Bounds on the maximum nonlinearity of permutations on the rings \({\mathbb {Z}}_p\) and \({\mathbb {Z}}_{2p}\)

In 2016, Y. Kumar et al. in the paper ‘Affine equivalence and non-linearity of permutations over \({\mathbb {Z}}_n\)’ conjectured that: For \(n\ge 3\), the nonlinearity of any permutation on \({\mathbb {Z}}_n\), the ring of integers modulo n, cannot exceed \(n-2\). For an odd prime p, we settle the above conjecture when \(n=2p\) and for \(p\equiv 3 \pmod {4}\) we prove the above conjecture with an improved upper bound. Further, we derive a lower bound on \(\max {\mathcal {N}}{\mathcal {L}}_n\) when n is an odd prime or twice of an odd prime where \(\max {\mathcal {N}}{\mathcal {L}}_n\) denotes the maximum possible nonlinearity of any permutation on \({\mathbb {Z}}_n\).