Paper 2021/821

On the hardness of the NTRU problem

Alice Pellet-Mary and Damien Stehlé


The 25 year-old NTRU problem is an important computational assumption in public-key cryptography. However, from a reduction perspective, its relative hardness compared to other problems on Euclidean lattices is not well-understood. Its decision version reduces to the search Ring-LWE problem, but this only provides a hardness upper bound. We provide two answers to the long-standing open problem of providing reduction-based evidence of the hardness of the NTRU problem. First, we reduce the worst-case approximate Shortest Vector Problem over ideal lattices to an average-case search variant of the NTRU problem. Second, we reduce another average-case search variant of the NTRU problem to the decision NTRU problem.

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Publication info
A minor revision of an IACR publication in ASIACRYPT 2021
Contact author(s)
alice pellet-mary @ math u-bordeaux fr
damien stehle @ ens-lyon fr
2021-10-05: revised
2021-06-16: received
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      author = {Alice Pellet-Mary and Damien Stehlé},
      title = {On the hardness of the NTRU problem},
      howpublished = {Cryptology ePrint Archive, Paper 2021/821},
      year = {2021},
      note = {\url{}},
      url = {}
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