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  1. $A$ sends $B$ the encryption $E_{pkA}(m)$.
  2. $B$ computes $R=xE_{pkA}(m) + y$ and sends $R$ back to $A$, but tells him nothing about the parameters $x$ and $y$.
  3. $A$ performs $D_{pkA}(R)$ and recovers the value $xm+y$.

Is this protocol possible using the Paillier cryptosystem?

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Yes (assuming $A$ knows the Paillier private key and $B$ knows the Paillier public key and the values of $x$ and $y$)

With Pallier, someone with the public key can:

  • Homomorpically multiply a ciphertext by a known value; that is, given $x$ and $E_{pkA}(m)$ compute $E_{pkA}(xm)$

  • Encrypt a known value; that is, given $y$, compute $E_{pkA}(y)$

  • Homomorphically add two encrypted values; that is, given $E_{pkA}(xm)$ and $E_{pkA}(y)$, compute $R = E_{pkA}(xm + y)$

The resulting $R$ value can be decrypted by $A$; $A$ cannot deduce anything other than the value $xm + y$

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