# Hash functions from superspecial genus-2 curves using Richelot isogenies

## TL;DR

This paper revisits a genus-2 curve-based hash function using Richelot isogenies, fixing security issues and analyzing its efficiency, while proposing the superspecial subgraph as the appropriate context for higher-dimensional isogeny-based hash functions.

## Contribution

It demonstrates how to repair Takashima's genus-2 hash function by imposing restrictions, clarifies its security, and advocates for using the superspecial subgraph for higher-dimensional isogeny constructions.

## Key findings

- The hash function can be secured by a simple restriction.
- Parallelization reduces the cost of square root extractions.
- The superspecial subgraph is the natural setting for higher-dimensional isogeny hash functions.

## Abstract

Last year Takashima proposed a version of Charles, Goren and Lauter's hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field Fp2. In a very recent paper Flynn and Ti point out that Takashima's hash function is insecure due to the existence of small isogeny cycles. We revisit the construction and show that it can be repaired by imposing a simple restriction, which moreover clarifies the security analysis. The runtime of the resulting hash function is dominated by the extraction of 3 square roots for every block of 3 bits of the message, as compared to one square root per bit in the elliptic curve case; however in our setting the extractions can be parallelized and are done in a finite field whose bit size is reduced by a factor 3. Along the way we argue that the full supersingular isogeny graph is the wrong context in which to study higher-dimensional analogues of Charles, Goren and Lauter's hash function, and advocate the use of the superspecial subgraph, which is the natural framework in which to view Takashima's Fp2-friendly starting curve.