  A restricted $d$th power of an ideal $I$ is obtained by restricting the
exponent vectors allowed to appear on the "natural" generating set of $I^d$,
for some integer $d$. In this paper, we study homological properties of
restricted powers of complete intersections. We construct an explicit minimal
free resolution for any restricted power of a complete intersection which
generalizes the $L$-complex construction of Buchsbaum and Eisenbud. We use this
resolution to compute an explicit basis for the Koszul homology which allows us
to deduce that the quotient defined by any restricted $d$th power of a complete
intersection is a Golod ring for $d \geq 2$. Finally, using techniques of
Miller and Rahmati, we show that the minimal free resolution of the quotient
defined by any restricted power of a complete intersection admits the structure
of an associative DG-algebra.
