# 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
##### References:
 [1] C. J. Colbourn and J. H. Dinitz, eds, The CRC Handbook of Combinatorial Designs, Second Edition, CRC Press, 1996. doi: 10.1201/9781420049954. [2] P. D'Arco, N. Nasr Esfahani and D. R. Stinson, All or nothing at all, Electronic Journal of Combinatorics, 23 (2016), paper 4.10, 24 pp. doi: 10.37236/6370. [3] D. A. Drake, Partial λ-geometries and generalized Hadamard matrices over groups, Canadian Journal of Mathematics, 31 (1979), 617-727.  doi: 10.4153/CJM-1979-062-1. [4] N. N. Esfahani, Generalizations of All-or-nothing Transforms and Their Application in Secure Distributed Storage, PhD thesis, University of Waterloo, 2021. [5] N. N. Esfahani, I. Goldberg and D. R. Stinson, Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets, IEEE Transactions on Information Theory, 64 (2018), 3136-3143.  doi: 10.1109/TIT.2017.2780233. [6] N. N. Esfahani and D. R. Stinson, Computational results on invertible matrices with the maximum number of invertible 2 × 2 submatrices, Australasian Journal of Combinatorics, 69 (2017), 130-144. [7] N. N. Esfahani and D. R. Stinson, On security properties of all-or-nothing transforms, Designs, Codes and Cryptography, 91 (2021), 2857-2867.  doi: 10.1007/s10623-021-00958-5. [8] N. N. Esfahani and D. R. Stinson, Asymmetric all-or-nothing transforms, To appear in Mathematical Cryptology. [9] G. O. Karame, C. Soriente, K. Lichota and S. Capkun, Securing cloud data under key exposure, IEEE Transactions on Cloud Computing, 7 (2019), 838-849.  doi: 10.1109/TCC.2017.2670559. [10] V. I. Levenshtein, Split orthogonal arrays and maximum independent resilient systems of functions, Designs, Codes and Cryptography, 12 (1997), 131-160.  doi: 10.1023/A:1008207230165. [11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977. [12] P. F. Oliveira, L. Lima, T. T. Vinhoza, J. Barros and M. Médard, Coding for trusted storage in untrusted networks, IEEE Transactions on Information Forensics and Security, 7 (2012), 1890-1899.  doi: 10.1109/TIFS.2012.2217331. [13] H. Pham, R. Steinwandt and A. Suárez Corona, Integrating classical preprocessing into an optical encryption scheme, Entropy, 21 (2019), 872.  doi: 10.3390/e21090872. [14] R. L. Rivest, All-or-nothing encryption and the package transform, Lecture Notes in Computer Science, (Fast Software Encryption 1997), 1267 (1997), 210–218. doi: 10.1007/BFb0052348. [15] D. R. Stinson, Combinatorial Designs: Constructions and Analysis, Springer-Verlag, New York, 2004. [16] D. R. Stinson, Something about all or nothing (transforms), Designs, Codes and Cryptography, 22 (2001), 133-138.  doi: 10.1023/A:1008304703074. [17] X. Wang, J. Cui and L. Ji, Linear (2, p, p)-AONTs exist for all primes p, Designs, Codes and Cryptography, 87 (2019), 2185-2197.  doi: 10.1007/s10623-019-00612-1. [18] Y. Zhang, T. Zhang, X. Wang and G. Ge, Invertible binary matrices with maximum number of 2-by-2 invertible submatrices, Discrete Mathematics, 340 (2017) 201–208. doi: 10.1016/j.disc.2016.08.016.

show all references

##### References:
 [1] C. J. Colbourn and J. H. Dinitz, eds, The CRC Handbook of Combinatorial Designs, Second Edition, CRC Press, 1996. doi: 10.1201/9781420049954. [2] P. D'Arco, N. Nasr Esfahani and D. R. Stinson, All or nothing at all, Electronic Journal of Combinatorics, 23 (2016), paper 4.10, 24 pp. doi: 10.37236/6370. [3] D. A. Drake, Partial λ-geometries and generalized Hadamard matrices over groups, Canadian Journal of Mathematics, 31 (1979), 617-727.  doi: 10.4153/CJM-1979-062-1. [4] N. N. Esfahani, Generalizations of All-or-nothing Transforms and Their Application in Secure Distributed Storage, PhD thesis, University of Waterloo, 2021. [5] N. N. Esfahani, I. Goldberg and D. R. Stinson, Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets, IEEE Transactions on Information Theory, 64 (2018), 3136-3143.  doi: 10.1109/TIT.2017.2780233. [6] N. N. Esfahani and D. R. Stinson, Computational results on invertible matrices with the maximum number of invertible 2 × 2 submatrices, Australasian Journal of Combinatorics, 69 (2017), 130-144. [7] N. N. Esfahani and D. R. Stinson, On security properties of all-or-nothing transforms, Designs, Codes and Cryptography, 91 (2021), 2857-2867.  doi: 10.1007/s10623-021-00958-5. [8] N. N. Esfahani and D. R. Stinson, Asymmetric all-or-nothing transforms, To appear in Mathematical Cryptology. [9] G. O. Karame, C. Soriente, K. Lichota and S. Capkun, Securing cloud data under key exposure, IEEE Transactions on Cloud Computing, 7 (2019), 838-849.  doi: 10.1109/TCC.2017.2670559. [10] V. I. Levenshtein, Split orthogonal arrays and maximum independent resilient systems of functions, Designs, Codes and Cryptography, 12 (1997), 131-160.  doi: 10.1023/A:1008207230165. [11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977. [12] P. F. Oliveira, L. Lima, T. T. Vinhoza, J. Barros and M. Médard, Coding for trusted storage in untrusted networks, IEEE Transactions on Information Forensics and Security, 7 (2012), 1890-1899.  doi: 10.1109/TIFS.2012.2217331. [13] H. Pham, R. Steinwandt and A. Suárez Corona, Integrating classical preprocessing into an optical encryption scheme, Entropy, 21 (2019), 872.  doi: 10.3390/e21090872. [14] R. L. Rivest, All-or-nothing encryption and the package transform, Lecture Notes in Computer Science, (Fast Software Encryption 1997), 1267 (1997), 210–218. doi: 10.1007/BFb0052348. [15] D. R. Stinson, Combinatorial Designs: Constructions and Analysis, Springer-Verlag, New York, 2004. [16] D. R. Stinson, Something about all or nothing (transforms), Designs, Codes and Cryptography, 22 (2001), 133-138.  doi: 10.1023/A:1008304703074. [17] X. Wang, J. Cui and L. Ji, Linear (2, p, p)-AONTs exist for all primes p, Designs, Codes and Cryptography, 87 (2019), 2185-2197.  doi: 10.1007/s10623-019-00612-1. [18] Y. Zhang, T. Zhang, X. Wang and G. Ge, Invertible binary matrices with maximum number of 2-by-2 invertible submatrices, Discrete Mathematics, 340 (2017) 201–208. doi: 10.1016/j.disc.2016.08.016.
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