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Detection of non-trivial preservable quotient spaces in S-Box(es)

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Abstract

Substitution Box (S-Box) is employed in block ciphers to ensure non-linearity. An n-bit bijective S-Box is a member of the Symmetric Group \(\mathbb {S}_{2^{n}}\). Ideally, an S-Box must follow a stringent cryptographic profile. Designing an S-Box is a transparent and justified process. The concerning point for an evaluator is the presence of vulnerabilities in the design of an S-Box, i.e., Kuznyechick. If a malicious designer keeps the non-trivial subspaces secret, it leads to sophisticated cryptanalytic attacks. This article investigates the behaviour of non-trivial subspaces in an S-Box and its Affine, Extended Affine (EA) and Carlet-Charpin-Zinoviev (CCZ) equivalence classes. This paper presents a novel algorithm for finding preservable quotient spaces in an S-Box, thus leveraging a way for shortlisting the potential candidates for an S-Box with backdoors. The proposed work emphasizes checking whether a target S-Box is a potential backdoor candidate. The backdoored designs proposed by KG Paterson, Carlo Harpes and Bannier are being identified and validated with the help of the proposed algorithm. Our findings establish that the additive linear structures responsible for the non-trivial subspace are not invariant under the EA and CCZ. Moreover, the analysis of \(3{-}bit\) permutations reveals that almost 23% population of \(\mathbb {S}_{2^{3}}\) preserve the quotient subspaces. Irrespective of the linear structures in its non-linear layer, the NIST Lightweight competitors do not preserve the quotient spaces in both the input and output space.

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Data Availability

No data has been generated for this research except for the analysis of \(\mathbb {S}_{2^3}\). The code for the empirical verification and further enhancement is uploaded to Github via https://github.com/dawood254/Detection-of-Non-Trivial-Subspaces-.git.

Notes

  1. Values are in hexadecimal format.

  2. The values in the S-0 table are given in Hexadecimal format, the lookup in S-0 is carried out similarly to DES [39].

  3. Values are in Hexadecimal Format, S-1(0D) = 1C, S-2(28) = 1F.

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Acknowledgements

We would like to express our sincere gratitude to the anonymous reviewers, who provided invaluable feedback and insightful comments on our manuscript. Their careful attention and constructive criticism helped us improve our work’s quality and clarity. We are grateful for their time and effort in reviewing our paper and for their commitment to maintaining the high standards of this journal. Without their contributions, this paper would not have been possible.

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Authors and Affiliations

  1. Department of Information Security, National University of Sciences and Technology (NUST), Islamabad, 44000, Pakistan

    Shah Fahd, Mehreen Afzal, Waseem Iqbal & Yawar Abbas

  2. Quaid-i-Azam University, Islamabad, Pakistan

    Dawood Shah

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  1. Shah Fahd
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Correspondence to Waseem Iqbal.

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Appendices

Appendices

1.1 A: Extended affine equivalent of [4, 27] mappings

See Tables 6, 7, 8, 9.

Table 6 Extended affine equivalent of S-0 3 [27]
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Table 7 Extended affine equivalent of S-1 4 [4]
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Table 8 Extended affine equivalent of S-2 5
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Table 9 Extended affine equivalent of S-3 [4]
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1.2 B: Comparison of non-trivial subspaces in parent and EA equivalent mappings

See Tables 10, 11, 12, 13, 14, 15.

Table 10 Behaviour of non trivial preservable subspaces S-0 3 and its Extended Affine Equivalent 8
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Table 11 Behaviour of non trivial preservable subspaces S-1 4 and its Extended Affine Equivalent 9
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Table 12 Behaviour of Non Trivial Preservable Subspaces of S-2 5 and its Extended Affine Equivalent 10
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Table 13 Behaviour of Non Trivial Preservable Subspaces of S-3 6 and its Extended Affine Equivalent 11
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Table 14 Behaviour of Non Trivial Preservable Subspaces in Finite Field Inversion Based S-Box(es)
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Table 15 Behaviour of Non Trivial Preservable Subspaces in NIST Lightweight Candidates S-Box(es) [42]
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Fahd, S., Afzal, M., Shah, D. et al. Detection of non-trivial preservable quotient spaces in S-Box(es). Neural Comput & Applic 35, 18343–18355 (2023). https://doi.org/10.1007/s00521-023-08654-2

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