I'm studying the paper Fully Homomorphic Encryption over the Integers by Marten van Dijk, Craig Gentry, Shai Halevi and Vinod Vaikuntanathan.
I have questions about the proof of Lemma A.1.
In page 6, the element of public key is in the form of $x=pq+r$ and the encryption is $$ c← [m+2r+2\sum x_i]_{x_0} $$
But in the proof of Lemma A.1., the public key is $x=pq+2r$ and the encryption is $$ c← [m+2r+\sum x_i]_{x_0} $$
This problem made me confused. I tried to proof with first definition, but I encounter with another problem. When I follow the same flow of the proof with first definition, I reached that the noise term is $$(m+2r+k*r_0+ \sum 2r_i )$$ What I need to next is to get the upper bound of a $|m+2r+k\cdot r_0+ \sum 2r_i|$. So I got $$|m+2r+k*r_0+ \sum 2r_i| \leq m+2\cdot2^{\rho\prime} + \tau \cdot 2^{\rho}+\tau \cdot 2^{\rho} =m+2\cdot2^{\rho\prime} +2 \tau2^{\rho}$$
If the definition of permitted circuit in this thesis is correct, at least $$m+2\cdot2^{\rho\prime} +2 \tau2^{\rho} \leq 2^{\rho \prime +2}$$ should be satisfied.
Can anybody help to get upper bound of noise?
Original question
I'm reading https://eprint.iacr.org/2009/616.pdf
I am reading the proof of Lemma A.1 which is in page 21.
In the page 22, the thesis tells that the noise is at most $(4\tau + 3)2^\rho < \tau2^{ \rho+3}$.
But I cannot follow the inequality.
Could you guys help me to understand the proof of Lemma A.1?