Hilbert series for systems of UOV polynomials

Multivariate public key cryptosystems (MPKC) are constructed based on the problem of solving multivariate quadratic equations (MQ problem). Among various multivariate schemes, UOV is an important signature scheme since it is underlying some signature schemes such as MAYO, QR-UOV, and Rainbow which was a finalist of NIST PQC standardization project. To analyze the security of a multivariate scheme, it is necessary to analyze the first fall degree or solving degree for the system of polynomial equations used in specific attacks. It is known that the first fall degree or solving degree often relates to the Hilbert series of the ideal generated by the system. In this paper, we study the Hilbert series of the UOV scheme, and more specifically, we study the Hilbert series of ideals generated by quadratic polynomials used in the central map of UOV. In particular, we derive a prediction formula of the Hilbert series by using some experimental results. Moreover, we apply it to the analysis of the reconciliation attack for MAYO.