Algorithm for binary less than in MPC Given binary shares $[a_{l-1}]_p,...,[a_0]_p, [b_{l-1}]_p,...,[b_0]_p$ such that $a = \sum_{i=0}^{l-1} a_i2^i,b = \sum_{i=0}^{l-1} b_i2^i $
for $a,b \in Z_p$, how to compute $a \overset{?}{\lt} b$ in MPC (the result is not closed)?
find the positions where $a$ and $b$'s bits differ: $c_i=a_i\oplus b_i=a_i+b_i-2a_ib_i$

compute partial or-sums: $d_{l-1}=c_{l-1},\quad d_i=d_{i+1}\vee c_i=a_i+b_i-a_i b_i$

isolate the first differing bit: $e_{l-1}=d_{l-1},\quad e_i=d_i-d_{i+1}$

the result is the corresponding bit in $b$: $\sum e_i b_i $