Character sums of quadratic forms over finite fields and the number of rational points for some classes of artin-schreier type curves

Coşgun, Ayhan
Exponential sums of quadratic forms over finite fields have many applications to various areas such as coding theory and cryptography. As an example to these applications, there is an organic connection between exponential sums of quadratic forms and the number of rational points of algebraic curves defined over finite fields. This connection is central in the application of algebraic geometry to coding theory and cryptography. In this thesis, different facts and techniques of theory of finite fields are combined properly in order to improve and generalize some of the results in the existing literature on evaluation of exponential sums of certain quadratic forms. These evaluations also correspond to the Walsh-Hadamard transforms of Boolean functions in characteristic two. As a result of these evaluations, the number of rational points are computed for some classes of Artin-Schreier type curves over finite fields.


On multiplication in finite fields
Cenk, Murat; Özbudak, Ferruh (2010-04-01)
We present a method for multiplication in finite fields which gives multiplication algorithms with improved or best known bilinear complexities for certain finite fields. Our method generalizes some earlier methods and combines them with the recently introduced complexity notion (M) over cap (q)(l), which denotes the minimum number of multiplications needed in F-q in order to obtain the coefficients of the product of two arbitrary l-term polynomials modulo x(l) in F-q[x]. We study our method for the finite ...
Value sets of Lattes maps over finite fields
Küçüksakallı, Ömer (Elsevier BV, 2014-10-01)
We give an alternative computation of the value sets of Dickson polynomials over finite fields by using a singular cubic curve. Our method is not only simpler but also it can be generalized to the non-singular elliptic case. We determine the value sets of Lattes maps over finite fields which are rational functions induced by isogenies of elliptic curves with complex multiplication.
Galois structure of modular forms of even weight
Gurel, E. (Elsevier BV, 2009-10-01)
We calculate the equivariant Euler characteristics of powers of the canonical sheaf on certain modular curves over Z which have a tame action of a finite abelian group. As a consequence, we obtain information on the Galois module structure of modular forms of even weight having Fourier coefficients in certain ideals of rings of cyclotomic algebraic integers. (c) 2009 Elsevier Inc. All rights reserved.
Free storage basis conversion over extension field
Harold, Ndangang Yampa; Akyıldız, Ersan; Department of Cryptography (2014)
The representation of elements over finite fields play a great impact on the performance of finite field arithmetic. So if efficient representation of finite field elements exists and conversion between these representations is known, then it becomes easy to perform computation in a more efficient way. In this thesis, we shall provide a free storage basis conversion in the extension field F_(q^p) of F_q between Normal basis and Polynomial basis and vice versa. The particularity of this thesis is that, our t...
Large sparse matrix-vector multiplication over finite fields
Mangır, Ceyda; Cenk, Murat; Manguoğlu, Murat; Department of Cryptography (2019)
Cryptographic computations such as factoring integers and computing discrete logarithms require solving a large sparse system of linear equations over finite fields. When dealing with such systems iterative solvers such as Wiedemann or Lanczos algorithms are used. The computational cost of both methods is often dominated by successive matrix-vector products. In this thesis, we introduce a new algorithm for computing a large sparse matrix-vector multiplication over finite fields. The proposed algorithm is im...
Citation Formats
A. Coşgun, “Character sums of quadratic forms over finite fields and the number of rational points for some classes of artin-schreier type curves,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.