# Two problem about noise management of BFV

I have stuck in two problems when understanding the noise management of BFV scheme, and I don't have any idea about the two problem, help me please.

Problem 1:
In the Lemma 3 , the paper demonstrates that the norm of the noise after multiplying two ciphertexts $$ct_0$$ and $$ct_1$$ is $$\left \|v_{mul}\right\|$$=$$E\cdot\delta_R\cdot(\delta_R+1.25)+E_{Relin}$$, and I think the norm of $$v_{mul}$$ should be the the sum of $$v_3$$'s norm in Lemma 2 and $$E_{Relin}$$ in the Relinearisation step. The norm of $$v_3$$ in Lemma 2 is $$2\cdot\delta_R\cdot t\cdot E\cdot (\delta_R \cdot \left \|s\right\|+1)+2\cdot t^{2}\cdot\delta_R^2\cdot(\left \|s\right\|+1)^2$$, and it has a significant gap with the norm of $$\left \|v_{mul}\right\|$$ with $$E_{Relin}$$ removed. I am very confused with the gap, do you know why?

Problem 2:
The paper gives a noise size roughly after L levels of multiplications in the content between Lemma 2 and Theorem 1, it's $$2\cdot B\cdot \delta_R^{2L+1}\cdot t^L$$, and it seems that it can't be derived from Lemma 3. Do you know how the noise size comes?

Problem 2 first. If you use $$E=2*\delta_R*B$$ as an initial noise and you perform $$L$$ multiplications (using the same ciphertext), then you multiply (see Lemma 2) the noise $$E$$ each time by $$t\cdot \delta_R(\delta_R+1.25)$$, ignoring the relinearization error as in the paper. In total, you have $$E\cdot t^L \cdot \delta_R^L (\delta_R + 1)^L$$ afterwards. Now, they estimated $$(\delta_R+1)^L \approx \delta_R^L$$, which yields the claim.
Problem 1 seems a bit trickier. For sure they assumed $$\| s \| = 1$$. Since in Lemma 3 $$t$$ doesn't show up, probably they assumed it to be small, too (which it should be because of the statement in Lemma 2). The crucial observation is to ask where the $$(\delta_R+1.25)$$ comes from. I'm not 100% sure, but I think you'll find the answer in the discussion above Lemma 2, especially $$\| r_a \| < (\delta_R + 1)^2/4$$ looks suspicious in that matter.