diff --git a/sec-lattices.tex b/sec-lattices.tex
index 4d112a8..5c725a4 100644
--- a/sec-lattices.tex
+++ b/sec-lattices.tex
@@ -96,8 +96,11 @@ The Learning With Errors problem $\mathsf{LWE}_{n,q,\chi}$ asks to distinguish~$
 If $q$ is a prime power, $B \geq \sqrt{n}\omega(\log n)$, $\gamma= \widetilde{\mathcal{O}}(nq/B)$, then there exists an efficient sampleable $B$-bounded distribution~$\chi$ ({i.e.}, $\chi$ outputs samples with norm at most $B$ with overwhelming probability) such that $\mathsf{LWE}_{n,q,\chi}$ is as least as hard as $\mathsf{SIVP}_{\gamma}$ (see, e.g., \cite{Reg05,Pei09,BLP+13}).
  % (see~\cite{Pei09,BLPRS13} for classical analogues).
 
-
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% subsection: lattice trapdoors %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Lattice Trapdoors}
+\label{sse:lattice-trapdoors}
 
 In this section, we state the different algorithms that use ``\textit{lattice trapdoors}''.
 A trapdoor for lattice $\Lambda$ is a \textit{short} basis of this lattice.
@@ -108,6 +111,8 @@ Thus, a vector sampled in $D_{\Lambda^\perp_{q}(\mathbf{A}), \sigma}$, which is
 
 Gentry {\em et al.}~\cite{GPV08} showed that Gaussian distributions with lattice support can be sampled efficiently given a sufficiently short basis of the lattice.
 
+\scbf{Notation.} Given a matrix $\mathbf{A}$, let $\widetilde{\mathbf{A}}$ be the Gram-Schmidt orthogonalization of $\mathbf{A}$.
+
 \begin{lemma}[{\cite[Le.~2.3]{BLP+13}}]
 \label{le:GPV}
 There exists a $\ppt$ (probabilistic polynomial-time) algorithm $\GPVSample$ that takes as inputs a