Random number bitwise sharing

This functionality, denoted as $[r \in_R \mathbb{F}_p]_B$, generates a (uniformly randomly shared) bit decomposition of a (uniformly random) value in $\mathbb{F}_p$. This is typically done in precomputation phase, and implemented as follows:

1. Set $l = \lceil \log_2(p)\rceil$, obtain a random bitstring $[a]_B$ of length $l$ via $[a_i \in_R \{0,1\}]$.
2. Calculate $[a <_B p]$ and open it, if it is $1$, output $[a]_B$, otherwise repeat process.