Noncommutative geometry and stochastic processes

The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that reproduce in a proper way a stochastic process on a noncommutative geometry. These processes are characterized by producing complex values and so, the corresponding Fokker-Planck equation resembles the Schroedinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schroedinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist. In four dimensions, the full set of Dirac matrices is needed and the corresponding stochastic process in a noncommutative geometry is easily recovered as is the Dirac equation in the Klein-Gordon form being it the Fokker--Planck equation of the process.Comment: 16 pages, 2 figures. Updated a reference. A version of this paper will appear in the proceedings of GSI2017, Geometric Science of Information, November 7th to 9th, Paris (France

Automatic Application of Software Countermeasures Against Physical Attacks

International audienceWhile the number of embedded systems is continuously increasing, securing software against physical attacks is costly and error-prone. Several works proposed solutions that automatically insert protections against these attacks in order to reduce this cost and this risk of error. In this chapter, we present a survey of existing approaches and classify them by the level at which they apply the countermeasure. We consider three different levels: the source code level, the compilation level, and the assembly/binary level. We explain the advantages and disadvantages of each level considering different criteria. Finally, we encourage future works to take compilation into account when designing tools, to consider the problem of combining countermeasures, as well as the interactions between countermeasures and compiler optimisations. Going one step further, we encourage future works to imagine how compilation could be modified or redesigned to optimise both performance and security