From 92d85feb2023d561f1b69af8310b977eb9e283ba Mon Sep 17 00:00:00 2001 From: Fabrice Mouhartem Date: Fri, 2 Feb 2018 16:35:09 +0100 Subject: [PATCH] Negligible functions --- chap-proofs.tex | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/chap-proofs.tex b/chap-proofs.tex index a0479ae..aecfe80 100644 --- a/chap-proofs.tex +++ b/chap-proofs.tex @@ -85,17 +85,24 @@ That's why we'll now define the principle of polynomial time reduction. A language $A \subseteq \bit^\star$ is \emph{polynomial-time reducible to} a language $B \subseteq \bit^\star$, denoted by $A \redto B$, if there is a \emph{polynomial-time computable} function $f: \bit^\star \to \bit^\star$ such that for every $x \in \bit^\star$, $x \in A$ if and only if $f(x) \in B$. \end{definition} -In other words, a polynomial reduction from $A$ to $B$ is the description of a polynomial time algorithm (also called ``\emph{the reduction}''), that uses an algorithm for $B$ to solve $A$. - \begin{figure} \centering \input fig-poly-red \caption{Illustration of a polynomial-time reduction~{\cite[Fig. 2.1]{AB09}}} \label{fig:poly-reduction} \end{figure} +In other words, a polynomial reduction from $A$ to $B$ is the description of a polynomial time algorithm (also called ``\emph{the reduction}''), that uses an algorithm for $B$ to solve $A$. +This is illustrated in Figure~\ref{fig:poly-reduction}. + We can notice that \textsf{P} and \textsf{PP} are both closed under polynomial-time reduction. +Namely, if a problem is easier than another problem in \textsf{P} (resp. \textsf{PP}), then this problem is also in \textsf{P} (resp. \textsf{PP}). +Until know, we mainly focus on the running time of the algorithms. +In cryptology, it is also important to consider the success probability of algorithms: +an attack is successful if the probability that it succeed is noticeable. +\index{Negligible function} +\textsc{Notation.} Let $f : \NN \to [0,1]$ be a function. The function $f$ is called \emph{negligible} if $f(n) = n^{-\omega(1)}$, and this is written $f(n) = \negl[n]$. Non-negligible functions are called \emph{noticeable} functions. And if $f = 1- \negl[n]$, $f$ is called \emph{overwhelming}. \section{Random-Oracle Model, Standard Model and Half-Simulatability}