CRYPTOQUANT. CRYPTOGRAPHY AND QUANTUM INFORMATION
The aim of this program is to found a research group dealing with the mathematical challenges of Information Technologies and their interactions with Computer Science, Physics or Engineering. Because of the interdisciplinary nature of the topic, we pretend to involve researchers from other centers and instutes in the Madrid area and also to make collaboration with scientist from other places in/outside Spain. The first subjects of our group are related to Public Key Criptography, Criptoanalysis and Quantic Information Theory.
Information Technologies present big challenges for Mathematics; for example, thanks to the development of electronic communications, the uprise of new tools and utilities together with the universal use of computers, there exists a huge traffic and storage of confidential information, that must be protected. All these problems are well know and it is also well known the central role played by the Mathematics in all the solutions proposed by modern Cryptography. Mathematics helps to design new algorithms and to test their security by criptoanalysis.
The subjects for our research group are related to Public Key Cryptography and to the analysis and criptoanalysis of criptographic primitives. There are many cryptographic primitives based upon boolean fuctions. For that reason the study of such functions is crucial. Techniques of spectral analysis are used in cryptography to study nonlinear propierties of boolean functions. Between public key criptosystems the cryptographic primitives which may substitute RSA are very interesting; specially those which use elliptic curves, lattices or multivalued functions. Multivalued cryptography is based upon multivalued maps and the security of such primitives comes from the impossibility of solving in finite time the systems of equations that describe such maps. These systems seem very promising because of their high speed and because they are not limited by Short algorithm and thus will not be affected by the quantum computer.
In respect to Quantum Information, one of the mayor breakthroughs of the last years is the possibility to build a quantum computer. The crucial change of the basic rules (Quantum Physics instead of the classical theory) needs a new theory which is still developping: Quantum Information and Computation. One of the most interesting features of this new discipline is its interdisciplinary nature. Mathematicians from different subjects (analysis, algebra, topology, geometry, cryptography...), physicists, computer scientists, engineers... should share their efforts, knowledgements and points of view. There is a common concept to all works, independently of their approach, devoted to quantum information and computation: entaglement, the characteristic notion that distinguishs quantum from classical computation. It is a key concept but however poorly understood. Between the most imteresting questions related to entaglement we cite the following: decide wheter a density matrix is entagled or not, topological entaglement and entaglement and quantum channels.