MDS Variable Generation and Secure Summation with User Selection
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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 ∈{1,2,⋯, K}, the minimum size of the randomness variable at each user is 1 + 1/2 + ⋯ + 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 + ⋯ + 1/(K-1) bits, whose achievability uses the generation of (K,n)-MDS variables for n ∈{1,2,⋯,K-1}.
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