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Articles 1 - 9 of 9
Full-Text Articles in Algebra
On Constructions Of Maximum Distance Separable Pascal-Like Rhotrices Over Finite Fields, Neetu Dhiman, Mansi Harish, Shalini Gupta, Arun Chauhan
On Constructions Of Maximum Distance Separable Pascal-Like Rhotrices Over Finite Fields, Neetu Dhiman, Mansi Harish, Shalini Gupta, Arun Chauhan
Applications and Applied Mathematics: An International Journal (AAM)
Cryptography and coding theory are the important areas where Maximum Distance Separable (MDS) matrices are used extensively. The Pascal matrix plays vital role in combinatorics, matrix theory and its properties provide interesting combinatorial identities. Pascal matrices also have a wide range of applications in cryptography. In this paper, we define Pascal-like rhotrix, and further, we construct MDS Pascal-like rhotrices over finite fields.
The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna
The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna
Senior Honors Theses
The ability to communicate securely across the internet is owing to the security of the RSA cryptosystem, among others. This cryptosystem relies on the difficulty of integer factorization to provide secure communication. Peter Shor’s quantum integer factorization algorithm threatens to upend this. A special case of the hidden subgroup problem, the algorithm provides an exponential speedup in the integer factorization problem, destroying RSA’s security. Robert McEliece’s cryptosystem has been proposed as an alternative. Based upon binary Goppa codes instead of integer factorization, his cryptosystem uses code scrambling and error introduction to hinder decrypting a message without the private key. This …
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
All Theses
In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …
Cryptography Through The Lens Of Group Theory, Dawson M. Shores
Cryptography Through The Lens Of Group Theory, Dawson M. Shores
Electronic Theses and Dissertations
Cryptography has been around for many years, and mathematics has been around even longer. When the two subjects were combined, however, both the improvements and attacks on cryptography were prevalent. This paper introduces and performs a comparative analysis of two versions of the ElGamal cryptosystem, both of which use the specific field of mathematics known as group theory.
Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick
Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick
Senior Honors Theses
Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow …
Basis Reduction In Lattice Cryptography, Raj Kane
Basis Reduction In Lattice Cryptography, Raj Kane
Honors Theses
We develop an understanding of lattices and their use in cryptography. We examine how reducing lattice bases can yield solutions to the Shortest Vector Problem and the Closest Vector Problem.
Distance In Matrices And Their Applications To Fuzzy Models And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Distance In Matrices And Their Applications To Fuzzy Models And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral
Branch Mathematics and Statistics Faculty and Staff Publications
In this book authors for the first time introduce the notion of distance between any two m n matrices. If the distance is 0 or m n there is nothing interesting. When the distance happens to be a value t; 0 < t < m n the study is both innovating and interesting. The three cases of study which is carried out in this book are 1. If the difference between two square matrices is large, will it imply the eigen values and eigen vectors of those matrices are distinct? Several open conjectures in this direction are given. 2. The difference between parity check matrix and the generator matrix for the same C(n, k) code is studied. This will help in detecting errors in storage systems as well as in cryptography.
The Square Discrete Exponentiation Map, A Wood
The Square Discrete Exponentiation Map, A Wood
Mathematical Sciences Technical Reports (MSTR)
We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.
Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett
Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett
Mathematical Sciences Technical Reports (MSTR)
Cryptographic algorithms are based on a wide variety of difficult problems in mathematics. One of these problems is finding a solution to a system of multivariate quadratic equations (MQ). A generalization of this problem is to find a solution to a system of higher order non-linear equations. Both of these problems are NP-hard over any field. Many cryptosystems such as AES, Serpent, Toyocrypt, and others can be reduced to some form of the MQ problem. In this paper we analyze the relinearization and XL algorithms for solving overdetermined systems of non-linear equations, as well as two variations of the XL …