The field F_2 has two values, 0 and 1: addition is parity (1+1=0), and multiplication is ordinary multiplication of bits. A monomial is a product of variables; its degree is the number of factors, including repetitions. A sparse representation lists the nonzero monomials rather than every possible coefficient. On Boolean inputs replace x^a by x when a>=1, then cancel pairs of identical monomials. For example, x+x+y becomes y over F_2, not 2x+y. The constant monomial is 1; a list with no monomials represents zero. The equation P(x)=0 asks whether the number of active listed monomials is even.