Inspired by fast correlation attacks on stream ciphers, we present a stream cipher-like construction for a public-key cryptosystem whose security relies on two problems: finding a low-weight multiple of a given polynomial and a Hidden Correlation problem. We obtain a weakly secure public-key cryptosystem we call TCHo (as for Trapdoor Cipher, Hardware Oriented). Using the Fujisaki-Okamoto construction, we can build an hybrid cryptosystem, TCHon-FO, resistant against adaptive chosen ciphertext attacks.