Jean-Marie Le Bars

Morgan Barbier, Hayat Cheballah, Jean-Marie Le Bars

The M{ö}bius transform is a crucial transformation into the Boolean world; it allows to change the Boolean representation between the True Table and Algebraic Normal Form. In this work, we introduce a new algebraic point of view of this transformation based on the polynomial form of Boolean functions. It appears that we can perform a new notion: the M{ö}bius computation variable by variable and new computation properties. As a consequence, we propose new algorithms which can produce a huge speed up of the M{ö}bius computation for sub-families of Boolean function. Furthermore we compute directly the M{ö}bius transformation of some particular Boolean functions. Finally, we show that for some of them the Hamming weight is directly related to the algebraic degree of specific factors.

Morgan Barbier, Hayat Cheballah, Jean-Marie Le Bars

Extensive studies of Boolean functions are carried in many fields. The Mobius transform is often involved for these studies. In particular, it plays a central role in coincident functions, the class of Boolean functions invariant by this transformation. This class -- which has been recently introduced -- has interesting properties, in particular if we want to control both the Hamming weight and the degree. We propose an innovative way to handle the Mobius transform which allows the composition between several Boolean functions and the use of Shannon or Reed-Muller decompositions. Thus we benefit from a better knowledge of coin-cident functions and introduce new properties. We show experimentally that for many features, coincident functions look like any Boolean functions.

Morgan Barbier, Jean-Marie Le Bars, Christophe Rosenberger

In this paper, we deal with the proof of ownership or legitimate usage of a digital content, such as an image, in order to tackle the illegitimate copy. The proposed scheme based on the combination of the watermark-ing and cancelable biometrics does not require a trusted third party, all the exchanges are between the provider and the customer. The use of cancelable biometrics permits to provide a privacy compliant proof of identity. We illustrate the robustness of this method against intentional and unintentional attacks of the watermarked content.