Since the appearance of public-key cryptography in Diffie-Hellman seminal paper, many schemes have been proposed, but many have been broken. Indeed, for many people, the simple fact that a cryptographic algorithm withstands cryptanalytic attacks for several years is considered as a kind of validation. But some schemes took a long time before being widely studied, and maybe thereafter being broken. A~much more convincing line of research has tried to provide ``provable`` security for cryptographic protocols, in a complexity theory sense: if one can break the cryptographic protocol, one can efficiently solve the underlying problem. Unfortunately, very few practical schemes can be proven in this so-called ``standard model`` because such a security level rarely meets with efficiency. A convenient but recent way to achieve some kind of validation of efficient schemes has been to identify some concrete cryptographic objects with ideal random ones: hash functions are considered as behaving like random functions, in the so-called ``random oracle model``, block ciphers are assumed to provide perfectly independent and random permutations for each key in the ``ideal cipher model``, and groups are used as black-box groups in the ``generic model``. In this paper, we focus on practical asymmetric protocols together with their ``reductionist`` security proofs. We cover the two main goals that public-key cryptography is devoted to solve: authentication with digital signatures, and confidentiality with public-key encryption schemes.