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106 Chapter 3. Bent functions and algebraic curvesThe genus of a hyperelliptic curve will be denoted by g. The above definition includes theelliptic curves, i.e. curves of genus 1, but it is sometimes understood that a hyperelliptic curveshould be of genus g ≥ 2, hence not an elliptic curve.A description of the different normal forms for hyperelliptic curves in even characteristic canbe found in the work of Enge [83]. For cryptographic applications, the curves are often chosen tobe imaginary hyperelliptic curves. This is also the kind of curves we will encounter in Chapter 4.An imaginary hyperelliptic curve of genus g can be described by an affine part given by thefollowing equation 14 :H : y 2 + h(x)y = f(x) ,where h(x) is of degree less than or equal to g and f(x) is monic of degree 2g + 1. In particular,its smooth projective model has only one point at infinity. Such a curve of genus 2 is depicted inFigure 3.7.Figure 3.7: The hyperelliptic curve H : y 2 = (x + 2)(x + 1)x(x − 2)(x − 5/2) of genus 2We finally define an interesting subclass of hyperelliptic curves.Definition 3.2.14 (Artin–Schreier curve). An Artin–Schreier curve is an imaginary hyperellipticcurve of genus g whose affine part is given by an equation of the formH : y 2 + x k y = f(x) ,where 0 ≤ k ≤ g and f(x) is monic of degree 2g + 1.3.2.4 Point countingThe main fact about elliptic and hyperelliptic curves defined over a finite field F q m that we willuse in Chapter 4 is the existence of algorithms to compute their cardinalities in polynomial timeand space in m. Moreover, those algorithms are quite efficient in small characteristic.14 Beware that the projective curve corresponding to the homogenization of the following equation will have asingularity at infinity as soon as g ≥ 2. The hyperelliptic curve is therefore the desingularization of that curve. It isa fact that it also has one and only one point at infinity, and that its affine part is described by the same equation.