diff --git a/doc/specification.tex b/doc/specification.tex
index 0466c9b0b71280d7e58d71101db4106c409d87f1..09a1936fa1f1f704a2ef29332bd292a626d6e578 100644
--- a/doc/specification.tex
+++ b/doc/specification.tex
@@ -345,7 +345,10 @@ $\textsf{field}(o)$ to access the field \textsf{field} of $o$.
\right\}
\end{gather*}
-A private key is a random number $x$ modulo $q$. The corresponding
+A private key is a number $x$ modulo $q$, chosen at random in the
+basic decryption mode, and computed after several interactions in the
+threshold mode.
+The corresponding
$\pklabel$ is $X=g^x$. A $\tpk$ is a bundle of this public key with a
\hyperref[common]{$\proof$} of knowledge computed as follows:
\begin{enumerate}
@@ -622,7 +625,8 @@ Trustee $\mathcal{T}_j$ fills $\textsf{vo}_j$ as follows:
\]
\item \textsf{public\_key} is set to a
\hyperref[trustee-keys]{\texttt{trustee\_public\_key}} structure
- built using $S_j$ as private key.
+ built using $S_j$ as private key, which computes the corresponding
+ public key and a proof of knowledge of $S_j$.
\end{itemize}
The administrator checks $\textsf{vo}_j$ as follows:
\begin{itemize}