The 32-bit swap after the sixteenth iteration ofthe des algorithm is needed to make the encryption processinvertible by simply running the ciphertext back through thealgorithm with the key order reversed. this was demonstrated inproblem 3.7. however, it still may not be entirely clear why the32-bit swap is needed. to demonstrate why, solve the followingexercises. first, some notation:
a||b
= the concatenation of the bit stringsa and b
ti(r||l)
= the transformation defined by the ithiteration of the encryption algorithm, for 1 (not=) i (not= ) 16
tdi(r||l)
= the transformation defined by the ithiteration of the decryption algorithm, for 1 (not = ) i (not =) 16
t17(r||l)
= l||r. thistransformation occurs after the sixteenth iteration of theencryption algorithm.

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