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Last year Takashima proposed a version of Charles, Goren and Lauter's hashfunction using Richelot isogenies, starting from a genus-2 curve that allowsfor all subsequent arithmetic to be performed over a quadratic finite fieldFp2. In a very recent paper Flynn and Ti point out that Takashima's hashfunction is insecure due to the existence of small isogeny cycles. We revisitthe construction and show that it can be repaired by imposing a simplerestriction, which moreover clarifies the security analysis. The runtime of theresulting hash function is dominated by the extraction of 3 square roots forevery block of 3 bits of the message, as compared to one square root per bit inthe elliptic curve case; however in our setting the extractions can beparallelized and are done in a finite field whose bit size is reduced by afactor 3. Along the way we argue that the full supersingular isogeny graph isthe wrong context in which to study higher-dimensional analogues of Charles,Goren and Lauter's hash function, and advocate the use of the superspecialsubgraph, which is the natural framework in which to view Takashima'sFp2-friendly starting curve.

Hash functions from superspecial genus-2 curves using Richelot isogenies