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Minimizing Even-Mansour Ciphers for Sequential Indifferentiability (Without Key Schedules)

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Progress in Cryptology – INDOCRYPT 2022 (INDOCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13774))

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Abstract

Iterated Even-Mansour (IEM) schemes consist of a small number of fixed permutations separated by round key additions. They enjoy provable security, assuming the permutations are public and random. In particular, regarding chosen-key security in the sense of sequential indifferentiability (seq-indifferentiability), Cogliati and Seurin (EUROCRYPT 2015) showed that without key schedule functions, the 4-round Even-Mansour with Independent Permutations and no key schedule \(\text {EMIP} _4(k,u) = k \oplus \textbf{p} _4\big ( k \oplus \textbf{p} _3\big ( k \oplus \textbf{p} _2( k\oplus \textbf{p} _1(k \oplus u) ) \big ) \big )\) is sequentially indifferentiable.

Minimizing IEM variants for classical strong (tweakable) pseudorandom security has stimulated an attractive line of research. In this paper, we seek for minimizing the \(\text {EMIP} _4\) construction while retaining seq-indifferentiability. We first consider \(\text {EMSP}\), a natural variant of \(\text {EMIP}\) using a single round permutation. Unfortunately, we exhibit a slide attack against \(\text {EMSP}\) with any number of rounds. In light of this, we show that the 4-round \(\text {EM2P}_4^{\textbf{p} _1,\textbf{p} _2} (k,u)=k\oplus \textbf{p} _1\big (k \oplus \textbf{p} _2\big (k\oplus \textbf{p} _2(k\oplus \textbf{p} _1(k\oplus u))\big )\big )\) using 2 independent random permutations \(\textbf{p} _1,\textbf{p} _2\) is seq-indifferentiable. This provides the minimal seq-indifferentiable IEM without key schedule.

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Notes

  1. 1.

    In comparison, Cogliati and Seurin’s simulator for \(\text {EMIP}_4\) completes all newly constituted pairs \(\big ((2,x_2,y_2),(3,x_3,y_3)\big )\) of records of \(P_2\) and \(P_3\).

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Acknowledgements

We sincerely thank the anonymous reviewers for their helpful comments. Chun Guo was partly supported by the National Natural Science Foundation of China (Grant No. 62002202).

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Authors and Affiliations

  1. Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education, Shandong University, Qingdao, Shandong, 266237, China

    Shanjie Xu, Qi Da & Chun Guo

  2. School of Cyber Science and Technology, Shandong University, Qingdao, Shandong, China

    Shanjie Xu, Qi Da & Chun Guo

  3. Shandong Research Institute of Industrial Technology, Jinan, Shandong, China

    Chun Guo

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  1. Shanjie Xu
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Correspondence to Chun Guo .

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  1. University of Hyogo, Hyogo, Japan

    Takanori Isobe

  2. Indian Institute of Technology Madras, Chennai, India

    Santanu Sarkar

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Xu, S., Da, Q., Guo, C. (2022). Minimizing Even-Mansour Ciphers for Sequential Indifferentiability (Without Key Schedules). In: Isobe, T., Sarkar, S. (eds) Progress in Cryptology – INDOCRYPT 2022. INDOCRYPT 2022. Lecture Notes in Computer Science, vol 13774. Springer, Cham. https://doi.org/10.1007/978-3-031-22912-1_6

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