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The McEliece Cryptosystem Resists Quantum Fourier Sampling Attacks

Quantum computers can break the RSA and El Gamal public-key cryptosystems, since they can factor integers and extract discrete logarithms. If we believe that quantum computers will someday become a reality, we would like to have \emph{post-quantum} cryptosystems which can be implemented today with classical computers, but which will remain secure even in the presence of quantum attacks. In this article we show that the McEliece cryptosystem over \emph{well-permuted, well-scrambled} linear codes resists precisely the attacks to which the RSA and El Gamal cryptosystems are vulnerable---namely, those based on generating and measuring coset states. This eliminates the approach of strong Fourier sampling on which almost all known exponential speedups by quantum algorithms are based. Specifically, we show that the natural case of the Hidden Subgroup Problem to which the McEliece cryptosystem reduces cannot be solved by strong Fourier sampling, or by any measurement of a coset state. We start with recent negative results on quantum algorithms for Graph Isomorphism, which are based on particular subgroups of size two, and extend them to subgroups of arbitrary structure, including the automorphism groups of linear codes. This allows us to obtain the first rigorous results on the security of the McEliece cryptosystem in the face of quantum adversaries, strengthening its candidacy for post-quantum cryptography.