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In real word RSA modules are so large that probability for finding $c_1$ which is not coprime with $n$ is approximately zero. Also if you founded such number then $p=gcd(c_1,n)\neq1$ so $p$ is a factor of $n$ and in this case attack is not necessary because $n$ is factored. $gcd(296,1073)=37\neq 1$ so $p=37,q=\frac{1073}{37}=29$ and $\phi(n)=1008$ Now ...

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You need an authentic channel from Alice to Bob to get a secret channel from Bob to Alice. This assumption is missing in a), so anyone in control of the communication channel can play man in the middle on any protocol. As long you don't have a secret channel from Alice to Bob or an authentic channel from Bob to Alice, Alice will never (= for any protocol) ...

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The problem is to reliably and efficiently find message $m$ (with $0\le m<n$) given RSA modulus $n$, distinct RSA public exponents $e_1$ and $e_2$ coprime to each others and to the unknown $\phi(n)$, and ciphertexts $c_1=m^{e_1}\bmod n$ and $c_2=m^{e_2}\bmod n$. WLoG, and per the corrected question, $y_1$ is negative when it is applied the extended ...

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With such a small block size there is no way to employ RSA padding modes such as PKCS#1 v1.5 padding or OAEP. You could however see the encryption as ECB mode encryption. In that case you could apply padding mechanisms that have been constructed for symmetric block ciphers. Those padding modes however have been defined for bytes rather than characters. ...

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This is an issue with any block cipher. One solution is to pad the message. This means that, first you split it into blocks and then you will have some remaining characters at the end that are not one whole block. So lets say that the block length is L and you have n characters. You can add at the end of your message L-n extra characters so that with those, ...

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