Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets
This addresses a theoretical problem in cryptography and coding theory for researchers, but it appears incremental as it builds on existing work on AONTs.
The paper tackles the problem of determining when (t, s, v)-all-or-nothing transforms exist over arbitrary alphabets, focusing on the case t=2, and obtains necessary and sufficient conditions for their existence, including connections to orthogonal arrays and resilient functions.
A $(t, s, v)$-all-or-nothing transform is a bijective mapping defined on $s$-tuples over an alphabet of size $v$, which satisfies the condition that the values of any $t$ input co-ordinates are completely undetermined, given only the values of any $s-t$ output co-ordinates. The main question we address in this paper is: for which choices of parameters does a $(t, s, v)$-all-or-nothing transform (AONT) exist? More specifically, if we fix $t$ and $v$, we want to determine the maximum integer $s$ such that a $(t, s, v)$-AONT exists. We mainly concentrate on the case $t=2$ for arbitrary values of $v$, where we obtain various necessary as well as sufficient conditions for existence of these objects. We consider both linear and general (linear or nonlinear) AONT. We also show some connections between AONT, orthogonal arrays and resilient functions.