Two problem about noise management of BFV

Question

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?

Answer 1

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.