2001 Combinatorial and Computational Mathematics  
For a long time, cryptology had been a mystic art more than a science, solving the confidentiality concerns with secret and private techniques. Automatic machines, electronic and namely computers modified the environment and the basic requirements. The main difference was the need of public mechanisms to allow large-scale communications with just a small secret shared between the interlocutors, but that furthermore resist against adversaries with more powerful computers. Unfortunately, the
more » ... ity remained heuristic: with a permanent fight between designers (the cryptographers) and breakers (the cryptanalysts). In 1976, Diffie and Hellman claimed the possibility of achieving confidentiality between two people without any common secret information. However, they needed quite new objects: (trapdoor) one-way functions. Hopefully, mathematics, with algorithmic number theory, have been realized to provide such objects. A new direction in cryptography was under investigations: asymmetric cryptography and provable security. In this paper we review the main problems that cryptography tries to solve, and how it achieves these goals thanks to the algorithmic number theory. After a brief history of the ancient and conventional cryptography, we review the Diffie-Hellman's suggestion with the apparent paradox. Then, we survey the solutions based on the integer factorization or the discrete logarithm, two problems that nobody knows how to efficiently solve.
doi:10.1142/9789812799890_0007 fatcat:if3nqipxibhgdgygksyr4qyeai