99 research outputs found
The Gaussian normal basis and its trace basis over finite fields
AbstractIt is well known that normal bases are useful for implementations of finite fields in various applications including coding theory, cryptography, signal processing, and so on. In particular, optimal normal bases are desirable. When no optimal normal basis exists, it is useful to have normal bases with low complexity. In this paper, we study the type k(⩾1) Gaussian normal basis N of the finite field extension Fqn/Fq, which is a classical normal basis with low complexity. By studying the multiplication table of N, we obtain the dual basis of N and the trace basis of N via arbitrary medium subfields Fqm/Fq with m|n and 1⩽m⩽n. And then we determine all self-dual Gaussian normal bases. As an application, we obtain the precise multiplication table and the complexity of the type 2 Gaussian normal basis and then determine all optimal type 2 Gaussian normal bases
Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields
Let be a finite Galois extension of fields with abelian Galois group
. A self-dual normal basis for is a normal basis with the
additional property that for .
Bayer-Fluckiger and Lenstra have shown that when , then
admits a self-dual normal basis if and only if is odd. If is an
extension of finite fields and , then admits a self-dual normal
basis if and only if the exponent of is not divisible by . In this
paper we construct self-dual normal basis generators for finite extensions of
finite fields whenever they exist.
Now let be a finite extension of \Q_p, let be a finite abelian
Galois extension of odd degree and let \bo_L be the valuation ring of . We
define to be the unique fractional \bo_L-ideal with square equal to
the inverse different of . It is known that a self-dual integral normal
basis exists for if and only if is weakly ramified. Assuming
, we construct such bases whenever they exist
Geometric non-vanishing
We consider -functions attached to representations of the Galois group of
the function field of a curve over a finite field. Under mild tameness
hypotheses, we prove non-vanishing results for twists of these -functions by
characters of order prime to the characteristic of the ground field and by
certain representations with solvable image. We also allow local restrictions
on the twisting representation at finitely many places. Our methods are
geometric, and include the Riemann-Roch theorem, the cohomological
interpretation of -functions, and some monodromy calculations of Katz. As an
application, we prove a result which allows one to deduce the conjecture of
Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function
fields whose -function vanishes to order at most 1 from a suitable
Gross-Zagier formula.Comment: 46 pages. New version corrects minor errors. To appear in Inventiones
Mat
On self-dual MRD codes
We determine the automorphism group of Gabidulin codes of full length and
characterise when these codes are equivalent to self-dual codes.Comment: Improved exposition according to the referees' comment
New results on linear permutation polynomials with coefficients in a subfield
Some families of linear permutation polynomials of with
coefficients in are explicitly described (via conditions
on their coefficients) as isomorphic images of classical subgroups of the
general linear group of degree over the ring
. In addition,
the sizes of some of these families are computed. Finally, several criteria to
construct linear permutation polynomials of (where is
a prime number) with prescribed coefficients in are given.
Examples illustrating the main results are presented
Decoding and constructions of codes in rank and Hamming metric
As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions.
We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x).
An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes.
Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles.
We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin
On interpolation-based decoding of a class of maximum rank distance codes
In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio
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