, less so for the heavier baryons, for the heaviest baryons we found its mass to actually increase as temperature was increased. We have also used our method to predict the dissociation points of some heavy baryons and have found that heavier baryons are able to survive for longer, as temperature is increased. Furthermore, all baryons grow quickly in size as temperature is increased, reaching infinite size at criticality

State of the Art Report : Verified Computation

This report describes the state of the art in verifiable computation. The problem being solved is the following: The Verifiable Computation Problem (Verifiable Computing Problem) Suppose we have two computing agents. The first agent is the verifier, and the second agent is the prover. The verifier wants the prover to perform a computation. The verifier sends a description of the computation to the prover. Once the prover has completed the task, the prover returns the output to the verifier. The output will contain proof. The verifier can use this proof to check if the prover computed the output correctly. The check is not required to verify the algorithm used in the computation. Instead, it is a check that the prover computed the output using the computation specified by the verifier. The effort required for the check should be much less than that required to perform the computation. This state-of-the-art report surveys 128 papers from the literature comprising more than 4,000 pages. Other papers and books were surveyed but were omitted. The papers surveyed were overwhelmingly mathematical. We have summarised the major concepts that form the foundations for verifiable computation. The report contains two main sections. The first, larger section covers the theoretical foundations for probabilistically checkable and zero-knowledge proofs. The second section contains a description of the current practice in verifiable computation. Two further reports will cover (i) military applications of verifiable computation and (ii) a collection of technical demonstrators. The first of these is intended to be read by those who want to know what applications are enabled by the current state of the art in verifiable computation. The second is for those who want to see practical tools and conduct experiments themselves

Security and Privacy for Modern Wireless Communication Systems

The aim of this reprint focuses on the latest protocol research, software/hardware development and implementation, and system architecture design in addressing emerging security and privacy issues for modern wireless communication networks. Relevant topics include, but are not limited to, the following: deep-learning-based security and privacy design; covert communications; information-theoretical foundations for advanced security and privacy techniques; lightweight cryptography for power constrained networks; physical layer key generation; prototypes and testbeds for security and privacy solutions; encryption and decryption algorithm for low-latency constrained networks; security protocols for modern wireless communication networks; network intrusion detection; physical layer design with security consideration; anonymity in data transmission; vulnerabilities in security and privacy in modern wireless communication networks; challenges of security and privacy in node–edge–cloud computation; security and privacy design for low-power wide-area IoT networks; security and privacy design for vehicle networks; security and privacy design for underwater communications networks