Related papers: Normal Bases using 1-dimensional Algebraic Groups
In this paper, a method for constructing a near optimal normal basis for algebraic extensions of a finite field is described. In each extension, except for the squares of basis elements, the product of two distinct normal basis elements can…
Let L be an abelian number field of degree n with Galois group G. In this paper we study how to compute efficiently a normal integral basis for L, if there is at least one, assuming that the group G and an integral basis for L are known.
We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of…
We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse…
We describe algorithms to compute fixed fields, splitting fields and towers of radical extensions without using polynomial factorisation in towers or constructing any field containing the splitting field, instead extending Galois group…
In this article, we present two methods to construct 2-uninorms on bounded lattices by using additive generators, which are further used for inducing uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively. We also provide some…
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
Normal basis is used in many applications because of the efficiency of the implementation. However, most space complexity reduction techniques for binary field multiplier are applicable for only optimal normal basis or Gaussian normal basis…
We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting…
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96],…
We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for…
In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloor\alpha p^\theta\rfloor$ with $\alpha>0$ and $\theta>1$.
Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of…
We study how to lift Markov bases and Gr\"obner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the…
It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $\alpha \in K$ whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$ is called…
We construct standard resolutions for analytic local modules on complex hypersurfaces using standard basis methods, with extensions to complete intersections. The algebraic version over arbitrary infinite fields is also suggested.…
Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer…
Algebraic methods are used to construct families of unramified abelian extensions of some families of number fields with specified Galois groups.