All or Nothing at All
This work addresses a theoretical problem in cryptography for researchers, focusing on incremental improvements in AONT security for general t.
The paper tackles the problem of constructing unconditionally secure all-or-nothing transforms (AONT) for general t, specifically t=2, by investigating binary matrices to maximize security probabilities. It provides upper bounds via quadratic programming and lower bounds from combinatorial constructions like symmetric BIBDs and cyclotomy, with results from exhaustive searches and random constructions for small s.
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.