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1 LSV, ENS de Ca han, CNRS UMR 8643 & INRIA Futurs 2 LIF, Université-Marseille 1 & CNRS UMR 6166 Abstra t. We show several results about uni ation problems in the equational theory ACUNh onsisting of the theory of ex lusive or with one homomorphism. These results are shown using only te hniques of automata and ombinations of uni ation problems. We show how to onstru t a most-general uni er for ACUNh-uni ation problems with onstants using automata. We also prove that the rstorder theory of ground terms modulo ACUNh is de idable if the signature does not ontain free nononstant fun tion symbols, and that the existential fragment is de idable in the general ase. Furthermore, we show a te hni al result about the set of most-general uni ers obtained for general uni ation problems. 1 Introdu tion In this paper we are interested in uni ation, disuni ation, and more generally in de iding the rst-order theory of terms modulo the equational theory ACUNh. This theory onsists of the following equational axioms: (A) x⊕ (y ⊕ z) = (x⊕ y)⊕ z (C) x⊕ y = y ⊕ x (U) x⊕ 0 = x (N) x⊕ x = 0 (h) h(x⊕ y) = h(x)⊕ h(y) Our interest in these problems was raised by our re ent work on the symboli veri ation of ryptographi proto ols modulo the equational theory ACUNh [DLLT06℄. The result of that paper is a omplete onstraint solving algorithm for the parti ular kind of symboli onstraints that orrespond to the existen e of an atta k against the se urity of a ryptographi proto ol, taking into a ount some properties of the ryptographi primitives des ribed by the equational theory ACUNh. The onstraint solving algorithm pro eeds by several su essive simpli ation steps. The ompleteness of these steps relies on the notion of a nonollapsing solution: A solution σ to a onstraint system C is nonollapsing