# RSA cryptography Algebra

This is a homework problem I am trying to do. I have done part 2i) as well as 2ii) and know how to do the rest. I am stuck on 2iii) and 2vii). I truly dont know 2vii because it could be some special case I am not aware of. As for 2iii) I tried to approach it the same way as I did 2ii in which case I said you could take the 2 equations and use substitution method to isolate each variable and plug it in to find your variable values but 2iii) that doesnt work so I dont know how to do it.

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For $s$ sufficiently small, we can go from $b^2=n+s^2$ to $b^2\approx n$. Take the square root and you approximately have the average of $p$ and $q$. Since $s$ is small so is their difference (relatively), so we can search around $\sqrt{n}$ for $p$ or $q$. The part (iv) means absolute difference and should have written that. Take the square root of your number and you will find $p$ and $q$ very close nearby.
For (vii), say cipher and plaintexts are equal, so $m\equiv m^e$ modulo $n$. There are only a certain # of $m$ that allow this: those with $m\equiv0$ or $1$ mod $n$ or $p|m$ and $q|(m-1)$ or vice-versa, by elementary number theory. If the scheme uses padding to avoid these messages, then no collision is possible between plain and cipher, but otherwise if you allow arbitrary numbers as messages it clearly is.