  McEliece and Niederreiter cryptosystems are robust and versatile
cryptosystems. These cryptosystems work with many linear error-correcting
codes. They are popular these days because they can be quantum-secure. In this
paper, we study the Niederreiter cryptosystem using non-binary quasi-cyclic
codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the
corresponding Niederreiter cryptosystem is resistant to the hidden subgroup
problem using weak quantum Fourier sampling. Though our work uses the weak
Fourier sampling, we argue that its conclusions should remain valid for the
strong Fourier sampling as well.
