Negligible functions

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}