  We continue a study of unconditionally secure all-or-nothing transforms
(AONT) begun in \cite{St}. An AONT is a bijective mapping that constructs s
outputs from s inputs. We consider the security of t inputs, when s-t outputs
are known. Previous work concerned the case t=1; here we consider the problem
for general t, focussing on the case t=2. We investigate constructions of
binary matrices for which the desired properties hold with the maximum
probability. Upper bounds on these probabilities are obtained via a quadratic
programming approach, while lower bounds can be obtained from combinatorial
constructions based on symmetric BIBDs and cyclotomy. We also report some
results on exhaustive searches and random constructions for small values of s.
