All or Nothing at All

TL;DR

This paper extends the study of unconditionally secure all-or-nothing transforms (AONT) to the case of multiple inputs, focusing on binary matrix constructions and probabilistic bounds for security when multiple outputs are known.

Contribution

It generalizes previous work on AONT security from one input to multiple inputs, providing bounds and constructions for binary matrices with desired properties.

Findings

Upper bounds on security probabilities via quadratic programming

Lower bounds from combinatorial constructions like symmetric BIBDs and cyclotomy

Results from exhaustive searches and random constructions for small s

Abstract

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.