Algorithm 2: Quantum key recovery attack.

Input: $i,j\in \left[m\right]$ and V

Quantum oracle: $|UVZ\rangle \to |UVZ+Z_U\rangle$

1: Set the quantum state to $|0V{(1_{ij})}_{i=j}\rangle \in Z_q^{mn}\times Z_q^{m^2}\times Z_q^{m^2}$.

2: Make a quantum Fourier transform on the first and third registers.

3: Make a quantum oracle query and obtain (by writing $Z^{\prime }=Z+Z_U$).

${\displaystyle \frac{1}{\sqrt{q^{mn}}}\frac{1}{\sqrt{q^{m^2}}}\sum _{U,Z^{\prime },}(\prod _{i=j}{\omega }^{Z_{ij}^{\prime }-Z_{U_{ij}}})|UV{Z}^{\prime }\rangle }$.

4: Discard the last two registers and apply the quantum Fourier transform.

5: Measure the first register and output $\alpha$.