A public-key cryptosystem invented by Pascal Paillier in 1999.
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How hard are discrete logarithms problems in $\mathbb Z^{*}_{n}$ and $\mathbb Z^{*}_{n^2}$, where $n$ is the RSA $n=pq$
Use the notations form the Wikipedia article Paillier Cryptosystem
, assume that the chipertext $c$ and $c^{\lambda} \mod n^2$ are both given, is it possible to compute $\lambda$ easily?
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Security relevance of random factor in Paillier
In the Paillier cryptosystem [1] the encryption of $m \in \mathbb{Z}_N$ with randomness $r \in \mathbb{Z}_n^*$ is $c = g^m r^n \bmod{n^2}$.
The additive-homomorphic property of the system shows that
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