In this paper, we prove new upper bounds on the complexity of the cert
ified write-all problem with respect to an adaptive adversary. Our str
ongest result is that for any epsilon > 0, there exists an O(p(1+epsil
on)) work algorithm for the p-processor p-memory cell write-all. We al
so give a randomized O(p(2) log p) work algorithm for a p-processor p(
2)-memory cell write-all.