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Designs, Codes and Cryptography

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04-02-2019

Linear (2, p, p)-AONTs exist for all primes p

Authors: Xin Wang, Jie Cui, Lijun Ji

Published in: Designs, Codes and Cryptography | Issue 10/2019

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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 Nasr Esfahani et al. (IEEE Trans Inf Theory 64:3136–3143, 2018) 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.

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Title: Linear (2, p, p)-AONTs exist for all primes p
Authors: Xin Wang
Jie Cui
Lijun Ji
Publication date: 04-02-2019
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 10/2019
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-019-00612-1

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