Let B_{n, m} be the set of all Boolean functions from {0, 1}ⁿ to {0, 1}^m, B_n = B_{n, 1} and U₂ = B₂∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U₂.1.A lower bound (Formula presented.) for a linear function f ∈ B_{n − 1,logn}. The lower bound follows from the following more general result: for any matrix A ∈ {0, 1}^{m × n} with n pairwise different non-zero columns and b ∈ {0, 1}^m,(Formula presented.)2.A lower bound (Formula presented.) for f ∈ B_{n, n}. Again, this is a consequence of the following result: for any f ∈ B_n satisfying a certain simple property,(Formula presented.) where g(f) ∈ B_{n, n} is defined as follows: g(f) = (f ⊕ x₁, … , f ⊕ x_n) (to get a 7n − o(n) lower bound it remains to plug in a known function f ∈ B_{n, 1} with (Formula presented.)).