# Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets

## TL;DR

This paper investigates the existence conditions of all-or-nothing transforms (AONT) over arbitrary alphabets, focusing on the case t=2, and explores their connections to orthogonal arrays and resilient functions.

## Contribution

It provides necessary and sufficient conditions for the existence of (t, s, v)-AONTs, especially for t=2, including both linear and nonlinear cases, and links them to other combinatorial objects.

## Key findings

- Derived conditions for AONT existence over arbitrary alphabets.
- Established connections between AONT, orthogonal arrays, and resilient functions.
- Analyzed both linear and nonlinear AONT cases.

## Abstract

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.