Summary: Quasigroups of huge order, like \(2^{256}\), 2\(^{512}\), \(2^{1024}\), that can be effectively constructed, have important applications in designing several cryptographic primitives. We propose an effective method for construction of such huge quasigroups of order \(r=2^{s2^t}\) for small fixed values of \(s\) and arbitrary values of \(t\); the complexity of computation of the quasigroup multiplication is \(\mathcal O(\log(\log(r)))=\mathcal O(t)\). Besides the computational effectiveness, these quasigroups can be constructed in such a way to have other desirable cryptographic properties: do not satisfy the commutative law, the associative law, the idempotent law, to have no proper subquasigroups, to be non-linear, etc. These quasigroups are constructed by complete mappings generated by suitable bijections of order \(2^s\) via extended Feistel network functions.