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Fix cw.tex
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Andre Henriques 2023-11-03 13:24:19 +00:00
parent 7674362b6e
commit 4db4524dff
1 changed files with 17 additions and 1 deletions
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cw/cw.tex
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@ -57,7 +57,23 @@
$$H(m')=E(K, IV \oplus m') = C_2$$ $$H(m')=E(K, IV \oplus m') = C_2$$
And if there the hashing function was not collision resistant that would imply And if there the hashing function was not collision resistant that would imply
$$C_1=C_2\Rightarrow D(C_1)=D(C_2) \Rightarrow m=m'$$ $$C_1=C_2\Rightarrow D(C_1)=D(C_2) \Rightarrow m=m'$$
and since $m !=m'$ the hash function is collision resistant, for messages with 1 block. 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
$$H(m)=H(m') \and m\ne m'$$