Bit decomposition

Previous constructions were intentionally circumventing bit-decomposition protocol, as it is more heavy than any of them. However, sometimes bit decomposition is necessary.

This functionality takes a shared value $[x]$ and produces a shared bitstring $[x]_B$.

1. $[r \in_R \mathbb{F}_p]_B$, calculate $[r] = \underset{i=0}{\overset{l}\sum}2^ir_i$
2. Open $b = (x+r) \mod p$, denote $b_i$ its bit decomposition. Denote $b'_i$ the bit decomposition of $p + b$.
3. Calculate $[t] = [r \leq b]$.
4. Set $[q_i] = [t] b'_i + [1-t] b_i$
5. $[x]_B = [q]_B - [r]_B$ //obtained using bitstring subtraction