Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some properties of ${\tau}$-adic expansions on hyperelliptic Koblitz
  curves   This paper explores two techniques on a family of hyperelliptic curves that
have been proposed to accelerate computation of scalar multiplication for
hyperelliptic curve cryptosystems. In elliptic curve cryptosystems, it is known
that Koblitz curves admit fast scalar multiplication, namely, the ${\tau}$-adic
non-adjacent form ($\tau$-NAF). It is shown that the $\tau$-NAF has the three
properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming
weight. These properties are not only of intrinsic mathematical interest, but
also desirable in some cryptographic applications. On the other hand,
G{\"u}nther, Lange, and Stein have proposed two generalizations of $\tau$-NAF
for a family of hyperelliptic curves, called \emph{hyperelliptic Koblitz
curves}. However, to our knowledge, it is not known whether the three
properties are true or not. We provide an answer to the question. Our
investigation shows that the first one has only the existence and the second
one has the existence and uniqueness. Furthermore, we shall prove that there
exist 16 digit sets so that one can achieve the second one.

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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th