# What is the difference between a permutation and a shuffle (transposition cipher)

A non-cryptographic definition of a permutation is "2a: the act or process of changing the lineal order of an ordered set of objects. 2b: an ordered arrangement of a set of objects

The Wikipedia article on Random permutation states that "A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards."

An ideal block cipher is a pseudorandom permutation.

Shuffling a deck of identical cards would result in output indistinguishable from the input. Applying an ideal block cipher to an all-zero (or all-one) plaintext block would yield a random ciphertext block, not the same all-zero (or all-one) input block!

What's an easy-to-understand difference between a shuffle (equivalently a transposition cipher) and a permutation in the sense meant by cryptographers?

• Looking to see if anyone has better/clearer/more intuitive answers. It's easy enough to state the difference using mathematical notation, but getting it across clearly without that seems a bit harder. Commented Nov 6, 2020 at 2:57
• I do not understand the distinction you are trying to make. Could you try to clarify it some? This is likely complicated because a transposition has a well-defined mathematical meaning which appears to be distinct from what you mean (in particular, transpositions are permutations which interchange exactly two elements, so the subset of transpositions is a subset of the set of all permutations. The subgroup generated by these elements is the full permutation group though). Commented Nov 6, 2020 at 6:01
• Swapped it to "transposition cipher" since that's the more precise term I was going for. Commented Nov 6, 2020 at 15:45