Lightweight multimedia encryption: Algorithms and performance analysis.  Page 45 
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2. Let ci denote the ith bit of C. For any integer 1 < k ≤ N, we have the following recursive equation: V (k) =X¯i V (k − ¯i), (2.4) where ck−¯i+1 ck−¯i+2 . . . ck is a ¯ibit Huffman code in one of all Huffman tables used in RHT encryption. 3. For any integer 1 < k ≤ N, let S = {s, s + 1, . . . s + L}, s + L ≤ k be any consecutive L + 1 integers. V (k) can be represented as a weighted sum of V (i) with indices i in S; namely, V (k) =Xi aiV (i), i ∈ S, (2.5) where ai’s are integer coefficients. The key to prove the above Lemma is the observation that the gap between two consec utive viable parsing positions must be less than L + 1 because L is the maximal code length of all Huffman tables. By tracing the viable parsing positions backward, we can derive the recursive equation of V (N) 2.4 and 2.5. A detailed proof is given in Appendix A.1. Next, we examine the size of V (N) with respect to bit stream length N. Equations (2.4) and (2.5) suggest that the size of R(N) will grow very quickly for large value of N. It is a wellknown fact that a function satisfying such a recursive relationship has a form of c aN with constant c > 0 and a > 1. Thus, we expect V (N) to have an exponential rate of growth with respect to N. However, the code length ¯i in (2.4) and coefficients ai in (2.5) are not constant values since they vary with the particular bit stream C and all Huffman tables. Thus, it is not easy to derive a determinate formula of V (N) as a function of N. We establish a lower bound of the size of R(N) below. 35
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Title  Lightweight multimedia encryption: Algorithms and performance analysis.  Page 45 
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Full text  2. Let ci denote the ith bit of C. For any integer 1 < k ≤ N, we have the following recursive equation: V (k) =X¯i V (k − ¯i), (2.4) where ck−¯i+1 ck−¯i+2 . . . ck is a ¯ibit Huffman code in one of all Huffman tables used in RHT encryption. 3. For any integer 1 < k ≤ N, let S = {s, s + 1, . . . s + L}, s + L ≤ k be any consecutive L + 1 integers. V (k) can be represented as a weighted sum of V (i) with indices i in S; namely, V (k) =Xi aiV (i), i ∈ S, (2.5) where ai’s are integer coefficients. The key to prove the above Lemma is the observation that the gap between two consec utive viable parsing positions must be less than L + 1 because L is the maximal code length of all Huffman tables. By tracing the viable parsing positions backward, we can derive the recursive equation of V (N) 2.4 and 2.5. A detailed proof is given in Appendix A.1. Next, we examine the size of V (N) with respect to bit stream length N. Equations (2.4) and (2.5) suggest that the size of R(N) will grow very quickly for large value of N. It is a wellknown fact that a function satisfying such a recursive relationship has a form of c aN with constant c > 0 and a > 1. Thus, we expect V (N) to have an exponential rate of growth with respect to N. However, the code length ¯i in (2.4) and coefficients ai in (2.5) are not constant values since they vary with the particular bit stream C and all Huffman tables. Thus, it is not easy to derive a determinate formula of V (N) as a function of N. We establish a lower bound of the size of R(N) below. 35 