In this thesis, we study two different primitives: lossy trapdoor functions and zero-knwoledge proof systems. The lossy trapdoor functions (LTFs) are function families in which injective functions and lossy ones are computationally indistinguishable. Since their introduction, they have been found useful in constructing various cryptographic primitives. We give in this thesis efficient constructions of a variant of LTF : Lossy Algebraic Filter. Using this variant, we can improve the efficiency of the KDM-CCA (Key-Depended-Message Chosen-Ciphertext-Attack) encryption schemes and fuzzy extractors. In the second part of this thesis, we investigate on constructions of zero-knowledge proof systems. We give the first logarithmic-size ring-signature with tight security using a variant of Groth-Kolhweiz Σ-protocol in the random oracle model. We also propose one new construction of lattice-based Designated-Verifier Non-Interactive Zero-Knowledge arguments (DVNIZK). Using this new construction, we build a lattice-based voting scheme in the standard model.