Interval test

This functionality, denoted as $[a < x < b]$ tests whether $[x]$ lives in a public interval $(a; b)$.

It is implemented as follows:

1. $[r \in_R \mathbb{F}_p]_B$, calculate $[r] = \underset{i=0}{\overset{l-1}\sum}2^i [r_i]$
2. Open $[r+x]$, and denote $a' = (r + x) - b; b' = (r + x) - a$
3. Now, we output $([a' <_B r] \wedge [r <_B b']) \vee ([a'+p <_B r ] \wedge [r <_B b'+p])$

Here, some of the public terms are trivial automatically, and do not need to be calculated (any checks that are out of the range $[0 .. p-1]$).