In this note we represent some results about computation of the group of S-units in hyperelliptic fields. These results can have various applications, in particular in hyperelliptic cryptosystems

Benyash-Krivets, Valery

Platonov, Vladimir

English

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784 Communications of the Moscow Mathematical Society
S-units in hyperelliptic fields
V.V. Benyash-Krivets and V.P. Platonov
In this note we present some results about computation of the group of S-units in
hyperelliptic fields.
Let k = Fq(x) be the field of rational functions of one variable over a finite field Fq of
characteristic p > 2, and let d(x) = a0x
2n+1+a1x
2n+ · · ·+a2n+1 ∈ Fq[x] be a square-free
polynomial, a0 ̸= 0. Let K = k(
√
d ). For an irreducible polynomial v ∈ Fq[x], we denote
by | · |v the corresponding valuation on k. Let (α, β) be a point on the curve y2 = d(x),
β ̸= 0. Then the valuation | · |x−α has two extensions to K. These extensions will be
denoted by | · |1 and | · |2. The non-Archimedean valuation | · |∞ has a unique extension
to K, which also will be denoted by | · |∞. The following two cases are the basic cases for
investigation of S-units: 1) S = {| · |∞, | · |1}; 2) S = {| · |∞, | · |1, | · |2}. Let OS be the ring
of S-integers in K, that is, the elements y ∈ K such that |y|v > 0 for all valuations | · |v
on K which do not belong to S. The set US of all invertible elements of OS is called the
group of S-units of the field K. By the generalized Dirichlet theorem on units (see [1],
Chap. IV, Theorem 9), the group US is the direct product of the group F ∗q and the free
Abelian group G of rank |S| − 1. The independent generators of the group G are called
fundamental S-units.
In the classical case of a quadratic extension L = Q(
√
d ) of Q one can find a funda-
mental unit of the field L by using the continued fraction expansion of
√
d [2]. However
the method of continued fractions does not work properly for function fields. A goal of
this note is to find an algorithm for computing the fundamental S-units in a hyperelliptic
field K in the two basic cases above.
The first proposition is of a technical character.
Proposition 1. Let y = f + g
√
d , where f, g ∈ Fq[x], f ̸= 0, g ̸= 0, (f, g)=1, and let
v ∈ Fq[x] be an irreducible polynomial. Then the following statements are true.
1. If | · |v has two extensions | · |v′ and | · |v′′ to K , then either |y|v′ = 0 or |y|v′′ = 0.
2. If v - d and | · |v has a unique extension | · |v′ to K , then |y|v′ = 0.
3. If v | d and v - f , then | · |v has a unique extension | · |v′ to K and |y|v′ = 0.
4. If v | d and v | f , then | · |v has a unique extension | · |v′ to K and |y|v′ = 1/2.
The following proposition gives a characterization of S-integers in K.
Proposition 2. Every element y ∈ OS admits a representation
y = (f + g
√
d )(x− α)−m,
where f, g ∈ Fq[x] and m > 0. Moreover, if m > 0 and x − α does not divide f and g
simultaneously, then x− α - f and x− α - g.
The proof of Proposition 2 is based on the following observation: if the denominator
of y is divisible by an irreducible polynomial v ̸= x − α, then for some extension | · |v′ of
| · |v we have |y|v′ < 0 by Proposition 1.
We note that not every element of type (f + g
√
d )(x − α)m is an S-integer. In what
follows it is important to us to determine the images of the norm map on S-units.
AMS 2000 Mathematics Subject Classification. Primary 11T99; Secondary 11G20, 11R27,
16U60.
DOI 10.1070/RM2007v062n04ABEH004435.
Communications of the Moscow Mathematical Society 785
Proposition 3. If ε = (f + g
√
d )/(x − α)m1 ∈ US , where f, g ∈ Fq[x], and if ε ̸∈ F ∗q ,
then f ̸= 0, g ̸= 0, (f, g) = 1, and NK/k(ε) = a(x− α)m, where a ∈ F ∗q and 0 ̸= m ∈ Z.
As in the case of S-integers, if an element ε ∈ K has the property that NK/k(ε) =
a(x − α)m, it does not follow that ε is an S-unit. However, the following proposition
shows that if the norm equation
f2 − g2d = a(x− α)m (1)
with m ̸= 0 fixed is soluble in polynomials f, g ∈ Fq[x], then either f + g
√
d ∈ US or
f − g√d ∈ US .
Proposition 4. Let y = f + g
√
d with f, g ∈ Fq[x], and let NK/k(y) = a(x − α)m,
where a ∈ F ∗q and m > 0. If |y|2 = 0, then y = f + g
√
d ∈ US , and if |y|2 > 0, then
(f + g
√
d )/(x− α)m ∈ US .
Theorem 1. Let S′ = {| · |∞, | · |1, | · |2} and S = {| · |∞, | · |1}. If ε is a fundamental
S-unit, then ε and x− α are independent fundamental S′-units.
The proof of Theorem 1 is based on the following observation: if δ is an arbitrary
S′-unit, then, as in the S-unit case, NK/k(δ) = a(x − α)m. By Proposition 4, we have
either δ ∈ US or δ/(x − α)m ∈ US . In the first case δ = bεt, while in the second case
δ = bεt(x− α)m, where b ∈ F ∗q and t ∈ Z. This means that ε and x− α are independent
fundamental S′-units.
Therefore, first of all we need to be able to compute a fundamental S-unit. With this
aim, we need to find a minimal positive integerm such that the equation (1) has a solution
in polynomials f, g ∈ Fq[x]. Then either f + g
√
d or f − g√d is a fundamental S-unit.
Let v = x− α, and let kv be a completion of k with respect to the valuation | · |v. The
field kv can be identified with the field Fq((v)) of formal power series. Since the valuation
| · |v has two extensions to K, we have
√
d ∈ Fq((v)). Let
√
d =
∑∞
i=0 div
i. The following
theorem provides an algorithm for finding a fundamental S-unit.
Theorem 2. Let n = (deg d− 1)/2. For an integer r > n consider the matrices
Dr =

dn+1 dn+2 . . . dr+1
dn+2 dn+3 . . . dr+2
. . . . . . . . . . . .
dn+r dn+r+1 . . . d2r
 and Hr =

dn+2 dn+3 . . . dr+2
dn+3 dn+4 . . . dr+3
. . . . . . . . . . . .
dn+r+1 dn+r+2 . . . d2r+1
 .
If m = 2r + 1 is odd, then the norm equation (1) has a solution in non-zero polynomials
f, g ∈ Fq[x] if and only if rankDr < r − n + 1, and if m = 2r is even, then (1) has
a solution f, g ∈ Fq[x] if and only if rankHr−1 < r − n.
If K is the function field of an elliptic curve, then Dr and Hr are square matrices. In
this case determinants of the special kind in Theorem 2 were first considered by Jacobi
in the 19th century. They are called ‘persymmetric determinants’ and the corresponding
matrices are known as ‘Hankel matrices’.
Remark. We have restricted ourselves here to valuations associated with points on hyper-
elliptic curves over a field Fq. In fact, similar results are valid for arbitrary valuations on
the field Fq(x).
786 Communications of the Moscow Mathematical Society
Bibliography
[1] A. Weil, Basic number theory, Grundlehren Math. Wiss., vol. 144, Springer, New York 1967.
[2] З.И. Боревич, И. Р. Шафаревич, Теория чисел, Наука, Москва 1964; English transl.,
Z. I. Borevich and I. R. Shafarevich, Number theory, Pure Appl. Math., vol. 20, Academic
Press, New York–London 1966.
V.V. Benyash-Krivets
Belorussian State University
E-mail : benyash@bsu.by
V.P. Platonov
Scientific Research Institute for Systems Analysis,
Russian Academy of Sciences
E-mail : platonov@niisi.ras.ru
Presented by V.M. Buchstaber
Accepted 16/JUL/07
Translated by THE AUTHORS


S-units in hyperelliptic fields

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S-units in hyperelliptic fields

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