Jacobian versus Infrastructure in Real Hyperelliptic Curves

Hyperelliptic curves of low genus are good candidates for curve-based cryptography. Hyperelliptic curves comes in two models: imaginary and real. The existence of two points at infinity in real models makes them more complicated than their imaginary counterparts. However, real models are more general than the other model, every imaginary hyperelliptic curve can be transformed into a real curve over the same base field Fq , while the reverse process requires a larger base field.

Real hyperelliptic curves have not received as much attention by the cryptographic community as imaginary models, but more recent research has shown them to be suitable for cryptography. Real models admit two structures, the Jacobian (a finite abelian group) and the infrastructure (almost group just fails associativity). In this talk, we explain these two structures and compare their arithmetic based on some recent research. We show that the Jacobian makes a better performance in the real model. We also confirm our claim with some numerical evidence for genus 2 and 3 hyperelliptic curves.

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