In this paper, we determine several new classes of polynomials which permute \(\mathbb {F}_{q^2}\) for \(q=5^m\) . Precisely, we determine the permutation trinomials over \(\mathbb {F}_{q^{2}}\) of the form \(x^r(x^{4(q-1)}+ax^{2(q-1)}+b)\) and \(x^r(x^{6(q-1)}+ax^{4(q-1)}+b)\in \mathbb {F}_5[x]\) for different choices of a, b and r. Furthermore, we explicitly show that these trinomials are not quasi-multiplicative (QM) equivalent to any known permutation trinomial over \(\mathbb {F}_{q^{2}}\) . To prove this we provide a new efficient and generic algorithm which can be used in verification of QM-equivalence between any two permutation trinomials over finite fields. Using these trinomials, we also determine new classes of permutation quadrinomials over \(\mathbb {F}_{q^2}\) of the form \(x^r(x^{6(q-1)}+ax^{4(q-1)}+ax^{2(q-1)}+1)\) for \(a \in \{1, \pm 2\}\) and arbitrary r, and \(x^r(\pm x^{8(q-1)}+x^{6(q-1)}+x^{2(q-1)}\pm 1)\) for different choices of r.
