Secure Summation with User Selection and Collusion

The secure summation problem is studied with user selection and collusion, where a server may select any $U$ out of $K$ users and compute the sum of the inputs from the selected users without learning any additional information even if the server colludes with any $T$ out of $K$ users. The optimal communication and randomness rate is characterized when either $U=2$ or $T=1$, i.e., to securely compute 1 bit of the selected sum, each user needs to send 1 bit to the server, each user needs to hold a key of $T+1$ bits when $U=2$ and $U/(U-1)$-bits when $T=1$, and all users need to hold key variables of ($\binom{T+2}{2}$) bits when $U=2$ and $U/(U-1)+U-1$ bits when $T=1$.