Lightweight multimedia encryption: Algorithms and performance analysis.  Page 72 
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the more pairs are needed and hence the greater the cryptanalyst’s effort to find out the correct key. The resistance to the knownplaintext attack thus counts heavily on the number and characteristics of alias keys for a general pair of plaintext/ciphertext. It is shown in lemma 3 that the number of alias keys Alias(A,C) is larger than p¼N/2 for at least one ciphertext A′. This is however a conservative worstcase estimate for two reasons. First, the actual size of C(N) is strictly less than ¡ N N/2¢, yet ¡ N N/2¢ was used in the derivation. Second, it is implicitly assumed (by the pigeon hole principle) that alias keys of different ciphertexts do not overlap. In fact, we can show by plausible reasoning that the number of alias keys far exceeds p¼N/2. It also grows exponentially with respect to the ciphertext length N. In analysis below, we consider the statistical average and denote the average number of alias keys for a general Nbit plaintext by A(N). We have the following property regarding A(N). Lemma 4 A(N) ∼ cN for sufficiently large N, where c > 1 is a constant. The key to prove this conclusion is the observation that if k1 is any one of the A(N) alias keys for a general plaintext A1 and k2 is any one of the A(N) alias keys for another plaintext A2 then a concatenation key k = k1k2 is alias key for the plaintext concatenation A1A2. A function A(N) satisfying this property must be of the form cN. Appendix A.4 gives a detailed proof. We emphasize that cN is not an accurate formula of A(N) but it depicts the asymptotic behavior of A(N) with respect to N. For a sufficiently large value of N (a long enough plaintext), the average number of alias keys for a general plaintext quickly becomes intractable since it is exponential with the plaintext length. This exponential growth rate of the number 62
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Title  Lightweight multimedia encryption: Algorithms and performance analysis.  Page 72 
Repository email  cisadmin@lib.usc.edu 
Full text  the more pairs are needed and hence the greater the cryptanalyst’s effort to find out the correct key. The resistance to the knownplaintext attack thus counts heavily on the number and characteristics of alias keys for a general pair of plaintext/ciphertext. It is shown in lemma 3 that the number of alias keys Alias(A,C) is larger than p¼N/2 for at least one ciphertext A′. This is however a conservative worstcase estimate for two reasons. First, the actual size of C(N) is strictly less than ¡ N N/2¢, yet ¡ N N/2¢ was used in the derivation. Second, it is implicitly assumed (by the pigeon hole principle) that alias keys of different ciphertexts do not overlap. In fact, we can show by plausible reasoning that the number of alias keys far exceeds p¼N/2. It also grows exponentially with respect to the ciphertext length N. In analysis below, we consider the statistical average and denote the average number of alias keys for a general Nbit plaintext by A(N). We have the following property regarding A(N). Lemma 4 A(N) ∼ cN for sufficiently large N, where c > 1 is a constant. The key to prove this conclusion is the observation that if k1 is any one of the A(N) alias keys for a general plaintext A1 and k2 is any one of the A(N) alias keys for another plaintext A2 then a concatenation key k = k1k2 is alias key for the plaintext concatenation A1A2. A function A(N) satisfying this property must be of the form cN. Appendix A.4 gives a detailed proof. We emphasize that cN is not an accurate formula of A(N) but it depicts the asymptotic behavior of A(N) with respect to N. For a sufficiently large value of N (a long enough plaintext), the average number of alias keys for a general plaintext quickly becomes intractable since it is exponential with the plaintext length. This exponential growth rate of the number 62 