On cocycles with values in the group SU(2)

In this paper we introduce the notion of degree for $C^1$-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a $C^1$-cocycle $\phi:S^1\to SU(2)$ over an irrational rotation by $\alpha$ has nonzero degree, then the skew product $S^1\times SU(2)\ni(x,g)\mapsto (x+\alpha,g\phi(x))\in S^1\times SU(2)$ is not ergodic and the group of essential values of $\phi$ is equal to the maximal Abelian subgroup of SU(2). Moreover, if $\phi$ is of class $C^2$ (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity. Possible values of degree are discussed, too.Comment: 30 page

Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t f(\phi_sx)ds)$, where $(\phi_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\mathscr{C}^{2+\epsilon}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(\Phi^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(\Phi^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(\Phi^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.Comment: 57 pages, 4 picture

Multi-Level Steganography: Improving Hidden Communication in Networks

The paper presents Multi-Level Steganography (MLS), which defines a new concept for hidden communication in telecommunication networks. In MLS, at least two steganographic methods are utilised simultaneously, in such a way that one method (called the upper-level) serves as a carrier for the second one (called the lower-level). Such a relationship between two (or more) information hiding solutions has several potential benefits. The most important is that the lower-level method steganographic bandwidth can be utilised to make the steganogram unreadable even after the detection of the upper-level method: e.g., it can carry a cryptographic key that deciphers the steganogram carried by the upper-level one. It can also be used to provide the steganogram with integrity. Another important benefit is that the lower-layer method may be used as a signalling channel in which to exchange information that affects the way that the upper-level method functions, thus possibly making the steganographic communication harder to detect. The prototype of MLS for IP networks was also developed, and the experimental results are included in this paper.Comment: 18 pages, 13 figure

How Hidden Can Be Even More Hidden?

The paper presents Deep Hiding Techniques (DHTs) that define general techniques that can be applied to every network steganography method to improve its undetectability and make steganogram extraction harder to perform. We define five groups of techniques that can make steganogram less susceptible to detection and extraction. For each of the presented group, examples of the usage are provided based on existing network steganography methods. To authors' best knowledge presented approach is the first attempt in the state of the art to systematically describe general solutions that can make steganographic communication more hidden and steganogram extraction harder to perform.Comment: 5 pages, 8 figure