Some Results on the Existence of $t$ -All-or-Nothing Transforms Over Arbitrary Alphabets

A (t, s, v)-all-or-nothing transform (AONT) 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)-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. This includes computer searches that establish the existence of (2, q, q)-AONT for all odd primes not exceeding 29. We also show some connections between AONT, orthogonal arrays, and resilient functions.