Hyperelliptic curve cryptography

Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insofar as the Jacobian of a hyperelliptic curve is an abelian group in which to do arithmetic, just as we use the group of points on an elliptic curve in ECC. == Definition == An (imaginary) hyperelliptic curve of genus g {\displaystyle g} over a field K {\displaystyle K} is given by the equation C : y 2 + h ( x ) y = f ( x ) ∈ K [ x , y ] {\displaystyle C:y^{2}+h(x)y=f(x)\in K[x,y]} where h ( x ) ∈ K [ x ] {\displaystyle h(x)\in K[x]} is a polynomial of degree not larger than g {\displaystyle g} and f ( x ) ∈ K [ x ] {\displaystyle f(x)\in K[x]} is a monic polynomial of degree 2 g + 1 {\displaystyle 2g+1} . From this definition it follows that elliptic curves are hyperelliptic curves of genus 1.

Source: Wikipedia — Hyperelliptic curve cryptography (CC BY-SA 4.0)

Hyperelliptic curve cryptography

Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insofar as the Jacobian of a hyperelliptic curve is an abelian group in which to do arithmetic, just as we use the group of points on an elliptic curve in ECC. == Definition == An (imaginary) hyperelliptic curve of genus g {\displaystyle g} over a field K {\displaystyle K} is given by the equation C : y 2 + h ( x ) y = f ( x ) ∈ K [ x , y ] {\displaystyle C:y^{2}+h(x)y=f(x)\in K[x,y]} where h ( x ) ∈ K [ x ] {\displaystyle h(x)\in K[x]} is a polynomial of degree not larger than g {\displaystyle g} and f ( x ) ∈ K [ x ] {\displaystyle f(x)\in K[x]} is a monic polynomial of degree 2 g + 1 {\displaystyle 2g+1} . From this definition it follows that elliptic curves are hyperelliptic curves of genus 1.

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Source: Wikipedia "Hyperelliptic curve cryptography" · CC BY-SA 4.0

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