From d5edfb6a44dc77da7aeda585424c28f1c9a79632 Mon Sep 17 00:00:00 2001
From: kklein
Date: Thu, 25 Oct 2018 11:36:53 +0200
Subject: [PATCH] simple butterfly is not enough
---
security_proof.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/security_proof.tex b/security_proof.tex
index 75ab871..3216bd7 100644
--- a/security_proof.tex
+++ b/security_proof.tex
@@ -291,7 +291,7 @@ This proves the claim.
\end{itemize}
\end{proof}
-It remains to prove that the statement of the Lemma remains true if we replace the random permutations $\pi$ by butterfly graphs $B_i$ with random permutations $\sigma_i$ on $\bin^{2\cdot w/2}$ \kknote{For now, blocks of size $w/2$ to be consistent with the above, but need to change notation...}, see Figure \ref{fig:B8}.\footnote{Note, that our graph $B_S$ in fact consists of two copies of the original butterfly graph, hence has length $2\log (2S)$. Still, by misuse of the name, we will refer to $B_S$ as the butterfly graph of size $S$.} More precisely, we need to prove that pebbling internal nodes of the permutations $\B_i$ doesn't help to decrease the number and structure of paths between input and output nodes of $\B_i$ from what one could achieve by pebbling only input and output nodes of $\B_i$.\footnote{Note, in the case of random permutations, we used a complete bipartite graph to represent the computation. Here, we show that the butterfly graph is in some sense close to a comlete bipartite graph.}
+It remains to prove that the statement of the Lemma remains true if we replace the random permutations $\pi$ by butterfly graphs $B_i$ with random permutations $\sigma_i$ on $\bin^{2\cdot w/2}$ \kknote{For now, blocks of size $w/2$ to be consistent with the above, but need to change notation...}, see Figure \ref{fig:B8}.\footnote{Note, that our graph $B_S$ in fact consists of two copies of the original butterfly graph, hence has length $2\log (2S)$. Still, by misuse of the name, we will refer to $B_S$ as the butterfly graph of size $S$. We also note, that the simple butterfly graph would not be sufficient for our needs, i.e., the claim of the Lemma below does not hold for simple butterfly graphs.} More precisely, we need to prove that pebbling internal nodes of the permutations $\B_i$ doesn't help to decrease the number and structure of paths between input and output nodes of $\B_i$ from what one could achieve by pebbling only input and output nodes of $\B_i$.\footnote{Note, in the case of random permutations, we used a complete bipartite graph to represent the computation. Here, we show that the butterfly graph is in some sense close to a comlete bipartite graph.}
\begin{figure}
\centering
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