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The goal of this investigation is effective method of key exchange whichbased on non-commutative group $G$. The results of Ko et al. \cite{kolee} isimproved and generalized. The size of a minimal generating set for thecommutator subgroup of Sylow 2-subgroups of alternating group is found. Thestructure of the commutator subgroup of Sylow 2-subgroups of the alternatinggroup ${A_{{2^{k}}}}$ is investigated and used in key exchange protocol whichbased on non-commutative group.  We consider non-commutative generalization of CDH problem \cite{gu2013new,bohli2006towards} on base of metacyclic group of Miller-Moreno type (minimalnon-abelian group). We show that conjugacy problem in this group isintractable. Effectivity of computation is provided due to using groups ofresidues by modulo $n$. The algorithm of generating (designing) common key innon-commutative group with 2 mutually commuting subgroups is constructed by us.

Commutator subgroups of Sylow 2-subgroups of alternating group and Miller-Moreno groups as bases of new Key Exchange Protocol