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In this work, we propose two McEliece variants: one from Moderate Density Parity-Check (MDPC) codes and another from quasi-cyclic MDPC codes. MDPC codes are LDPC codes of higher density (and worse error-correction capability) than what is usually adopted for telecommunication applications. However, in cryptography we are not necessarily interested in correcting many errors, but only a number which ensures an adequate security level. By this approach, we reduce under certain hypotheses the security of the scheme to the well studied decoding problem. Furthermore, the quasi-cyclic variant provides extremely compact-keys (for 80-bits of security, public-keys have only 4801 bits).