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

A <inline-formula> <tex-math notation="LaTeX">$(t, s, v)$ </tex-math></inline-formula>-all-or-nothing transform (AONT) is a bijective mapping defined on <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-tuples over an alphabet of size <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>, which satisfies the condition that the values of any <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> input co-ordinates are completely undetermined, given only the values of any <inline-formula> <tex-math notation="LaTeX">$s-t$ </tex-math></inline-formula> output co-ordinates. The main question we address in this paper is: for which choices of parameters does a <inline-formula> <tex-math notation="LaTeX">$(t, s, v)$ </tex-math></inline-formula>-AONT exist? More specifically, if we fix <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>, we want to determine the maximum integer <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> such that a <inline-formula> <tex-math notation="LaTeX">$(t, s, v)$ </tex-math></inline-formula>-AONT exists. We mainly concentrate on the case <inline-formula> <tex-math notation="LaTeX">$t=2$ </tex-math></inline-formula> for arbitrary values of <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>, where we obtain various necessary as well as sufficient conditions for existence of these objects. This includes computer searches that establish the existence of <inline-formula> <tex-math notation="LaTeX">$(2, q, q)$ </tex-math></inline-formula>-AONT for all odd primes not exceeding 29. We also show some connections between AONT, orthogonal arrays, and resilient functions.