I am reading the Wikipedia article on Broadcast attack. In the proof, the editor made an assumption that $M$ is less than all $N$. Why is this assumption safe?
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Because RSA doesn't work at all if the message is larger than the modulus.
In practical confidentiality applications, the message typically just consists of a symmetric session key and some identifying metadata, so there's plenty of room for it (plus some randomized padding to defend against the attack) within the key sizes that are needed for security anyway.
The Broadcast Attack is an attack on the RSA cryptosystem. The Wikipedia page on RSA has the following:
So it looks like that's just the standard protocol? For a more in-depth answer about why, I would have to substantially refresh my understanding of how RSA works.