Contemporary Mathematics
Volume 574, 2012
http://dx.doi.org/10.1090/conm/574/11429
Construction of a k-complete addition law on Jacobians of
hyperelliptic curves of genus two
Christophe Ar` ene and Romain Cosset
Abstract. In this paper we explain how to construct Fq-complete addition
laws on the Jacobian of an hyperelliptic curve of genus 2 classically embedded
in
P15,
and provide the first explicit example of such addition laws in the
genus 2 case. This is a generalization to abelian surfaces of the arithmetic
completeness of addition laws on elliptic models firstly revealed with Edwards
curves in 2007.
1. Introduction
Cryptographic protocols using abelian varieties, specifically elliptic curves and
abelian surfaces, have been studied intensively for at least the past two decades.
They are based on the discrete logarithm problem and pairings for which the com-
putation of the addition of two points is central. In particular, one pays attention to
two aspects. Obviously, the number of operations needed to compute the equations
must be as small as possible. It appears that their domain of definition has also to
be taken into account. Indeed, with the development of embedded cryptosystems,
the theoretical resistance of the discrete logarithm problem is no longer sufficient
to ensure the protocol security, we also have to deal with physical attacks. For
instance the implementation of the usual formulæ on the Weierstraß model of an
elliptic curve is vulnerable against side-channel attacks due to the use of different
formulæ for a generic addition or a doubling. We refer to [LM05] for a possible
alternative on genus 2 curve cryptosystems.
In this paper we only consider this second problem. Lange and Ruppert [LR85]
first considered this problem geometrically. They worked on the existence of com-
plete sets of addition laws, i.e. for all P , Q in A
(
k
)
there is an addition law defined
at (P, Q). See the definition of completeness in Definition 1.4 . Then Bosma and
Lenstra [BL95] proved for an elliptic curve given in Weierstraß form that such a
set has always cardinality greater than one, a fact generalized in [AKR11] to any
abelian variety with a projective embedding. This motivates to restrict the additon
laws to the points defined on a non algebraically closed base field, specifically a
finite field. An addition law is said to be k-complete if it is defined over (A×A)(k).
2010 Mathematics Subject Classification. Primary 68-04; Secondary 11-04.
Key words and phrases. Theta functions, Jacobian, genus 2 curve, addition law, complete-
ness, embedding, line bundle, finite field.
The authors acknowledge the financial support by grant ANR-09-BLAN-0020-01 from the
French ANR and the AXA Research Fund for the Ph.D. grant of the first author.
c 2012 American Mathematical Society
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