  We give some optimal size generating sets for the group generated by shifts
and local permutations on the binary full shift. We show that a single
generator, namely the fully asynchronous application of the elementary cellular
automaton 57 (or, by symmetry, ECA 99), suffices in addition to the shift. In
the terminology of logical gates, we have a single reversible gate whose shifts
generate all (finitary) reversible gates on infinitely many binary-valued wires
that lie in a row and cannot (a priori) be rearranged. We classify pairs of
words $u, v$ such that the gate swapping these two words, together with the
shift and the bit flip, generates all local permutations. As a corollary, we
obtain analogous results in the case where the wires are arranged on a cycle,
confirming a conjecture of Macauley-McCammond-Mortveit and Vielhaber.
