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\begin{center}\framebox{\parbox{\boxlength}{\bf
CS 681: Computational Number Theory and Algebra \hfill Lecture 11
\\
Lecturer: Manindra Agrawal \hfill
Notes by: Ashwini Aroskar
%\\
\begin{flushright}
%date
August 30, 2004.
\end{flushright}
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\vspace{5mm}

\section{Recall}

\begin{fact}
$Res(f,g) = 0$ \quad iff \quad $ gcd(f,g) > 1$ \label{first-fact}
\end{fact}

\begin{fact}
There exists $y$ $ \epsilon$ $ F_q$ such that $ gcd(e(x)-y, f(x))
>
1$
 \label{first-fact}
\end{fact}
We want $y$ $ \epsilon$ $ F_q$ such that $Res(e(x)-y,f(x))=0$ \\
$Res(e(x)-y,f(x))$ is a polynomial in $y$ over $F_q$ of degree
$\leq 2d-1$\\
Let this polynomial be $g(y)$.\\
\\
If we can find a root of $g$ in $F_q$ then we can factorize $f$.\\
\\
Let $\hat{g} (y) = gcd(g(y), y^q-y)$\\
All roots of $g(y)$ in $F_q$ are roots of $\hat{g}(y)$ too.\\
\\
Now, the remaining problem is to find roots of a given polynomial
over a finite field $F_q$.\\
No polynomial time algorithm is known for this problem.\\


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\section{A Randomized polynomial time algorithm for root finding}
Let $f(x)$ be a square-free polynomial over $F_q$ of degree $d$
and such that $f$ factors completely over $F_q$.\\
\\
Let $f(x) = \Pi_{i=1}^d (x-\alpha_i)$\\
Note that $\alpha_i \not= \alpha_j$.\\
\\
Let $f_{ss}(x)=f(x^2+\beta)=\Pi_{i=1}^d (x^2+\beta-\alpha_i)$\\
\\
If there exist $\alpha_i$ and $\alpha_j$ such that
$x^2+\beta-\alpha_i$ is reducible and $x^2+\beta-\alpha_j$ is
irreducible, then $f_{ss}$ can be factored. \\
Using factors of $f_{ss}$, $f$ can be factored.\\
\\
Fix $\{\alpha_i,\alpha_j\} = \{\alpha_1, \alpha_2\}$\\
\\
$Prob[x^2+\beta-\alpha_1 $ and $ x^2+\beta-\alpha_2$ are both
reducible
or irreducible$]$\\
$= Prob[$both $\alpha_1-\beta$ and $\alpha_2-\beta$ are squares in
$F_q$ or neither is$]$\\
$= Prob[ \beta \epsilon F_q : (\alpha_1-\beta)^{\frac{q-1}{2}} =
(\alpha_2-\beta)^{\frac{q-1}{2}} ]$\\
$= \frac{1}{\vert F_q \vert}($ number of roots of polynomial
$(\alpha_1-z)^{\frac{q-1}{2}}-(\alpha_2-z)^{\frac{q-1}{2}} )$\\
$\leq \frac{q-1}{2q} < \frac{1}{2}$\\
\\
Choose $k$ values of $\beta$.\\
$Prob[$no value of $\beta$ helps factor $f_{ss}] < \frac{1}{2^k}$\\
\\
Repeating this algorithm makes the probability of error very
small.\\
Roots of $f$ can be computed using repeated applications of the
algorithm.\\
\\
There exist randomized polynomial time algorithms for factoring
multivariate polynomials in compact representation.\\
\\
A polynomial over the field of rationals can be factored in
polynomial time.

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