The permutation polynomial with the formula b(x q + ax + δ)i + c(x q + ax + δ)j + uxq + vx and their compositional inverses over 𝔽 q ²

The field containing exactly q elements is referred to as 𝔽 q , where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(x q +ax+δ) i +c(x q +ax+δ) j +ux q +vx and its compositional inverse over F q is examined in this work, where b, c ∈ 𝔽 q , a, δ, u and v ∈ 𝔽 q 2 with a 1+q = 1, (av q + u)(au − v) ≠ 0 and u q + v = a(u + av q ).