Lightweight multimedia encryption: Algorithms and performance analysis.  Page 66 
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can be evaluated accordingly. Second, we examine the randomness of the input and the output streams from an information theory viewpoint. That is, we prove that there is no entropy change between the input and the output streams. 3.5.1 Key Space Analysis The security of the RPB encryption scheme relies on the size of the key space. For a given Nbit ciphertext C = RPB(A, p, r), the key space of the RPB scheme is the total number of different ways to decrypt C using all possible partition sequence p and rotation sequence r. As mentioned before, if the ciphertext is C = RPB(A, p, r), then the plaintext is A = RPB(C, p, p − r). Thus, the key space is equivalent to the total number of different ways to encrypt A using all possible p and r. We have the following definition. Definition 5 Let A = (a1a2 . . . aN) be a bit stream of length N. Two RPBs of A, RPB(A, p1, r1) and RPB(A, p2, r2), are said to be different if they achieve a different order of ai’s in the resulting stream C. The total number of different RPBs is denoted by R(N). The key space of a complete permutation of A = (a1a2 . . . aN) is N!. Clearly, R(n) < N! because a lot of these permutations cannot be achieved by applying RPB operation due to two reasons. First, the block rotation in RPB operation prohibits some particular permutations to be produced. For example, in a simple case A = (a1a2a3a4), the permutation (a4a3a2a1) cannot be a result of any RPB operation. Actually R(4) = 12 while the number of complete permutation is 4! = 24. Second, the upper bound of the partitioned block size reduces the number of different RPBs. Because we require pi < B, it is impossible that an RPB starts with ai for i > B + 1. While an exact expression of R(N) may be difficult to obtain, we derive a recursive relationship of R(N) and establish a lower bound for R(N) as given in the following lemma. 56
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Title  Lightweight multimedia encryption: Algorithms and performance analysis.  Page 66 
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Full text  can be evaluated accordingly. Second, we examine the randomness of the input and the output streams from an information theory viewpoint. That is, we prove that there is no entropy change between the input and the output streams. 3.5.1 Key Space Analysis The security of the RPB encryption scheme relies on the size of the key space. For a given Nbit ciphertext C = RPB(A, p, r), the key space of the RPB scheme is the total number of different ways to decrypt C using all possible partition sequence p and rotation sequence r. As mentioned before, if the ciphertext is C = RPB(A, p, r), then the plaintext is A = RPB(C, p, p − r). Thus, the key space is equivalent to the total number of different ways to encrypt A using all possible p and r. We have the following definition. Definition 5 Let A = (a1a2 . . . aN) be a bit stream of length N. Two RPBs of A, RPB(A, p1, r1) and RPB(A, p2, r2), are said to be different if they achieve a different order of ai’s in the resulting stream C. The total number of different RPBs is denoted by R(N). The key space of a complete permutation of A = (a1a2 . . . aN) is N!. Clearly, R(n) < N! because a lot of these permutations cannot be achieved by applying RPB operation due to two reasons. First, the block rotation in RPB operation prohibits some particular permutations to be produced. For example, in a simple case A = (a1a2a3a4), the permutation (a4a3a2a1) cannot be a result of any RPB operation. Actually R(4) = 12 while the number of complete permutation is 4! = 24. Second, the upper bound of the partitioned block size reduces the number of different RPBs. Because we require pi < B, it is impossible that an RPB starts with ai for i > B + 1. While an exact expression of R(N) may be difficult to obtain, we derive a recursive relationship of R(N) and establish a lower bound for R(N) as given in the following lemma. 56 