CCZ equivalence of power functions

Let \(F\simeq {{\mathrm{GF}}}(p^n)\) be a finite field of characteristic p and \(p_k\) and \(p_\ell \) be power functions on F defined by \(p_k(x)=x^k\) and \(p_\ell (x)=x^\ell \) respectively. We show, that \(p_k\) and \(p_\ell \) are CCZ equivalent, if and only if there exists a positive integer \(0\le a< n\), such that \(\ell \equiv p^a k \pmod {p^n-1}\) or \(k\ell \equiv p^a \pmod {p^n-1}\).