In this thesis, we study algebraic coding theory based McEliece-type cryptosystems over quasi-cyclic codes. The main goal of this thesis is to construct a cryptosystem that resists quantum Fourier sampling making it quantum secure. We propose a new variant of Niederreiter cryptosystem over rate $\frac{m-1}{m}$ quasi-cyclic codes which is secure against quantum Fourier sampling due to indistinguishability of the hidden subgroup. The proof of indistinguishability is achieved due to two constraints over automorphism group; small size and large minimal degree. Apart from this cryptosystem, we also present a class of $\frac{1}{m}$ quasi-cyclic codes, with small size and large minimal degree of the automorphism group.