  In secure summation, $K$ users, each holds an input, wish to compute the sum
of the inputs at a server without revealing any information about {\em all the
inputs} even if the server may collude with {\em an arbitrary subset of users}.
In this work, we relax the security and colluding constraints, where the set of
inputs whose information is prohibited from leakage is from a predetermined
collection of sets (e.g., any set of up to $S$ inputs) and the set of colluding
users is from another predetermined collection of sets (e.g., any set of up to
$T$ users). For arbitrary collection of security input sets and colluding user
sets, we characterize the optimal randomness assumption, i.e., the minimum
number of key bits that need to be held by the users, per input bit, for weakly
secure summation to be feasible, which generally involves solving a linear
program.
