  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.
