This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on
B
U
BU
. The calculations are carried out in
H
∗
(
B
U
,
Z
(
p
)
)
H^{\ast }(BU,{{\mathbf {Z}}_{(p)}})
and
H
∗
(
B
U
,
Z
(
p
)
)
{H_{\ast }}(BU,{{\mathbf {Z}}_{(p)}})
where
p
p
local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any
H
H
map (or infinite loop map)
B
U
→
B
U
BU \to BU
, and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.