  All-or-nothing transforms have been defined as bijective mappings on all
s-tuples over a specified finite alphabet. These mappings are required to
satisfy certain "perfect security" conditions specified using entropies of the
probability distribution defined on the input s-tuples. Alternatively, purely
combinatorial definitions of AONTs have been given, which involve certain kinds
of "unbiased arrays". However, the combinatorial definition makes no reference
to probability definitions.
  In this paper, we examine the security provided by AONTs that satisfy the
combinatorial definition. The security of the AONT can depend on the underlying
probability distribution of the s-tuples. We show that perfect security is
obtained from an AONT if and only if the input s-tuples are equiprobable.
However, in the case where the input s-tuples are not equiprobable, we still
achieve a weaker security guarantee. We also consider the use of randomized
AONTs to provide perfect security for a smaller number of inputs, even when
those inputs are not equiprobable.
