  All-or-nothing transforms (AONT) were proposed by Rivest as a message
preprocessing technique for encrypting data to protect against brute-force
attacks, and have numerous applications in cryptography and information
security. Later the unconditionally secure AONT and their combinatorial
characterization were introduced by Stinson. Informally, a combinatorial AONT
is an array with the unbiased requirements and its security properties in
general depend on the prior probability distribution on the inputs $s$-tuples.
Recently, it was shown by Esfahani and Stinson that a combinatorial AONT has
perfect security provided that all the inputs $s$-tuples are equiprobable, and
has weak security provided that all the inputs $s$-tuples are with non-zero
probability.
  This paper aims to explore on the gap between perfect security and weak
security for combinatorial $(t,s,v)$-AONTs. Concretely, we consider the typical
scenario that all the $s$ inputs take values independently (but not necessarily
identically) and quantify the amount of information
$H(\mathcal{X}|\mathcal{Y})$ about any $t$ inputs $\mathcal{X}$ that is not
revealed by any $s-t$ outputs $\mathcal{Y}$. In particular, we establish the
general lower and upper bounds on $H(\mathcal{X}|\mathcal{Y})$ for
combinatorial AONTs using information-theoretic techniques, and also show that
the derived bounds can be attained in certain cases. Furthermore, the
discussions are extended for the security properties of combinatorial
asymmetric AONTs.
