Practical Security in Public-Key Cryptography

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. Moreover, for a long time the security proofs have only been performed in an asymptotic framework, which provides some confidence in the scheme but for very huge parameters only, and thus for unpractical schemes.A recent trend consists in providing very efficient reductions, with a practical meaning: with usual parameters (such as 1024-bit RSA moduli) the computational cost of any attack is actually 272, given the state of the art about classical problems (e.g. integer factoring).In this paper, we focus on practical schemes 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.