Modular Arithmetic, Groups, Finite Fields and Probability

2015 ◽  
pp. 3-25
Author(s):  
Nigel P. Smart
Keyword(s):  
Finite Fields ◽  
Modular Arithmetic ◽  
Arithmetic Groups

Number Theory and Finite Fields

2008 ◽  
pp. 33-50
Author(s):  
Manuel Mogollon
Keyword(s):  
Number Theory ◽  
Finite Fields ◽  
Digital Signatures ◽  
Public Key ◽  
Modular Arithmetic ◽  
Math Concepts

Mathematics plays an important role in encryption, public-key, authentication, and digital signatures. Knowing certain basic math concepts such as counting techniques, permutations, plotting a curve, raising a number to a power, modular arithmetic, and congruence would help to understand the material in this book.

scholarly journals NOTE ON A PAPER OF J. HOFFSTEIN

2007 ◽  
Vol 49 (2) ◽  
pp. 243-255 ◽  
Author(s):  
S. J. PATTERSON
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Field ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Arbitrary Order ◽  
Theta Functions ◽  
Theta Series ◽  
Arithmetic Groups ◽  
Recent Developments

AbstractThe concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

10. Vector spaces

2015 ◽  
pp. 126-137
Author(s):  
Peter M. Higgins
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Fields ◽  
Finite Abelian Group ◽  
Scalar Multiplication ◽  
Modular Arithmetic ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Cyclic Groups ◽  
Factorization Of Polynomials

‘Vector spaces’ discusses the algebra of vector spaces, which are abelian groups with an additional scalar multiplication by a field. Every finite abelian group is the direct product of cyclic groups. Any finite abelian group can be represented in one of two special ways based on numerical relationships between the subscripts of the cyclic groups involved. In one representation, all the subscripts are powers of primes; in the alternative, each subscript is a divisor of its successor. It concludes by bringing together the ideas of modular arithmetic, the construction of the complex numbers, factorization of polynomials, and vector spaces to explain the existence of finite fields.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields

Maximum Distance Separable Hankel Rhotrices over Finite Fields

2018 ◽  
Vol 43 (1-4) ◽  
pp. 13-45
Author(s):  
Prof. P. L. Sharma ◽  
◽  
Mr. Arun Kumar ◽  
Mrs. Shalini Gupta ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Maximum Distance ◽  
Maximum Distance Separable

Notes on three-pass protocol

2020 ◽  
Vol 25 (4) ◽  
pp. 4-9
Author(s):  
Yerzhan R. Baissalov ◽  
Ulan Dauyl
Keyword(s):  
Finite Fields ◽  
Matrix Algebras ◽  
Associative Structures

The article discusses primitive, linear three-pass protocols, as well as three-pass protocols on associative structures. The linear three-pass protocols over finite fields and the three-pass protocols based on matrix algebras are shown to be cryptographically weak.

Sign in / Sign up

Export Citation Format

Share Document