Real hyperelliptic curve explained

. The general formula of Hyperelliptic curve over a finite field is given by $$C\; :\; y^2\; +\; h(x)\; y\; =\; f(x)\; \backslash in\; K[x,y]$$where satisfy certain conditions. 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.

Definition

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$$C(K)\; =\; \backslash left\backslash \; \backslash cup\; S$$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.

Example

Let

where$$f(x)\; =\; x^6\; +3x^5\; -\; 5x^4\; -\; 15x^3\; +\; 4x^2\; +\; 12x\; =\; x(x-1)(x-2)(x+1)(x+2)(x+3)$$over . Since and has degree 6, thus is a curve of genus g = 2.

The homogeneous version of the curve equation is given by$$Y^2\; Z^4\; =\; X^6\; +\; 3\; X^5\; Z\; -\; 5\; X^4\; Z^2\; -\; 15\; X^3\; Z^3\; +\; 4X^2\; Z^4\; +\; 12X\; Z^5.$$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 degree 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 non-singular (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$$D\; =\; \backslash sum\_$$ where and for almost all .

The degree of $D=\; \backslash sum\_$ is defined by $$\backslash deg(D)\; =\; \backslash sum\_.$$

is said to be defined over if $D^\backslash sigma\; =\; \backslash sum\_n\_P\; P^\backslash sigma\; =\; D$ for all automorphisms σ of over . The set of divisors of defined over forms an additive abelian group under the addition rule $$\backslash sum\; a\_P\; P\; +\; \backslash sum\; b\_P\; P\; =\; \backslash sum\; .$$

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$$\backslash mathrm(G)=\backslash sum\_\; \_P(G)P.$$If the function has a pole at a point then }_P (G) 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 $D=\backslash sum\_^r\; P\_i-r\backslash infty\_2$, where }_q), , and if The representative of is then called semi reduced. If satisfies the additional condition then the representative is called reduced.^[1] Notice that is allowed for some i. It follows that every degree 0 divisor class contain a unique representative with $$D=\; D\_x-\backslash deg(D\_x)\; \backslash infty\_2+v\_1\; (D)(\backslash infty\_1-\backslash infty\_2),$$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$$D\_\backslash infty=\; \backslash frac(\backslash infty\_1+\backslash infty\_2).$$

If

is odd then $$D\_\backslash infty=\; \backslash frac\; \backslash infty\_1+\backslash frac\; \backslash infty\_2.$$For example, let the affine parts of two divisors be given by and then the balanced divisors are and

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.^[2]

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 .^[3] Here:

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, we obtain $x=\; \backslash frac$ and $y=\; \backslash frac$. For g = 2, we have $y\; =\; \backslash frac$.

For example, let

then $x=\; \backslash frac$ and $y=\; \backslash frac$, we obtain $$\backslash left(\backslash frac\backslash right)^2=\backslash frac\; \backslash left(\backslash frac\; +1\backslash right)\backslash left(\backslash frac\; +2\backslash right)\backslash left(\backslash frac\; +3\backslash right)\backslash left(\backslash frac\; -1\backslash right)\backslash left(\backslash frac\; -2\backslash right).$$

To remove the denominators this expression is multiplied by

, then:$$y\text{'}^2=(x\text{'}+1)(2x\text{'}+1)(3x\text{'}+1)(4x\text{'}+1)(1)(1-x\text{'})$$giving the curve $$C\text{'}\; :\; y\text{'}^2\; =\; \backslash bar(x\text{'})$$ where $$\backslash bar(x\text{'})\; =\; (x\text{'}+1)\; (2x\text{'}+1)\; (3x\text{'}+1)\; (4x\text{'}+1)\; (1)\; (1-x\text{'})\; =\; -24x\text{'}^5-26x\text{'}^4\; +\; 15x\text{'}^3\; +\; 25x\text{'}^2\; +\; 9x\text{'}+1\; .$$ is an imaginary quadratic curve since has degree .

Notes and References

1. Erickson . Stefan . Jacobson . Michael J., Jr. . Stein . Andreas . 10.3934/amc.2011.5.623 . 4 . Advances in Mathematics of Communications . 2855275 . 623–666 . Explicit formulas for real hyperelliptic curves of genus 2 in affine representation . 5 . 2011.
2. Jacobson . Michael J. Jr. . Scheidler . Renate . Renate Scheidler . Stein . Andreas . 10.2478/v10127-010-0030-9 . Tatra Mountains Mathematical Publications . 2791633 . 31–65 . Cryptographic aspects of real hyperelliptic curves . 47 . 2010.
3. Galbraith . Steven D. . Lin . Xibin . Morales . David J. Mireles . Galbraith . Steven D. . Paterson . Kenneth G. . Pairings on hyperelliptic curves with a real model . https://eprint.iacr.org/2008/250 . 10.1007/978-3-540-85538-5_18 . 2733918 . 265–281 . Springer . Lecture Notes in Computer Science . Pairing-Based Cryptography – Pairing 2008, Second International Conference, Egham, UK, September 1–3, 2008. Proceedings . 5209 . 2008.