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Permutation Group

I cannot seem to figure out how the operation table for this permutation is formed. Is it multiplying each index and doing modulus? I can't seem to figure out.

This is a Table 4.2 found in "Cryptography and Network Security" by Forouzan

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It is the permutation group operation, which is the result of first applying the first permutation (listed on the left), and then the second permutation (listed on top).

For example, the permutation listed as [3, 1, 2] is the permutation that maps the element 1 to 3, the element 2 to 1, and the element 3 to 2.

And, the permutation listed as [1, 3, 2] is the permutation that maps the element 1 to 1, the element 2 to 3 and the element 3 to 2.

So, if we combine these two permutations, applying [3, 1, 2] first, we can compute this by tracking what the permutations do to the specific elements. For example, it maps the element 1 to 3 (which is what the first permutation does), and then mapping the resulting 3 to 2 (which is what the second permutation does). Similarly, 2 is mapped to 1 (again, the first permutation), and then the second permutation maps the resulting 1 to 1. And, the element 3 is mapped to 2 by the first permutation, and that is then mapped to 3 by the second.

Hence, the resulting permutation is [2, 1, 3], which is what is listed in the [3, 1 2] row and the [1, 3, 2] column.

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