Linear $(2,p,p)$-AONTs do Exist

TL;DR

This paper proves the existence of linear (2,p,p)-AONTs, introduces an infinite class of improved linear AONTs over prime power alphabets, and explores their connections with orthogonal arrays.

Contribution

It demonstrates the existence of linear (2,p,p)-AONTs, provides the first infinite class of such AONTs surpassing previous constructions, and establishes new relationships with orthogonal arrays.

Findings

Linear (2,p,p)-AONTs exist for prime p.

An infinite class of improved linear AONTs is constructed.

A recursive method for general AONTs and their link to orthogonal arrays is presented.

Abstract

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 that if any $s-t$ of the $s$ outputs are given, then the values of any $t$ inputs are completely undetermined. When $t$ and $v$ are fixed, to determine the maximum integer $s$ such that a $(t,s,v)$-AONT exists is the main research objective. In this paper, we solve three open problems proposed in [IEEE Trans. Inform. Theory 64 (2018), 3136-3143.] and show that there do exist linear $(2,p,p)$-AONTs. Then for the size of the alphabet being a prime power, we give the first infinite class of linear AONTs which is better than the linear AONTs defined by Cauchy matrices. Besides, we also present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays.