diff --git a/report/report.tex b/report/report.tex
index e1be200..93cb1a1 100644
--- a/report/report.tex
+++ b/report/report.tex
@@ -350,6 +350,11 @@ The ``IO adjacency'' term is an additional term in the signatures of order
 above $0$, indicating what input and output pins of the circuit group
 containing the current gate are adjacent to it.
 
+The default order of signature used in all computations, unless more is useful,
+is 2, after a few benchmarks.
+
+\todo{explain range of $n$}
+
 \paragraph{Efficiency.} Every circuit memoizes all it can concerning its
 signature: the inner signature, the IO adjacency, the signatures of order $n$
 already computed, etc.
@@ -373,7 +378,33 @@ would be to try to multithread this computation.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Group equality}
-\todo{}
+
+Given two circuit group gates, the task of group equality is to determine
+whether the two groups are structurally equivalent, as discussed above.
+
+Group equality itself is handled as a simple backtracking algorithm, trying to
+establish a match (an isomorphism, that is, a permutation of the gates of one
+of the groups) between the two groups given.
+
+The gates of the two groups are matched by equal signatures, equal number of
+inputs and outputs, based on the signature of default order (that is, 2). A few
+checks are made, \eg{} every matching group must have the same size on both
+sides (if not, then, necessary, the two groups won't match). Then, the worst
+case of number of permutations to check is evaluated.
+
+If this number is too high, the signature order will be incremented, and the
+matching groups re-created accordingly, until a satisfyingly low number of
+permutations is reached (or the diameter of the circuit is reached, meaning
+that increasing the order of signature won't have any additional impact). This
+order increase ``on-demand'' proved itself very efficient, effectively lowering
+the number of permutations examined to no more than $4$ in studied cases.
+
+Once a permutation is judged worth to be examined, the group equality is run
+recursively on all its matched gates. If this step succeeds, the graph
+structure is then checked. If both steps succeeds, the permutation is correct
+and an isomorphism has been found; if not, we move on to the next permutation.
+
+\todo{Anything more to tell here?}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Pattern-match}
@@ -381,6 +412,11 @@ would be to try to multithread this computation.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Performance}
+
+\todo{}
+
+\subsection{Corner cases}
+
 \todo{}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%