Hyperelliptic curve cryptography

Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insomuch as the algebraic geometry construct of a hyperelliptic curve with an appropriate group law provides an Abelian group on which to do arithmetic.

The use of hyperelliptic curves in cryptography came about in 1989 from Neal Koblitz. Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA). Because the arithmetic on hyperelliptic curves is more complicated than that on elliptic curves, a properly implemented cryptosystem based on hyperelliptic curves can be more secure than elliptic curve based cryptosystems that have the same key size.

The hyperelliptic curves used are typically of the sort $y^{2}=f(x)\,$ where the degree of $f$ is 5 -- a genus two hyperelliptic curve, or 7 -- a genus three hyperelliptic curve.

Taking the idea of hyperelliptic curve cryptography to the next level, tori can be used in torus based cryptography. These systems are even more complicated (computationally) than hyperelliptic curve based cryptosystems.