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I'm aware that RSA and ECC can be reduced to the Abelian Hidden Subgroup Problem (HSP), which is what makes them vulnerable to Shor's algorithm. I'm curious whether similar reductions exist for lattice-based cryptographic problems such as the Inhomogeneous Short Integer Solution (ISIS) problem or Learning With Errors (LWE).

I know that, in 2003, Oded Regev showed some connections between the dihedral HSP and LWE using coset sampling techniques, but I'm specifically interested in whether there are any known reductions from standard lattice problems (such as SIS or LWE) to any variant of the HSP, whether Abelian or non-Abelian.

I'm asking here since I haven't been able to find a clear answer to this topic anywhere, so any additional information or insight would be greatly appreciated.

Rodrigo de Azevedo
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beep
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1 Answers1

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Depending on what you mean by a variant of the hidden subgroup problem, Brakerski et al have shown that there is a quantum polynomial time reduction from LWE to to a relaxed version of the dihedral coset problem which they term ``eDCP'' (extrapolated dihedral cost problem).

Daniel S
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