In this paper (A simple provably secure key exchange by Ding et al.), I am trying to understand the the correctness(which is given on page number 8) of the key exchange technique based on LWE. To understand the correctness of scheme, there is very important Lemma 2(on page number 6). The Lemma is as follows:

Lemma 2. Let q>8 be an odd integer, the function E defined above is robust extractor with respect to S with error tolerance $\frac{q}{4}$-2

I have confusion in the proof of Lemma2.

1. How author derieve the condition that Lemma2 is true for q$\gt$8?
2. How |y+ $\sigma$. $\frac{(q-1)}{2}$ mod q| $\le$ $\frac{q}{4}$+1 ?
3. At the end of proof of Lemma2, the author writes that Our robust extractor enjoys a very nice property which says that for uniformly random x$\in$$ \mathbb{Z} _q$, E(x, $\sigma$) is uniform in {0,1} even conditioned on $\sigma$, where $\sigma$ <-- S(x). This property becomes Lemma3 of this paper.

Can anyone answer above three questions and thus the proof of Lemma2 and Lemma3? Thanks!

In this paper (A simple provably secure key exchange by Ding et al.), I am trying to understand the correctness  (which is given on page number 8) of the key exchange technique based on LWE.

To understand the correctness of the scheme, there is a very important lemma (on page number 6). The lemma is as follows:

Lemma 2. Let $q>8$ be an odd integer, the function $E$ defined above is a robust extractor with respect to $S$ with error tolerance $\frac{q}{4}-2$.

I'm confused by the proof of Lemma 2:

1. How does the author derive the condition that Lemma 2 is true for $q\gt 8$?
2. How does $|y+ \sigma$ $\frac{(q-1)}{2} \mod q| \le \frac{q}{4}+1$?
3. At the end of the proof of Lemma 2, the author writes that

Our robust extractor enjoys a very nice property which says that for uniformly random $x\in$$ \mathbb{Z} _q$, $E(x, \sigma$) is uniform in $\{0,1\}$ even conditioned on $\sigma$, where $\sigma \leftarrow S(x)$.

This property becomes Lemma 3 of this paper.

Can anyone answer the above three questions and thus the proof of Lemma 2 and Lemma 3? Thanks!