Hyperelliptic curve cryptography (HECC) is a candidate to standardization which is a competitive alternative to elliptic curve cryptography (ECC). We extend Regev's algorithm to this setting. For genus-two curves relevant to cryptography, this yields a quantum attack up to nine times faster than the state-of-the-art. This implies that HECC is slightly weaker than ECC. In a more theoretical direction we show that Regev's algorithm obtains its full speedup with respect to Shor's when the genus is high, a setting which is already known to be inadequate for cryptography.