if its iteration graph $\Gamma(f)$ is strongly connected, then
the output of $\chi_{\textit{14Secrypt}}$ follows
a law that tends to the uniform distribution
-if and only if its Markov matrix is a doubly stochastic matrix.
-
-
+if and only if its Markov matrix is a doubly stochastic one.
In~\cite[Section 4]{DBLP:conf/secrypt/CouchotHGWB14},
we have presented a general scheme which generates
function with strongly connected iteration graph $\Gamma(f)$ and
For instance, the iteration graph $\Gamma(f^*)$
(given in Figure~\ref{fig:iteration:f*})
is the $3$-cube in which the Hamiltonian cycle
-$000,100,101,001,011,111,110,010,000$
+$000,100,101,001,011,111,$ $110,010,000$
has been removed.
\end{xpl}
\begin{thrm}
The iteration graph $\Gamma(f)$ issued from
the ${\mathsf{N}}$-cube where an Hamiltonian
-cycle is removed is strongly connected.
+cycle is removed, is strongly connected.
\end{thrm}
Moreover, if all the transitions have the same probability ($\frac{1}{n}$),
This section ends with the idea of removing a Hamiltonian cycle in the
$\mathsf{N}$-cube.
In such a context, the Hamiltonian cycle is equivalent to a Gray code.
-Many approaches have been proposed a way to build such codes, for instance
+Many approaches have been proposed as a way to build such codes, for instance
the Reflected Binary Code. In this one and
for a $\mathsf{N}$-length cycle, one of the bits is exactly switched
$2^{\mathsf{N}-1}$ times whereas the others bits are modified at most
$\left\lfloor \dfrac{2^{\mathsf{N-1}}}{\mathsf{N}-1} \right\rfloor$ times.
It is clear that the function that is built from such a code would
-not provide a uniform output.
+not provide an uniform output.
The next section presents how to build balanced Hamiltonian cycles in the
$\mathsf{N}$-cube with the objective to embed them into the
