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prof. Alexei Rudakov -- -- -- Topics for PROJECTS/DIPLOMA for 2004/05. -- -- -- -- -- |
Security of ElGamal and related schemesWhat it means to prove security and how one can do it? In a sense it means to consider idealization of the "real world", make formal assumtions about the security goals and an attacker resources, then there are several idealized models of the cryptographic computations -- "generic algorithms", random choices, random oracle hash functions. To understand some of it with the focus on ElGamal and its generalizations could be content of the project.
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"Teleporting a photon" experiments had been done with success several years ago (see the enthusiastic article by A.Zelinger in Scientific American, April 2000, also: www.quantum.at). And the teleportation is only one of the ways of exploiting the phenomena of quantum entanglement, whose existence bothered Eistein back in 1930ties, but it was surely proven by series of experiments in mid 90ties, more than 50 year later. One does not need the deep knowledge of quantum mechanics to begin to study entanglement. The axiomatics of quantum mechanics uses quite basic properties of vector spaces and linear operators, and with them one can make models for quantum systems that demonstrate entanglement and teleportation effects. It is intresting to study the related mathematics too: the various properties of tensors and projector operators in the tensor products of (finite-dimensional) vector spaces. One needs first of all to be at ease with C (complex numbers) and the general concepts of the hermitian vector space over C, then with the notions of the unitary and hermitian linear operators acting on the space. Then it is quite possible to read the axomatic description of the quantum system / evolution / measurement. This is more or less all what needed to start reading various articles discussing models and experiments around Einstein-Podolsky-Rosen paradox, teleporting, entanglement, destributed entanglement, hyperentanglement, etc (one should not aim to read them all, just some).
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Some year ago prof.Victor Kac (MIT) and myself started to study the irreducible representations of the algebras E(3,6), E(3,8), E(5,10). The work is far from over, but for E(3,6) we have got pretty much of the picture (see Comm.Math.Phys. 222(2001), 611-661, or math-ph/0012050, math-ph/0012049, math-ph/030045). It is interesting also to look at the minimal non-irreducible representations, so called extensions of one irreducible by another, which could be written by a short exact sequence. It is currently an open question , and seemingly not so very hard one. There is the general technic to study these sequences, the calculating of so call Ext-group, or Ext-functor, that could be approached by several different ways. With the detailed information about the irriducible representations availible, it seems possible to expand our knowledge and get to know these Ext's also. |
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