How can I convert a mathematical formula into a logical formula?
Considering you are working in $GF(2)$, every multiplication is equivalent to a AND and addition to a XOR.


  $GF(2)$ also known as $&lt;\mathbb{Z}_2,+,\times&gt;$ is the Galois Field (GF) of two elements: $0$ and $1$.
  
  Because $GF(2)$ is a field, 
  
  
  addition has an identity element ($0$) and an inverse for every element;
  multiplication has an identity element ($1$) and an inverse for every element but $0$;
  addition and multiplication are commutative and associative;
  multiplication is distributive over addition.
  
  
  One of the most interesting property of $GF(2)$ is its link with the logical functions over booleans. An analogy can therefore be done between the two:
  
  
  The field addition ($+$) correspond to the logical XOR operation ($\oplus$), also $\forall x \in GF(2), x + x = 0$.
  The field multiplication ($\times$) correspond to the logical AND operation ($\land$).
  
  
  Remarks: $0 + 1 = 1$ ($0 \oplus 1 = 1$) and $1 + 1 = 0$ ($1 \oplus 1 = 0$), therefore $x \mapsto x + 1$ is equivalent to the logical NOT. With this in mind, every booleans operations have an equivalent in $GF(2)$.