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（即单射性）的计算问题。该问题的一个特例是判定二元域$\mathbb{F}_2$上的映射$F:\mathbb{F}_{2}^n\rightarrow\mathbb{F}_{2}^n$的可逆性。进一步地，该问题可扩展至有限域$\mathbb{F}$（而非$\mathbb{F}_2$）上表述。已知在该特例下，有限域上的代数条件等价于$F$的Koopman算子的可逆性，如文献\cite{RamSule}所示。本文针对二元布尔代数$B_0$上的布尔映射$F:B_0^n\rightarrow B_0^n$这一特例，基于布尔方程的\emph{蕴含项}导出了可逆性条件。随后将该条件推广至$n$元一般映射的情形。因此，该条件以蕴含项替代Koopman算子，给出了定义在二元域$\mathbb{F}_2$上映射$F$可逆性问题的另一种解答。有限域上判定映射$F$的可逆性（或计算其$GOE$）问题与可满足性问题（SAT）或有限域上多项式方程相容性判定问题存在本质区别。因此，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可逆性问题。类似地，即使对二元域$\mathbb{F}_2$上的映射，可满足性或多项式可解性算法似乎也不适用于计算$GOE(F)$。