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