Geodesic
Fast synthesis of invertible circuits based on permutation group theory
Prikladnaâ diskretnaâ matematika, no. 2 (2014), pp. 101-109

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Various algorithms for the synthesis of invertible logic circuits are considered and their main characteristics are presented. A new fast synthesis algorithm based on permutation group theory is proposed. This algorithm allows to synthesize schemes with the gate complexity $\mathrm O(n2^m)$ and with the time complexity $\mathrm O(n2^m)$ without using additional inputs. Here, $n$ is the number of scheme's inputs and $m$ is the upper bound for $\log k$, where $k$ is the number of non-fixed points of the given invertible transformation.
Keywords: invertible logic, synthesis algorithm, permutation groups. 
@article{PDM_2014_2_a8,
     author = {D. V. Zakablukov},
     title = {Fast synthesis of invertible circuits based on permutation group theory},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {101--109},
     publisher = {mathdoc},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2014_2_a8/}
}
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D. V. Zakablukov. Fast synthesis of invertible circuits based on permutation group theory. Prikladnaâ diskretnaâ matematika, no. 2 (2014), pp. 101-109. http://geodesic.mathdoc.fr/item/PDM_2014_2_a8/