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11. Find the value of respectively.
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in (4.13) and (4.14),
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12. Consider the CMEA cipher with block size n = 3. Suppose that we choose plaintext blocks of the form (po,pl,p2) = (1 - x,1 - x,O), where x E {0,1,2,.. . ,255}. Show that
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b. If x is odd, then ciphertext co is odd.
13. For the CMEA cipher with block size n = 3, suppose that
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a. Apply the algorithm in Table 4.6 to determine the resulting ciz. phertext (co,el, c ) b. Describe a chosen plaintext attack that uses this result to determine T(O). 14. The purpose of this problem is to determine the probability that the attack on the CMEA cipher will succeed. Let n = 3 be the block size. In the CMEA attacks discussed in this chapter, we first determine a putative T(O), then for each j = 1 , 2 , . . . ,255, we attempt to recover T ( j ) .If for any j we find xj - j # C and (xj@ 1)- j # C , then we know that the putative T ( 0 )is incorrect. If this does not occur, then we assume T ( 0 ) is correct and for each j we have recovered either T ( j )or T ( j )@ 1. Let z E { T ( j ) , T ( j ) 1) be the recovered value. If xj - j E C but j @
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# C, then we know that x j = T ( j ) and, similarly, if E C but xj - ,j # C , then we know that xj @ 1 = T ( j ) . However, if ( z j e 1 ) - j E C and xj - j E C, then we cannot, immediately determine the value of T ( j )from xj.
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Let A be the set of j E { 0 ) 1 , 2 , ...,255} for which T ( j )cannot he uniquely determined. Also, let U be the set of indices for which T ( j ) has been uniquely determined. Then A U U = {0,1,2, . . . ,255}, and A and U are disjoint. Note that 0 E U .
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a. Determine E(IA1) and E ( I U ( ) , where is the cardinality of the set X , and E is the expected value. Write a program to empirically
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verify your results. b. Let a = IAJ and u = 256 - a = IUI. Let k E A. What is the probability that we can find some C E U and an index j such that l@ (zj V 1) = k . Note that if no such e and j can be found, the CMEA chosen plaintext attack described in this chapter cannot resolve the ambiguity in the low-order bit of T ( k ) . 15. For the SCMEA cipher, find the equations for c1 and c2 that correspond to the equation for cg in (4.18). 16. The results in Table 4.8 refer to the known plaintext attack on SCMEA. Empirically determine the analogous results for the known plaintext attack on CMEA. 17. Implement the CMEA known plaintext attack in a way that minimizes the amount of known plaintext required. Empirically determine the minimum number of known plaintext blocks required to correctly determine the key. Your results should be based on at least 1000 successful attacks. Hint: A successful attack may need to be repeated multiple times to determine the precise minimum number of known plaintext blocks required. 18. For the Akelarre cipher, let X O XI, X 2 ,X s be the input to round T and , Z,,Z1,22, Z, be the output of round T , let X << l denote a left rota< tion of X by l. Also, let AR(X, Y )be the addition--rotation structure. R,ecall that the inputs to the addition-rotation structure are two 32bit words and tht: output consists of two 32-bit words. Next, define (Uo, U1, U2, U s ) = ( X o , X I X 2 ,X s ) << lT, where l is the rotation in , < , round T , and define (To, I )= AR(U0 @ U2, U1 @ U s ) . T
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