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A New Cryptographic Analysis of 4-bit S-Boxes

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Information Security and Cryptology (Inscrypt 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9589))

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Abstract

An exhaustive search of all 16! bijective 4-bit S-boxes has been conducted by Markku-Juhani et al. (SAC 2011). In this paper, we present an improved exhaustive search over all permutation-xor equivalence classes. We put forward some optimizing strategies and make some improvements on the basis of their work. For our program, it only takes about one-sixth of the time of the experiment by Markku-Juhani et al. to get the same results. Furthermore, we classify all those permutation-xor equivalence classes in terms of a new classification criterion, which has been come up with by Wentao Zhang et al. (FSE 2015). For some special cases, we calculate the distributions of permutation-xor equivalence classes with respect to their differential bound and linear bound. It turns out that only in three special cases, there exist S-boxes having a minimal differential bound \(p=1/4\) and a minimal linear bound \(\epsilon =1/4\), which imply the optimal S-boxes.

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Acknowledgements

The research presented in this paper is supported by the National Natural Science Foundation of China (No. 61379138), and the “Strategic Priority Research Program” of the Chinese Academy of Sciences (No. XDA06010701).

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Ling Cheng, Wentao Zhang & Zejun Xiang

Authors
  1. Ling Cheng
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  2. Wentao Zhang
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  3. Zejun Xiang
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Corresponding author

Correspondence to Ling Cheng .

Editor information

Editors and Affiliations

  1. Chinese Academy of Sciences, SKLOIS Institute of Engineering, Beijing, China

    Dongdai Lin

  2. Center for Security Informatics, Indiana University at Bloomington, Indiana, USA

    XiaoFeng Wang

  3. Computer Science, Columbia University, New York, New York, USA

    Moti Yung

Appendix A

Appendix A

figure c
figure d

See Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14

Table 4. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =0 , \mathop {\mathrm {CarL1_S}}\nolimits =2\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 5. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =0 , \mathop {\mathrm {CarL1_S}}\nolimits =3\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 6. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =0 , \mathop {\mathrm {CarL1_S}}\nolimits =4\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 7. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =1 , \mathop {\mathrm {CarL1_S}}\nolimits =1\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 8. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =1 , \mathop {\mathrm {CarL1_S}}\nolimits =2\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 9. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =1 , \mathop {\mathrm {CarL1_S}}\nolimits =3\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 10. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =2 , \mathop {\mathrm {CarL1_S}}\nolimits =0\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 11. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =2 , \mathop {\mathrm {CarL1_S}}\nolimits =1\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 12. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =2 , \mathop {\mathrm {CarL1_S}}\nolimits =2\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 13. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =3 , \mathop {\mathrm {CarL1_S}}\nolimits =1\) in relation to differential bound p and linear bound \(\epsilon \)
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Table 14. Distribution of S-boxes with \(\mathop {\mathrm {CarD1_S}}\nolimits =4 , \mathop {\mathrm {CarL1_S}}\nolimits =0\) in relation to differential bound p and linear bound \(\epsilon \)
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Cheng, L., Zhang, W., Xiang, Z. (2016). A New Cryptographic Analysis of 4-bit S-Boxes. In: Lin, D., Wang, X., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2015. Lecture Notes in Computer Science(), vol 9589. Springer, Cham. https://doi.org/10.1007/978-3-319-38898-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-38898-4_9

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