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Given a divisible finite field extension KjF, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) = Z2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield KjF, and (3) D is the ordinary quaternion division algebra and F(p 1) is divisible  

Classic public key cryptographic systems are based on difficult mathematical problems that will lose their security with the advent of high-speed quantum computers. Isogeny-based cryptography is one of the five main methods in post-quantum cryptography, the hard problem of which is finding large-degree isogenies between elliptic curves. The most important advantage of isogeny-based protocols is the shorter public key length and the main problem is their low speed and low performance compared to other cryptographic systems. Due to the fact that these protocols are implemented on a finite field using basic operations, such as multiplication, squaring, addition, and subtraction, improving the... 

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≃ Z_2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and F(√-1) is divisible  

Public-key cryptography is one of the most practical cryptographic systems today, featuring no need for pre-established secure channel for key exchange. In recent years, major operations in PKE domain such as exponentiation, pairing, ECC and isogeny have been subject of research and numerous studies towards temporal or spatial optimizations. Inversion, being one of the most important operations in this area, requires heavy processing and is considered very time consuming. Hence, aiming to increase performance and speed in PKE processors, addressing Inversion operation in finite fields for improved area and speed is deemed necessary. In cryptographic systems, binary fields are very convenient... 

” The story of the "Weil conjectures" is a marvelous example of mathematical imagination, and one of the most striking instances exhibiting the fundamental unity of mathematics.”In 1949,Andre Weil stated some conjectures on the zeta function of Algebraic varieties over finite fields .These conjectures were analogue of the properties of Riemann zeta function ,in particular Riemann hypothesis.In fact ,Weil built a bridge between Diophantine structure on varieties over finite fields (Counting of rational points on varieties) and cohomological structure of them over the field of complex numbers(topology of variety).In this thesis, first we state Weil’s motivations for these conjectures and state... 

The design of deterministic sensing matrices is an important issue in compressive sensing in sparse signal processing. Various designs using finite field structures, combinatorics, and coding theory have been presented. The contribution of this thesis is designing many codes with large minimum distance using algebraic curves. Here, we initially design a algebraic-geometric code over a maximal curve in a Galoi field Fpm Afterwards, we map the code to the field Fp using trace map. This code has a large minimum distance. Using this code, we design a matrix with low coherence. One of the main issues in presented designs is that the number of matrix rows is considered to specific integers near...