A hyperelliptic curve is a class of algebraic curves. Hyperelliptic curves exist for every genus . The general formula of Hyperelliptic curve over a finite field is given by
where satisfy certain conditions. There are two types of hyperelliptic curves: real hyperelliptic curves and imaginary hyperelliptic curves which differ by the number of points at infinity. In this page, we describe more about real hyperelliptic curves, these are curves having two points at infinity while imaginary hyperelliptic curves have one point at infinity.
A real hyperelliptic curve of genus g over K is defined by an equation of the form where has degree not larger than g+1 while must have degree 2g+1 or 2g+2. This curve is a non singular curve where no point in the algebraic closure of satisfies the curve equation and both partial derivative equations: and . The set of (finite) –rational points on C is given by
Where is the set of points at infinity. For real hyperelliptic curves, there are two points at infinity, and . For any point , the opposite point of is given by ; it is the other point with x-coordinate a that also lies on the curve.
over . Since and has degree 6, thus is a curve of genus g = 2.
The homogeneous version of the curve equation is given by
It has a single point at infinity given by (0:1:0) but this point is singular. The blowup of has 2 different points at infinity, which we denote and . Hence this curve is an example of a real hyperelliptic curve.
In general, every curve given by an equation where f has even degee has two points at infinity and is a real hyperelliptic curve while those where f has odd degree have only a single point in the blowup over (0:1:0) and are thus imaginary hyperelliptic curves. In both cases this assumes that the affine part of the curve is nonsingular (see the conditions on the derivatives above)
Arithmetic in a real hyperelliptic curve
In real hyperelliptic curve, addition is no longer defined on points as in elliptic curves but on divisors and the Jacobian. Let be a hyperelliptic curve of genus g over a finite field K. A divisor on is a formal finite sum of points on . We write
- where and for almost all .
The degree of is defined by
is said to be defined over if for all automorphisms σ of over . The set of divisors of defined over forms an additive abelian group under the addition rule
The set of all degree zero divisors of defined over is a subgroup of .
We take an example:
Let and . If we add them then . The degree of is and the degree of is . Then,
For polynomials , the divisor of is defined by
- . If the function
has a pole at a point then is the order of vanishing of at . Assume are polynomials in ; the divisor of the rational function is called a principal divisor and is defined by . We denote the group of principal divisors by , i.e. . The Jacobian of over is defined by . The factor group is also called the divisor class group of . The elements which are defined over form the group . We denote by the class of in .
There are two canonical ways of representing divisor classes for real hyperelliptic curves which have two points infinity . The first one is to represent a degree zero divisor by such that , where ,, and if The representative of is then called semi reduced. If satisfies the additional condition then the representative is called reduced. Notice that is allowed for some i. It follows that every degree 0 divisor class contain a unique representative with
where is divisor that is coprime with both
- and , and .
The other representation is balanced at infinity. Let , note that this divisor is -rational even if the points and are not independently so. Write the representative of the class as , where is called the affine part and does not contain and , and let . If is even then
If is odd then
For example, let the affine parts of two divisors be given by
then the balanced divisors are
Transformation from real hyperelliptic curve to imaginary hyperelliptic curve
Let be a real quadratic curve over a field . If there exists a ramified prime divisor of degree 1 in then we are able to perform a birational transformation to an imaginary quadratic curve. A (finite or infinite) point is said to be ramified if it is equal to its own opposite. It means that , i.e. that . If is ramified then is a ramified prime divisor.
The real hyperelliptic curve of genus with a ramified -rational finite point is birationally equivalent to an imaginary model of genus , i.e. and the function fields are equal . Here:
- and … (i)
In our example where , h(x) is equal to 0. For any point , is equal to 0 and so the requirement for P to be ramified becomes . Substituting and , we obtain , where , i.e. .
From (i), we obtain and . For g=2, we have
For example, let then and , we obtain
To remove the denominators this expression is multiplied by , then:
giving the curve
- where .
is an imaginary quadratic curve since has degree .
- ^ Erickson, Michael J. Jacobson, Jr., Ning Shang, Shuo Shen, and Andreas Stein, Explicit formulas for real hyperelliptic curves of genus 2 in affine representation
- ^ D. Galbraith, Michael Harrison, and David J. Mireles Morales, Efficient Hyperelliptic Arithmetic Using Balanced Representation for Divisors
- ^ M. J. JACOBSON, JR., R. SCHEIDLER, AND A. STEIN, Cryptographic Aspects of Real Hyperelliptic Curves
- ^ D. Galbraith, Xibin Lin, and David J. Mireles Morales, Pairings on Hyperelliptic Curves with a Real Model