The HFE cryptosystem is one of the most popular multi- variate schemes. Especially in the area of digital signatures, the HFEv- variant offers short signatures and high performance. Recently, an instance of the HFEv- signature scheme called GeMSS was selected as one of the alternative candidates for signature schemes in the third round of the NIST Post-Quantum Crypto (PQC) Standardization Project.
In this paper, we propose a new key recovery attack on the HFEv- signature scheme. Our attack shows that both the Minus and the Vinegar modification do not enhance the security of the basic HFE scheme significantly. This shows that it is very difficult to build a secure and efficient signature scheme on the basis of HFE. In particular, we use our attack to show that the proposed parameters of the GeMSS scheme are not as secure as claimed.