Let d be the doubling map for M = M(2t). (1) The photo booth transformation for 2t×2t pixel pictures is the permutation f of M(2t)2 with f(x,y) = (d(x),d(y)).

(2) The doubling map d for M(2^t)
is a permutation of order t.

An immediate consequence of (1) and (2) is:

Corollary. The photo booth transformation for 2t×2t pixel pictures is a permutation of order t.

Thus: The photo booth transformation for 256×256 pixel pictures
has order 8, since 256 = 2⁸.