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$n$个布尔变量的布尔映射$F$的可逆性（或单射性）的计算问题。该问题的一个特例是判定二元域$\mathbb{F}_2$上映射$F:\mathbb{F}_{2}^n\rightarrow\mathbb{F}_{2}^n$的可逆性。进一步，该问题可推广至任意有限域$\mathbb{F}$（而非仅限于$\mathbb{F}_2$）。如文献\cite{RamSule}所示，有限域上该特例中$F$可逆性的代数条件等价于其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$F:B_0^n\rightarrow B_0^n$，从布尔方程\emph{蕴含项}的角度推导出其可逆性条件。该条件随后被推广至$n$变量一般映射的情形。因此，该条件从蕴含项视角（而非Koopman算子）给出了二元域$\mathbb{F}_2$上映射$F$可逆性判定问题的另一种解答。有限域上判定映射$F$可逆性（或计算其$GOE$）的问题似乎不同于可满足性问题（SAT）或有限域上多项式方程相容性判定问题。因此，现有的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$\mathbb{F}_2$上的映射，可满足性算法或多项式可解性算法似乎也无法用于计算$GOE(F)$。