Lightweight multimedia encryption: Algorithms and performance analysis.  Page 74 
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To solve this problem, we may examine it from another angle. That is, X containing only one key is equivalent to saying that ¯X , the complementary set of X, contains R(N)−1 keys. By treating ¯X , we can convert this problem to a variant of the classical coupon collector problem. Please refer to Appendix A.5 for a complete proof. Although the number of plaintext/ciphertext pairs needed to uniquely determine the correct key is linear with N, the total complexity to mount an attack is still formidable due to the exponential growth rate of alias key number A(N) with plaintext length N. It was mentioned before that in the black box structure, a cryptanalyst is only allowed to manipulate raw data M but prohibited from directly manipulating A, which is the input to the RPB unit. In other words, it is difficult for a cryptanalyst to obtain input A with an arbitrarily desired characteristics for advanced attacks. Thus, we can claim that, for sufficiently large N, which is true for multimedia data, the cascaded compressionRPB scheme has strong resistance against the knownplaintext attack. 3.5.6 Numerical Evaluation of A(N) As demonstrated above, the resistance of RPB encryption scheme to knownplaintext attack is attributed to the existence of multiple alias keys that encipher a given plaintext to the same ciphertext. We have proved that the average number of alias keys for a general Nbit stream also grows exponentially with N, that is, A(N) ∼ cN for c > 1, but did not give the exact value or a range of c. Table 3.2 provides a rough idea of how large A(N) is for a different pair of c and N. The size of N is typically of order 104 in bits. From the table, we see that 1.0110000 ≈ 1.636 × 1043 ≈ 2143. Thus, even if c is very small, in practice the size of A(N) is already impossible to attack with commonly available computing power. 64
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Title  Lightweight multimedia encryption: Algorithms and performance analysis.  Page 74 
Repository email  cisadmin@lib.usc.edu 
Full text  To solve this problem, we may examine it from another angle. That is, X containing only one key is equivalent to saying that ¯X , the complementary set of X, contains R(N)−1 keys. By treating ¯X , we can convert this problem to a variant of the classical coupon collector problem. Please refer to Appendix A.5 for a complete proof. Although the number of plaintext/ciphertext pairs needed to uniquely determine the correct key is linear with N, the total complexity to mount an attack is still formidable due to the exponential growth rate of alias key number A(N) with plaintext length N. It was mentioned before that in the black box structure, a cryptanalyst is only allowed to manipulate raw data M but prohibited from directly manipulating A, which is the input to the RPB unit. In other words, it is difficult for a cryptanalyst to obtain input A with an arbitrarily desired characteristics for advanced attacks. Thus, we can claim that, for sufficiently large N, which is true for multimedia data, the cascaded compressionRPB scheme has strong resistance against the knownplaintext attack. 3.5.6 Numerical Evaluation of A(N) As demonstrated above, the resistance of RPB encryption scheme to knownplaintext attack is attributed to the existence of multiple alias keys that encipher a given plaintext to the same ciphertext. We have proved that the average number of alias keys for a general Nbit stream also grows exponentially with N, that is, A(N) ∼ cN for c > 1, but did not give the exact value or a range of c. Table 3.2 provides a rough idea of how large A(N) is for a different pair of c and N. The size of N is typically of order 104 in bits. From the table, we see that 1.0110000 ≈ 1.636 × 1043 ≈ 2143. Thus, even if c is very small, in practice the size of A(N) is already impossible to attack with commonly available computing power. 64 