Test math
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cw/cw.tex
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70
cw/cw.tex
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%%% Preamble
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\documentclass[11pt, a4paper]{article}
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\usepackage[english]{babel} % English language/hyphenation
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\usepackage{url}
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\usepackage{tabularx}
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\usepackage{pdfpages}
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\usepackage{float}
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\usepackage{graphicx}
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\graphicspath{ {../images for report/} }
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\usepackage[margin=2cm]{geometry}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks,
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citecolor=black,
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filecolor=black,
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linkcolor=black,
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urlcolor=black
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}
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\usepackage{cleveref}
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%%% Custom headers/footers (fancyhdr package)
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\usepackage{fancyhdr}
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\pagestyle{fancyplain}
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\fancyhead{} % No page header
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\fancyfoot[L]{} % Empty
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\fancyfoot[C]{\thepage} % Pagenumbering
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\fancyfoot[R]{} % Empty
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\renewcommand{\headrulewidth}{0pt} % Remove header underlines
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\renewcommand{\footrulewidth}{0pt} % Remove footer underlines
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\setlength{\headheight}{13.6pt}
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% numeric
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\usepackage[style=ieee,sorting=none,backend=biber]{biblatex}
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\addbibresource{../main.bib}
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% Write the approved title of your dissertation
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\title{Automated image classification with expandable models}
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% Write your full name, as in University records
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\author{Andre Henriques, 6644818}
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\date{}
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%%% Begin document
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\begin{document}
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\section*{5}
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\subsection*{5.1}
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The hash function is collision resistante for $n=1$, since if the block size is one the hash function is the encryption. Therefore:
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if the message is only one block long:
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$$H=E$$
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$$m\ne m'$$
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$$H(m)=E(K, IV \xor m) = C_1$$
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$$H(m')=E(K, IV \xor m') = C_2$$
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And if there the hashing function was not collision resistant that would imply
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$$C_1=C_2\implies D(C_1)=D(C_2) \impies m=m'$$
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and since $m !=m'$ the hash function is collision resistant, for messages with 1 block.
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\end{document}
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