# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021068
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## Rectangular, range, and restricted AONTs: Three generalizations of all-or-nothing transforms

 David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

* D.R. Stinson's research is supported by NSERC discovery grant RGPIN-03882

Received  November 2021 Early access January 2022

All-or-nothing transforms (AONTs) were originally defined by Rivest [14] as bijections from $s$ input blocks to $s$ output blocks such that no information can be obtained about any input block in the absence of any output block. Numerous generalizations and extensions of all-or-nothing transforms have been discussed in recent years, many of which are motivated by diverse applications in cryptography, information security, secure distributed storage, etc. In particular, $t$-AONTs, in which no information can be obtained about any $t$ input blocks in the absence of any $t$ output blocks, have received considerable study.

In this paper, we study three generalizations of AONTs that are motivated by applications due to Pham et al. [13] and Oliveira et al. [12]. We term these generalizations rectangular, range, and restricted AONTs. Briefly, in a rectangular AONT, the number of outputs is greater than the number of inputs. A range AONT satisfies the $t$-AONT property for a range of consecutive values of $t$. Finally, in a restricted AONT, the unknown outputs are assumed to occur within a specified set of "secure" output blocks. We study existence and non-existence and provide examples and constructions for these generalizations. We also demonstrate interesting connections with combinatorial structures such as orthogonal arrays, split orthogonal arrays, MDS codes and difference matrices.

Citation: Navid Nasr Esfahani, Douglas R. Stinson. Rectangular, range, and restricted AONTs: Three generalizations of all-or-nothing transforms. Advances in Mathematics of Communications, doi: 10.3934/amc.2021068
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