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Andre Henriques 2023-11-07 17:19:17 +00:00
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This q pairs work because when the oracle selects b=0: This q pairs work because when the oracle selects b=0:
There will be no colisions: There will be no colisions:
$$\forall i,k : i \ne j \and P_0[i] \neq p E_0[i] \implies E_0[i] \neq E_0[j]$$ $$\forall i,k : i \ne j \and P_0\[i\] \neq p E_0\[i\] \implies E_0\[i\] \neq E_0\[j\]$$
there if you don't find any colissions you can assume that the the oracle selected b=0 there if you don't find any colissions you can assume that the the oracle selected b=0
if the oracle selects b=1: if the oracle selects b=1:
If q is big enough, there will be colissions: If q is big enough, there will be colissions:
$$\exists i,k : i \ne j \and P\[i\] \eq P\[j\] \and R\[i\] = R\[j\] \implies E_1\[i\] = E_1\[j\]$$ $$\exists i,k : i \ne j \and P_1\[i\] \eq P_1\[j\] \and R\[i\] = R\[j\] \implies E_1\[i\] = E_1\[j\]$$
where $R$ is the list of random values generated for each pair where $R$ is the list of random values generated for each pair
\subsection*{4.3} \subsection*{4.3}