  We investigate shift-invariant vectorial Boolean functions on $n$~bits that
are lifted from Boolean functions on $k$~bits, for $k\leq n$. We consider
vectorial functions that are not necessarily permutations, but are, in some
sense, almost bijective. In this context, we define an almost lifting as a
Boolean function for which there is an upper bound on the number of collisions
of its lifted functions that does not depend on $n$. We show that if a Boolean
function with diameter $k$ is an almost lifting, then the maximum number of
collisions of its lifted functions is $2^{k-1}$ for any $n$. Moreover, we
search for functions in the class of almost liftings that have good
cryptographic properties and for which the non-bijectivity does not cause major
security weaknesses. These functions generalize the well-known map $\chi$ used
in the Keccak hash function.
