In this paper, we propose a unification algorithm for the theory $E$ which combines unification algorithms for $E_{std}$ and $E_{ACUN}$ (ACUN properties, like XOR) but compared to the more general combination methods uses specific properties of the equational theories for further optimizations. Our optimizations drastically reduce the number of non-deterministic choices, in particular those for variable identification and linear orderings. This is important for reducing both the runtime of the unification algorithm and the number of unifiers in the complete set of unifiers. We emphasize that obtaining a ``small set of unifiers is essential for the efficiency of the constraint solving procedure within which the unification algorithm is used. The method is implemented in the CL-Atse tool for security protocol analysis.
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