Definition related to homomorphic encryption

In the definition of leveled FHE scheme, what is the mining of same decryption circuit? I have read several papers about this, but I don't understand this notion.

In addition, if the scheme {$E^d$} is leveled FHE (notation based on Gentry thesis), why the scheme $E$ works correctly for circuits of some depth?

• Could you add links to documents which you make reference? – Raoul722 Aug 6 '16 at 9:51
• I don't remember if I have already saw the term "same decryption circuit"... Your question doesn't seem a hard one, but it is not clear. If you clarify it, for instance by linking some papers, I think you will get the answers you want. (: – Hilder Vitor Lima Pereira Aug 6 '16 at 13:53

In the 'AND' operation : \begin{align*} c_i\cdot c_j =&\left(p\cdot q_i + 2\cdot n_i + b_i\right)\cdot\left(p\cdot q_j + 2\cdot n_j + b_j\right)\\ =&p\cdot\hat{q} + 2\cdot\hat{n} + \left(b_i\cdot b_j\right)\\\\ {\#bits(\hat{n})} < & \#bits(\frac{p}{2}) \end{align*} \\
If this noise constraint isn't preserved, then the decrypt operation no longer holds. $$Dec(c_k) = b_k = \left[p\cdot q_k + 2\cdot n_k + b_k\right]_p(mod\;2)$$