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Provable Security of (Tweakable) Block Ciphers Based on Substitution-Permutation Networks

  • Benoît Cogliati
  • Yevgeniy Dodis
  • Jonathan Katz
  • Jooyoung Lee
  • John Steinberger
  • Aishwarya Thiruvengadam
  • Zhe Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10991)

Abstract

Substitution-Permutation Networks (SPNs) refer to a family of constructions which build a wn-bit block cipher from n-bit public permutations (often called S-boxes), which alternate keyless and “local” substitution steps utilizing such S-boxes, with keyed and “global” permutation steps which are non-cryptographic. Many widely deployed block ciphers are constructed based on the SPNs, but there are essentially no provable-security results about SPNs.

In this work, we initiate a comprehensive study of the provable security of SPNs as (possibly tweakable) wn-bit block ciphers, when the underlying n-bit permutation is modeled as a public random permutation. When the permutation step is linear (which is the case for most existing designs), we show that 3 SPN rounds are necessary and sufficient for security. On the other hand, even 1-round SPNs can be secure when non-linearity is allowed. Moreover, 2-round non-linear SPNs can achieve “beyond-birthday” (up to \(2^{2n/3}\) adversarial queries) security, and, as the number of non-linear rounds increases, our bounds are meaningful for the number of queries approaching \(2^n\). Finally, our non-linear SPNs can be made tweakable by incorporating the tweak into the permutation layer, and provide good multi-user security.

As an application, our construction can turn two public n-bit permutations (or fixed-key block ciphers) into a tweakable block cipher working on wn-bit inputs, 6n-bit key and an n-bit tweak (for any \(w\ge 2\)); the tweakable block cipher provides security up to \(2^{2n/3}\) adversarial queries in the random permutation model, while only requiring w calls to each permutation, and 3w field multiplications for each wn-bit input.

Keywords

Substitution-permutation networks Tweakable block ciphers Domain extension of block ciphers Beyond-birthday-bound security 

Notes

Acknowledgments

The work of Aishwarya Thiruvengadam was done while at the University of Maryland. Benoît Cogliati was partially supported by the European Union’s H2020 Programme under grant agreement number ICT-644209. The work of Yevgeniy Dodis was done in part while visiting the University of Maryland, and was supported by gifts from VMware Labs and Google, as well as NSF grants 1619158, 1319051, and 1314568. The work of Jonathan Katz and Aishwarya Thiruvengadam was performed under financial assistance award 70NANB15H328 from the U.S. Department of Commerce, National Institute of Standards and Technology. Jooyoung Lee was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT), No. NRF-2017R1E1A1A03070248.

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Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Benoît Cogliati
    • 1
  • Yevgeniy Dodis
    • 2
  • Jonathan Katz
    • 3
  • Jooyoung Lee
    • 4
  • John Steinberger
    • 6
  • Aishwarya Thiruvengadam
    • 5
  • Zhe Zhang
    • 6
  1. 1.University of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.New York UniversityNew YorkUSA
  3. 3.University of MarylandCollege ParkUSA
  4. 4.KAISTDaejeonKorea
  5. 5.University of CaliforniaSanta BarbaraUSA
  6. 6.Tsinghua UniversityBeijingChina

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