Lightweight multimedia encryption: Algorithms and performance analysis.  Page 119 
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Appendix A Proof of Theorems A.1 Proof of Lemma 1 Proof : For any given bit stream C, we can parse C using the parsing attack algorithm described in Sec. 2.6.2. We view any L+1 consecutive integers s, s+1, . . . s+L as a window of length L+1 and examine the corresponding bit positions in this window as shown in Fig. A.1. Since L is the maximal code length of all Huffman tables used in RHT encryption, the parsing position must fall into this window at least once. Hence, there is a viable parsing for at least one length s + ® between s and s + L. According to Def. 3 we have V (s + ®) ≥ 1, 0 ≤ ® ≤ L, which is the one given by (2.3). bit stream 0 1 0 … 1 0 … 1 1 1 0 … position 1 2 3 … s s+1 … s+L–1 s+L V (s +a ) ³1 Figure A.1: Illustration of the viable parsing window. Let P(k) denote all viable kparsing schemes. Suppose the last parsed code length is ¯i. Then, it is clear that any parsing scheme in P(k) comes from a viable k−¯iparsing plus the last parsed code ck−¯i+1 ck−¯i+2 . . . ck , ending at kth bit ck. Thus, P(k) can be classified according to the last code ck−¯i+1 ck−¯i+2 . . . ck with length ¯i. Note that for different ¯i, the corresponding viable parsing schemes are distinct. For example, a viable k−2parsing plus 2bit code “11” is different from a viable k−4parsing plus 4bit code “1011”, though both of them lead to a viable kparsing ending at bit ck. This is illustrated in Fig. A.2. Therefore, this categorization of P(k) is allinclusive and mutualexclusive. The number of all viable kparsings is the sum of all viable k−¯iparsings. That is V (k) =X¯i V (k − ¯i) which is given by (2.4). Note that Eq. (2.4) is a recursive relationship that can be applied to V (k − ¯i) as well. Starting at bit position k, V (k) is represented as a sum of V (k−¯i). By repeatedly applying Eq. 2.4 backward, V (k) can be reduced to a weighted sum of more previous viable parsing V (i)’s with smaller length i. Due to the same reason, in the proof of Eq. (2.3), the length 109
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Title  Lightweight multimedia encryption: Algorithms and performance analysis.  Page 119 
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Full text  Appendix A Proof of Theorems A.1 Proof of Lemma 1 Proof : For any given bit stream C, we can parse C using the parsing attack algorithm described in Sec. 2.6.2. We view any L+1 consecutive integers s, s+1, . . . s+L as a window of length L+1 and examine the corresponding bit positions in this window as shown in Fig. A.1. Since L is the maximal code length of all Huffman tables used in RHT encryption, the parsing position must fall into this window at least once. Hence, there is a viable parsing for at least one length s + ® between s and s + L. According to Def. 3 we have V (s + ®) ≥ 1, 0 ≤ ® ≤ L, which is the one given by (2.3). bit stream 0 1 0 … 1 0 … 1 1 1 0 … position 1 2 3 … s s+1 … s+L–1 s+L V (s +a ) ³1 Figure A.1: Illustration of the viable parsing window. Let P(k) denote all viable kparsing schemes. Suppose the last parsed code length is ¯i. Then, it is clear that any parsing scheme in P(k) comes from a viable k−¯iparsing plus the last parsed code ck−¯i+1 ck−¯i+2 . . . ck , ending at kth bit ck. Thus, P(k) can be classified according to the last code ck−¯i+1 ck−¯i+2 . . . ck with length ¯i. Note that for different ¯i, the corresponding viable parsing schemes are distinct. For example, a viable k−2parsing plus 2bit code “11” is different from a viable k−4parsing plus 4bit code “1011”, though both of them lead to a viable kparsing ending at bit ck. This is illustrated in Fig. A.2. Therefore, this categorization of P(k) is allinclusive and mutualexclusive. The number of all viable kparsings is the sum of all viable k−¯iparsings. That is V (k) =X¯i V (k − ¯i) which is given by (2.4). Note that Eq. (2.4) is a recursive relationship that can be applied to V (k − ¯i) as well. Starting at bit position k, V (k) is represented as a sum of V (k−¯i). By repeatedly applying Eq. 2.4 backward, V (k) can be reduced to a weighted sum of more previous viable parsing V (i)’s with smaller length i. Due to the same reason, in the proof of Eq. (2.3), the length 109 