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Owen Williams
Owen Williams

Introduction To Cryptography With Coding Theory !!TOP!!

With its conversational tone and practical focus, this text mixes applied and theoretical aspects for a solid introduction to cryptography and security, including the latest significant advancements in the field. Assumes a minimal background. The level of math sophistication is equivalent to a course in linear algebra. Presents applications and protocols where cryptographic primitives are used in practice, such as SET and SSL. Provides a detailed explanation of AES, which has replaced Feistel-based ciphers (DES) as the standard block cipher algorithm. Includes expanded discussions of block ciphers, hash functions, and multicollisions, plus additional attacks on RSA to make readers aware of the strengths and shortcomings of this popular scheme. For engineers interested in learning more about cryptography.

Introduction to Cryptography with Coding Theory

With its lively, conversational tone and practical focus, this new edition mixes applied and theoretical aspects for a solid introduction to cryptography and security, including the latest significant advancements in the field.

\r \tWith its lively, conversational tone and practical focus, this new edition mixes applied and theoretical aspects for a solid introduction to cryptography and security, including the latest significant advancements in the field.

Cryptography deals with communication over non-secure channels. Coding theory deals with communication over noisy channels. In real life they are often used together. The design of good crypto- and coding systems uses a lot of math, mainly number theory and algebra. Assessing the strength of these systems usually requires probability, information theory, complexity theory. We will discuss the most important topics of cryptography and coding theory, picking up the necessary math along the way. The topics will hopefully include:

  • A few resources:Here is a video I've made where I talk about LaTeX and producing documents with it: Introduction to LaTeX and Sage Math Cloud. (Again, note that "Sage Math Cloud" is simply the old name for Cocalc. The video does not show it in great detail, but might be enough to get you started.) Note it was done for a different course, so disregard any information not about LaTeX itself.

  • TUG's Getting Started: some resources, from installation to first uses.

  • A LaTeX Primer by D. R. Wilkins: a nice introduction. Here is a PDF version.Art of Problem Solving LaTeX resources. A very nice and simple introduction! (Navigate with the links under "LaTeX" bar on top.)

  • LaTeX Symbol Lookup: Draw a symbol and the app will try to identify it and give you its LaTeX code.

  • LaTeX Wikibook: A lot of information.

  • LaTeX Cheat Sheet.

  • Cheat Sheet for Math.

  • List of LaTeX symbols. Comprehensive List of Math Symbols.

  • Constructions: a very nice resource for more sophisticated math expressions.

Back to the TOP.

Long time before computers and global communication networks existed, people were asking, how information can be transmitted in a safe way. Based on the probability theory 50 years ago, Shannon justified the modern communication theory, which forms one of the bases for the modern coding theory and cryptology.

New results of the complexity theory on the one hand and concrete security needs in global nets on the other hand led to a stormy upward development of cryptology in the last years. By the digitization new setting of tasks were added like the digital signature. This group of topics has itself to emphasis in the activity spectrum of computer scientists and mathematicians developed with a shift from the military to the commercial range (e-Commerce).

The aim of this course is to introduce some foundational algebraictechniques used in cryptography and coding theory, as well as to presentsome important and useful protocols.In particular we discuss many modern cryptosystems, includingAES, RSA and EC-based protocols, as to understand their properties andlimitations.

The main aim of cryptography algorithms is to prevent unauthorized people from attaining data that transmitted from one node to another or stored in any environment, even if it is captured, making it impossible to decrypt. Today, basis of many different types of encryption methods is based on classical encryption algorithms. Although many methods which have more complex mathematical infrastructure are tried to solve the data security problem become important by the advancement of technology. The hardware implementation difficulties of these complex methods have led the researchers to the different areas. One of these areas is ANN (Artificial Neural Networks). In the study area called "Neural Cryptography" which is formed by the combination of cryptology and ANN, ANN models are used in both encryption and cryptanalysis phase. In this study, we prepared a Neural Cryptography application and have tried to determine which data is encrypted by which classical method with using ANN.

The homotopy type of an aspherical simplicial or CW complex is uniquely determined by its fundamental group, so homotopy invariants of an aspherical space are invariants of its fundamental group. I will describe asphericity as it relates to relative group presentations and present applications to the theory of cyclically presented groups. Through work of several authors over the past two decades, details are emerging of an interface where aspherical relative presentations transition to non-aspherical ones. Focusing on this interface, I will describe joint work with Gerald Williams in which we discover infinite families of efficient finite groups that admit short presentations and whose orders involve all Mersenne numbers, as well as other conjecturally infinite families of rational primes.

The Homotopy Analysis Method is an innovative new way (Liao, 1992) to get analytical solutions to nonlinear dierential equations. We begin with a brief introduction to the concept of homotopy from topology. From there, the Homotopy Analysis Method is discussed in detail. We describe how the idea of homotopy is applied to introduce a parameter into ordinary/partial differential equations that do not have one to begin with. The homotopy between a linear operator and a nonlinear operator allows us to use perturbation on this parameter to obtain analytical solutions to these equations with small error. The idea is applied to an equation governing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. We also look at the Hasegawa-Mima equation, a very difficult PDE that governs the electric potential due to a drift wave in a plasma. Future work is discussed.

The discovery of the Jones polynomial lead to a vast family of invariants called the quantum invariants. Quantum invariants deeply connect many domains of mathematics such as quantum groups, hyperbolic geometry, knot theory and number theory. In this talk I will talk about quantum invariants and some of their connections with the geometry of the knot complement. Furthermore, I will describe some recent connections with number theory.

We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. Another reason that we propose this invariant is that we deal here with an elementary, interesting an new mathematics and after the Colloquium everybody can take part in developing the topic inventing new results and connections to other disciplines of mathematics (and likely statistical mechanics and combinatorial biology).

Spatial-temporal patterns appear often in historical ecosystem data, and the cause of the patterns can be attributed to various internal or external forces. We demonstrate that in spatial ecological models, spatial-temporal patterns can arise as a result of self-organization of the ecosystem. By using bifurcation theory, we show that the spatial-temporal patterns are generated with the effect of diffusion, advection, chemotaxis or time delay.

Now planar functions have applications in classical cryptographic systems, quantum cryptographic systems, wireless communication, and coding theory. Commutative semifields are equivalent to those planar functions that are known as Dembowski-Ostrom polynomials (DO polynomials). Here I will introduce how Joanne Hall and I have developed methods of constructing families of planar functions and commutative semifields of order \(p^2r\) for any odd prime \(p\) and any positive integer \(r\). These families yield a more general construction which includes some other families of known planar functions while at the same time creates new classes of planar functions. Subsequently these were used to construct mutually unbiased bases, a structure of importance in quantum information theory.

Regrettably, some of the most common assumptions needed for today's cryptographic solutions are no longer justifiable in a so-called post-quantum scenario. In particular, popular constructions involving elliptic curves are not available in this setting. Post-quantum cryptography is of interest when cryptographic solutions are expected to guarantee security for many years. The cryptographic community is currently trying to identify mathematical platforms for efficient post-quantum solutions of basic cryptographic tasks like public-key encryption or digital signatures. The second part of the talk will discuss some of the current approaches, including in particular attempts that invoke tools from group theory. 041b061a72


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