Protocol Specification. We now describe our protocol for group key agreement un- der the assumptions that (i) the clients have been loaded with master keys according to a distribution satisfying Proposition 1 and (ii) the network is fully connected. In Section V we will consider extensions that support dynamic master key exchange and extensions that account for multi-hop network topologies. Let G = {g1, . . . , gt} be a set of t clients that wish to agree upon a group key for a session with unique identifier u. Client gi possesses the set of master keys with indices tabulated by Ki. We define the occupancy set Oj ⊆ G for each master i=1 key index j ∈ KG = ∪t Ki as the subset of G that possess the key kj – i.e., gi ∈ Oj if and only if j ∈ Ki. Provided that the occupancy sets O = {Oj}j∈KG are known to all group members, then Algorithm 1 can be run in parallel at each client to establish the desired group key. Algorithm 1 determines the transmissions used for group key agreement by applying a simple greedy heuristic to the minimal connected spanning subhypergraph problem implied by the distribution of the master keys indexed by KG (hyperedges) on the clients in G (vertices). To clarify the notation used in Algorithm 1, we revisit the second example of Section II wherein the t = 5 clients G = {v1, v5, v6, v9, v11} wish to establish a group key for a session with unique identifier w. In this example, the occupancy sets with at least two elements are: O1 = {v1, v6} , O4 = {v1, v9} , O6 = {v5, v6, v11} , O9 = {v6, v9} , O11 = {v1, v11} , O12 = {v9, v11} , (3) O15 = {v1, v5} , O19 = {v5, v9} . The largest occupancy set is O6 so Algorithm 1 begins by setting j0 = 6, C = O6, and l = 1. Clients v5, v6, and Input: Occupancy sets O = {Oj}j∈KG , group G = {g1, . . . , gt}, and a common PRF φ(). Output: Group key sj0 ,u for session with unique identifier u. j0 ← index of largest occupancy set in O, C ← Oj0 , l ← 1; if gi ∈ Oj0 then compute the group key sj0 ,u ← φ (kj0 , u); end while C ƒ= G do jl ← index of an occupancy set Ojl ∈ O satisfying Ojl ∩ C ƒ= ∅ that maximizes |Ojl \ C|; il ← a client in Ojl ∩ C; if gi = il then l l compute the lth one-time pad sj ,u ← φ (kj , u); compute the bit-wise sum ml,u = sj0 ,u ⊕ sjl,u; transmit ml,u to all clients in Ojl \ C; else if gi ∈ Ojl \ C then
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