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title{Authentication Codes}
author{Abigail A. Villa\ Skezeer John B. Paz}
institute{Ph.D. Mathematics \ De La Salle University\ texttt{abigail_villa@dlsu.edu.ph // skezeer_john_paz@dlsu.edu.ph}}
date{November 27, 2017}
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AtBeginSection[]  % The commands within the following {} will be executed at the start of each section.
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frametitle{Outline} % This is the title of the outline.
tableofcontents[pausesections]  % This will display the table of contents and highlight the current section.
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begin{document}

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titlepage
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section{Orthogonal Array} % Since this is the start of a new section, our recurring outline will appear here.
subsection[PRELIMS]{Preliminaries}

begin{frame}
frametitle{PRELIMINARIES}
textbf{Latin Square}\ pause
A Latin square of order $n$ with entries from an $n-set~X$ is an $n times n$ array $L$ in which every cell contains an element of $X$ such that every row of $L$ is a permutation of $X$ and every column of $L$ is a permutation of $X$.
begin{block}{Example}
    A Latin square of order 4.\
    [
    begin{array}{|cccc|}
    hline
    1&2&3&4\
    4&1&2&3\
    3&4&1&2\
    2&3&4&1\
    hline
    end{array}
    ]
end{block}
end{frame}

begin{frame}
textbf{Quasigroup}\
textbf{Definition:} Let $X$ be a finite set of cardinality $n$, and let $circ$ be a binary operation defined on $X~(i.e., circ: Xtimes Xlongrightarrow X)$. We say that the pair $(X, circ)$ is a quasigroup of order $n$ provided that the following two properties are satisfied:\
begin{enumerate}
    item For every $x,yin X$, the equation $xcirc z=y$ has a unique solution for $zin X$.\
    item For every $x,yin X$, the equation $zcirc x=y$ has a unique solution for $zin X$.
end{enumerate}
The textit{operation table} of a binary operation $circ$ defined on $X$ is the $|X| times |X|$ array $A=(a_{x,y})$, where $a_{x,y}=xcirc y$

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begin{frame}
textbf{Idempotent/Symmetric quasigroup}\
textbf{Definition:} Suppose $(X,circ)$ is a quasigroup. We say that $(X,circ)$ is an textit{idempotent quasigroup} if $xcirc x =x$ for all $xin X$, and we say that $(X,circ)$ is a textit{symmetric quasigroup} if $xcirc y= y circ x$ for al $x,y in X$.
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These concepts can also be defined for Latin squares in the obvious way: A textit{symmetric Latin square} $L=(mathscr{l}_{x,y})$ .
 end{frame}

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