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This supplementary file consists of three sections. In Section I, a theorem is presented to prove that the number of points on a hyper-sphere over finite field GF(p) is at least pN1 for a given hyper-sphere determined by C = (c0; c1; : : : ; cN) 2 GF(p)N+1 and R 2 GF(p), where p is a prime. In Section II, a concrete algorithm to find a point on a hyper-sphere is constructed. In Section III, two lemmas and a theorem are proposed and proven, then the security of the proposed group key management scheme is proven formally.