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DiTRU: A Resurrection of NTRU over Dihedral Group

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Progress in Cryptology - AFRICACRYPT 2024 (AFRICACRYPT 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14861))

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Abstract

NTRU-like cryptosystems are among the most studied lattice-based post-quantum candidates. While most NTRU proposals have been introduced over a commutative ring of quotient polynomials, other rings can be used. Noncommutative algebra has been endorsed as a direction to build new variants of NTRU a long time ago. The first attempt to construct a noncommutative variant was due to Hoffstein and Silverman motivated by more resistance to lattice attack. The scheme has been built over the group ring of a dihedral group. However, their design differed from standard NTRU and soon was found vulnerable to algebraic attacks. In this work, we revive the group ring NTRU over the dihedral group as an instance of the GR-NTRU framework.

Unlike many proposals of noncommutative variants in the literature, our work focuses on putting the scheme into practice. We clear all the aspects that make our scheme implementable by proposing an efficient inversion algorithm over the new setting of the noncommutative ring, describing the decryption failure model, and analyzing the lattice associated with our instantiation. Finally, we discuss the best-known attacks against our scheme and provide an implementation targeting 128-bit, 192-bit, and 256-bit levels of security as proof of its practicality.

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Notes

  1. 1.

    Starting from \(N=113\), the results are averaged over at least 20 trials (only) since the time taken by one trial becomes extensively high. For \(N=127\) with DiTRU lattice, we recorded the trials that found the key with \(\beta \le 65\).

  2. 2.

    May [41] proposed an MITM attack on NTRU-type cryptosystems with a complexity \(O(|\mathcal {T}|^{0.3})\)(classic). However, it cannot be combined with hybrid attacks; therefore, we do not use it in our cost estimations.

  3. 3.

    In NTRUPrime, the authors conclude that according to the submission to NIST standardization process, a cryptosystem achieves levels of security corresponding to AES-128, AES-192, and AES-256, if the classical (pre-quantum) Core-SVP model assign at least \(2^{125}, 2^{181}, \) and \(2^{254}\), respectively to the selected parameter sets.

  4. 4.

    Our implementation is based on NTRU submissions to the first and third round of NIST competition with the required modifications to the dihedral group setup.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Ali Raya & Sugata Gangopadhyay

  2. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Vikas Kumar

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  1. Ali Raya
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Correspondence to Ali Raya , Vikas Kumar or Sugata Gangopadhyay .

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  1. Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Serge Vaudenay

  2. University of Birmingham, Birmingham, UK

    Christophe Petit

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Raya, A., Kumar, V., Gangopadhyay, S. (2024). DiTRU: A Resurrection of NTRU over Dihedral Group. In: Vaudenay, S., Petit, C. (eds) Progress in Cryptology - AFRICACRYPT 2024. AFRICACRYPT 2024. Lecture Notes in Computer Science, vol 14861. Springer, Cham. https://doi.org/10.1007/978-3-031-64381-1_16

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