fmouhart/thesis
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

proof-reading

Browse Source
This commit is contained in:
Fabrice Mouhartem
2018-04-13 18:52:15 +02:00
parent 9dd62d0c1d
commit 9d478c72fc
11 changed files with 180 additions and 155 deletions
Download Patch File Download Diff File Expand all files Collapse all files

8
chap-proofs.tex
View File

@ -93,8 +93,14 @@ That's why we'll now define the principle of polynomial time reduction.
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 $B$ in a black-box manner to solve $A$.
This is illustrated in Figure~\ref{fig:poly-reduction}.
To write down that a TM has black-box access to a TM $M_O$ that computes function $O$, we sometimes use the \textit{oracle} terminology.
\begin{definition}[Oracle machine] \index{Turing machine!Oracle machine}
A Turing Machine $M$ is said to have \textit{oracle access} to a function $O(\cdot)$ if it has access to the result of $O(x)$ for any input $x$ of its choice in constant time. We denote the output of $M$ on input $x$ with oracle $O$ by $M^O(x)$.
\end{definition}
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}).
Namely, if a problem is easier than another problem in \textsf{P} (resp. \textsf{PP}), then the former 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: