  In this paper we prove that in prime characteristic there is a functor
$-_{p-Leib}$ from the category of diassociative algebras to the category of
restricted Leibniz algebras, generalizing the functor from associative algebras
to restricted Lie algebras. Moreover we define the notion of restricted
universal enveloping diassociative algebra $Ud_{p}(g)$ of a restricted Leibniz
algebra $g$ and we show that $Ud_{p}$ is left adjoint to the functor
$-_{p-Leib}$. We also construct the restricted enveloping algebra, which
classifies the restricted Leibniz modules.
  In the last section we put a restricted pre-Lie structure on the tensor
product of a Leibniz algebra by a Zinbiel algebra.
