Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Improvement on the Hasse-Weil Bound and applications to Character
  Sums, Cryptography and Coding   The Hasse-Weil bound is a deep result in mathematics and has found wide
applications in mathematics, theoretical computer science, information theory
etc. In general, the bound is tight and cannot be improved. However, for some
special families of curves the bound could be improved substantially. In this
paper, we focus on the Hasse-Weil bound for the curve defined by $y^p-y=f(x)$
over the finite field $\F_q$, where $p$ is the characteristic of $\F_q$.
Recently, Kaufman and Lovett \cite[FOCS2011]{KL11} showed that the Hasse-Weil
bound can be improved for this family of curves with $f(x)=g(x)+h(x)$, where
$g(x)$ is a polynomial of degree $\ll \sqrt{q}$ and $h(x)$ is a sparse
polynomial of arbitrary degree but bounded weight degree. The other recent
improvement by Rojas-Leon and Wan \cite[Math. Ann. 2011]{RW11} shows that an
extra $\sqrt{p}$ can be removed for this family of curves if $p$ is very large
compared with polynomial degree of $f(x)$ and $\log_pq$.
  In this paper, we show that the Hasse-Weil bound for this special family of
curves can be improved if $q=p^n$ with odd $n$ which is the same case where
Serre \cite{Se85} improved the Hasse-Weil bound. However, our improvement is
greater than Serre's one for this special family of curves. Furthermore, our
improvement works for small $p$ as well compared with the requirement of large
$p$ by Rojas-Leon and Wan. In addition, our improvement finds interesting
applications to character sums, cryptography and coding theory. The key idea
behind is that this curve has the Hasse-Witt invariant $0$ and we show that the
Hasse-Weil bound can be improved for any curves with the Hasse-Witt invariant
$0$. The main tool used in our proof involves Newton polygon and some results
in algebraic geometry.

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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: econ
0
The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph