🤖 AI Summary
本文解决了群环中单位逆问题，通过基变换将其分解为小矩阵块，证明了在特定条件下该问题是多项式时间可解的，并提出了新的安全假设。
📝 Abstract
Several public-key schemes base their security on the belief that inverting a unit of a group ring is hard. A recent result showed that this belief is false on a quantum computer when the group is abelian. To restore security, designers moved to non-abelian groups, especially dihedral groups, believing that the hardness of the dihedral hidden subgroup problem (HSP) would protect the scheme. This paper shows that unit inversion is a different problem and does not require an HSP solver. Instead, it can be solved by a change of basis that splits the group ring into small matrix blocks. We prove that unit inversion is polynomial-time, classically and quantumly, when an efficient generalized Fourier transform exists, the group ring is semisimple, and the largest matrix block has polynomial size. Dihedral group rings satisfy these conditions because their irreducible representations have dimension at most 2 and an efficient Fourier transform exists. We give an explicit reversible quantum circuit for the block-inversion step and validate it in a register-level simulator. We also identify the exact structural boundary where the method stops and propose a candidate construction in the surviving regime under a new, clearly stated security assumption. The constructive results are supported by reproducible software artifacts and experiments.