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Why we don't use additive groups? Is it a security thing? Yes, it's a security consideration. If we used the additive group $(\Bbb Z_N,+)$ rather than $(\Bbb Z_N^*,*)$ for RSA, public encryption would go $M\mapsto C=e\,M\bmod N$ rather than $M\mapsto C=M^e\bmod N$. Problem is, decryption would be trivial since anyone with the public key $(N,e)$ could ...

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Your question is not clear. Which additive group would you like to use? RSA is hard because the group ${\mathbb Z}_N^*$ has unknown order (assuming the factorization of $N$ is unknown). Which additive group has that property?

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By construction, RSA moduli are the product of two large primes.

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What does the order of $U(N)$ has to do with RSA key generation? The usual notation is $\Bbb Z_N^*$ for the multiplicative group modulo $N$, that the question names $U(N)$, and $\Phi(N)$ or equivalently $\varphi(N)$ for its order (number of elements), as given by Euler's totient function. $\forall x\in\Bbb Z_N^*$, it holds $x^{\Phi(N)}\equiv1\pmod N$. This ...

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