MDS Variable Generation and Secure Summation With User Selection

A collection of K random variables are called $(K,n)$ -MDS if any n of the K variables are independent and determine all remaining variables. In the MDS variable generation problem, K users wish to generate variables that are $(K,n)$ -MDS using a randomness variable owned by each user. We show that to generate 1 bit of $(K,n)$ -MDS variables for each $n \in \{1,2,\cdots , K\}$ , the minimum size of the randomness variable at each user is $1 + 1/2 + \cdots + 1/K$ bits. An intimately related problem is secure summation with user selection, where a server may select an arbitrary subset of K users and securely compute the sum of the inputs of the selected users. We show that to compute 1 bit of an arbitrarily chosen sum securely, the minimum size of the key held by each user is $1 + 1/2 + \cdots + 1/(K-1)$ bits, whose achievability uses the generation of $(K,n)$ -MDS variables for $n \in \{1,2,\cdots ,K-1\}$ .