This paper addresses the computational problem of deciding invertibility (or one to one-ness) of a Boolean map $F$ in $n$-Boolean variables. This problem has a special case of deciding invertibilty of a map $F:\mathbb{F}_{2}^n\rightarrow\mathbb{F}_{2}^n$ over the binary field $\mathbb{F}_2$. Further the problem can be extended and stated over a finite field $\mathbb{F}$ instead of $\mathbb{F}_2$. Algebraic condition for invertibility of $F$ in this special case over a finite field is well known to be equivalent to invertibility of the Koopman operator of $F$ as shown in \cite{RamSule}. In this paper a condition for invertibility is derived in the special case of Boolean maps $F:B_0^n\rightarrow B_0^n$ where $B_0$ is the two element Boolean algebra in terms of \emph{implicants} of Boolean equations. This condition is then extended to the case of general maps in $n$ variables. Hence this condition answers the special case of invertibility of the map $F$ defined over the binary field $\mathbb{F}_2$ alternatively, in terms of implicants instead of the Koopman operator. The problem of deciding invertibility of a map $F$ (or that of finding its $GOE$) over finite fields appears to be distinct from the satisfiability problem (SAT) or the problem of deciding consistency of polynomial equations over finite fields. Hence the well known algorithms for deciding SAT or of solvability using Grobner basis for checking membership in an ideal generated by polynomials is not known to answer the question of invertibility of a map. Similarly it appears that algorithms for satisfiability or polynomial solvability are not useful for computation of $GOE(F)$ even for maps over the binary field $\mathbb{F}_2$.

翻译：本文研究了判定n元布尔映射F是否可逆（或单射）的计算问题。该问题的特例是判定定义在二元域F_2上的映射F：F_2^n→F_2^n的可逆性。进一步地，该问题可扩展至一般有限域F（替代F_2）。在有限域上的特例中，如文献\cite{RamSule}所示，F的可逆性代数条件等价于其Koopman算子的可逆性。本文针对布尔代数B_0（二值布尔代数）上的布尔映射F：B_0^n→B_0^n，基于布尔方程的蕴含项导出了可逆性条件。该条件随后被扩展至n变量的一般映射情形。因此，该条件以蕴含项替代Koopman算子，为定义在二元域F_2上的映射F的可逆性判定提供了另一种方案。判定有限域上映射F的可逆性（或计算其GOE）的问题，不同于可满足性问题（SAT）或判定有限域上多项式方程相容性的问题。因此，目前已知的SAT判定算法或基于Grobner基检验多项式生成理想成员性的可解性算法，均无法直接回答映射的可逆性问题。类似地，即使对于二元域F_2上的映射，可满足性或多项式可解性算法似乎也无助于计算GOE(F)。