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Is there a name for the subset of involution functions used in most reciprocal ciphers?

I've been saying things like

A pairing is a function f that pairs up each element x with some other element y, such that

  • f(x) = y and x != y and f(y) = x, for each and every x.

Or in other words,

  • f(x) != x for each and every x, and
  • f(f(x)) == x for each and every x

(In particular, the identity function is not a pairing).

My understanding is that the Enigma machine and nearly all other reciprocal ciphers always use such a function to transform each plaintext letter to the corresponding ciphertext letter. (I suspect because of the misunderstanding that "when we encrypt a plaintext letter, obviously the encrypted ciphertext should not be the same letter").

I've been calling such a function a pairing, but I've recently discovered that term is usually used to mean something quite different.

What term should I use instead of "pairing"?

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