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% Write the approved title of your dissertation
\title{Automated image classification with expandable models}

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\author{Andre Henriques, 6644818}

\date{}

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\begin{document}
	\section*{5}
	\subsection*{5.1}
	The hash function is collision resistante for $n=1$, since if the block size is one the hash function is the encryption. Therefore:
	if the message is only one block long:
	$$H=E$$
	$$m\ne m'$$
	$$H(m)=E(K, IV \oplus m) = C_1$$
	$$H(m')=E(K, IV \oplus m') = C_2$$
	And if there the hashing function was not collision resistant that would imply
	$$C_1=C_2\Rightarrow D(C_1)=D(C_2) \Rightarrow m=m'$$
	and since $m\ne m'$ the hash function is collision resistant, for messages with 1 block.

	For if the block size is bigger than one we can say
	$$H(m)=E(m)_{\text{Last Block}}$$
	$$E(m)=E(K, m)$$
	$$\exists a,b,c,d : m = a||b \and m' = c||d$$
	where a,b,c,d are the size of one block
	$$H(m)=E(b \oplus E(a \oplus IV)) = C_1$$
	$$H(m')=E(d \oplus E(c \oplus IV)) = C_2$$
	since it's possible to have: 
	$$b \oplus E(a \oplus IV) = d \oplus E(c \oplus IV)$$
	with:
	$$a \ne b \ne c \ne d$$
	therefore
	$$H(m)=H(m') \and m\ne m'$$

	
	

	

	


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