From a68fca947b67ba9d2496fde1960d9cb5f9a92e05 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Mon, 13 Jan 2020 14:35:04 +0100
Subject: [PATCH] Hash: Better explanation of P[M|C] in the proof of Lemma G
---
06-hash/hash.tex | 3 ++-
1 file changed, 2 insertions(+), 1 deletion(-)
diff --git a/06-hash/hash.tex b/06-hash/hash.tex
index d5b0eb7..7e12db3 100644
--- a/06-hash/hash.tex
+++ b/06-hash/hash.tex
@@ -330,7 +330,8 @@ Each term can be handled separately:
$$\vbox{\halign{\hfil $\displaystyle{#}$&$\displaystyle{{}#}$\hfil&\quad #\hfil\cr
\Pr[M\mid\lnot C] &\le d/m^2 &by $(2,d)$-independence of~${\cal G}$ \cr
\Pr[\lnot C] &\le 1 &trivially \cr
- \Pr[M\mid C] &\le d/m &because $(2,d)$-independence implies $(1,d)$-independence \cr
+ \Pr[M\mid C] &\le d/m &for $i_1\ne i_2$: the left-hand side is~0, \cr
+ {} & {} &for $i_1=i_2$: $(2,d)$-independence implies $(1,d)$-independence \cr
\Pr[C] &\le c/r &by $c$-universality of~${\cal F}$ \cr
}}$$
So $\Pr[M] \le d/m^2 + cd/mr$. We want to write this as $c' / m^2$, so
--
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