Paper 2024/905

On the Semidirect Discrete Logarithm Problem in Finite Groups

Christopher Battarbee, Sorbonne University
Giacomo Borin, IBM Research Europe, University of Zurich
Ryann Cartor, Clemson University
Nadia Heninger, University of California, San Diego
David Jao, University of Waterloo
Laura Maddison, University of Ottawa
Edoardo Persichetti, Florida Atlantic University
Angela Robinson, National Institute of Standards and Technology
Daniel Smith-Tone, National Institute of Standards and Technology
Rainer Steinwandt, University of Alabama in Huntsville

We present an efficient quantum algorithm for solving the semidirect discrete logarithm problem (SDLP) in any finite group. The believed hardness of the semidirect discrete logarithm problem underlies more than a decade of works constructing candidate post-quantum cryptographic algorithms from nonabelian groups. We use a series of reduction results to show that it suffices to consider SDLP in finite simple groups. We then apply the celebrated Classification of Finite Simple Groups to consider each family. The infinite families of finite simple groups admit, in a fairly general setting, linear algebraic attacks providing a reduction to the classical discrete logarithm problem. For the sporadic simple groups, we show that their inherent properties render them unsuitable for cryptographically hard SDLP instances, which we illustrate via a Baby-Step Giant-Step style attack against SDLP in the Monster Group. Our quantum SDLP algorithm is fully constructive for all but three remaining cases that appear to be gaps in the literature on constructive recognition of groups; for these cases SDLP is no harder than finding a linear representation. We conclude that SDLP is not a suitable post-quantum hardness assumption for any choice of finite group.

Available format(s)
Attacks and cryptanalysis
Publication info
Group-Based CryptographySemidirect Discrete Logarithm ProblemPost-Quantum Cryptography
Contact author(s)
christopher battarbee @ lip6 fr
giacomo borin @ ibm com
rcartor @ clemson edu
nadiah @ cs ucsd edu
djao @ uwaterloo ca
lmadd036 @ uottawa ca
epersichetti @ fau edu
angela robinson @ nist gov
daniel-c smith @ louisville edu
rs0141 @ uah edu
2024-07-10: revised
2024-06-06: received
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      author = {Christopher Battarbee and Giacomo Borin and Ryann Cartor and Nadia Heninger and David Jao and Laura Maddison and Edoardo Persichetti and Angela Robinson and Daniel Smith-Tone and Rainer Steinwandt},
      title = {On the Semidirect Discrete Logarithm Problem in Finite Groups},
      howpublished = {Cryptology ePrint Archive, Paper 2024/905},
      year = {2024},
      note = {\url{}},
      url = {}
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