Abstract
Let f=(f₀,f₁,…, f_ν-1) be a collection of one-to-one functions from some space~X into itself such that the sets f_j(X) are disjoint. If w=w₁w₂… w_k is a word on the alphabet {0,1,…,ν-1}, let Φ_f,w = f_w₁∘ f_w₂∘…∘ f_w_k. Given a function~F of which we know that it can be written as Φ_f,w, it is easy to recover~w. We give some examples of this situation where everything can be scrambled up by using some private key to get a new system g=(g₁,g₂,…,g_ν-1) on another set~Y in such a way that the images of the g_j are no longer disjoint. We define a cryptosystem whose public key is~g. The message to be encrypted is a word~w and the associated cryptogram is Φ_g,w. The private key allows to recover Φ_f,w from Φ_g,w.