The problem of factoring Boolean polynomials has significant applications in both classical and quantum computing technology. In this paper we have developed novel algorithms for factoring both ESOP and SOP expressions. Our aim is to optimize the AND-count. The AND-count plays a key role in determining the number of AND and Toffoli gates required to implement a reversible function with classical and quantum circuits, respectively. The first type of algorithms that we develop, are graphical. We reduce the problem of Boolean factoring to covering a bipartite graph with bicliques, and so optimizing the number of bicliques required to cover the bipartite graph, leads to reduced number of factors, and hence AND-count. The second type of algorithm is algebraic, and is derived from multivariate Horner method. We have compared the performances of our algorithms with existing popular methods like EXORCISM-4 and EPOEM2, on random functions of up to 12 variables. We have observed that our multivariate Horner method is substantially faster, while our biclique-based method achieves the maximum AND-count reduction. In fact, compared to EXORCISM-4 our biclique based method achieves up to 5 times reduction in AND-count.

翻译：布尔多项式因子分解问题在经典计算和量子计算技术中均有重要应用。本文提出了针对ESOP（异或-或积）表达式与SOP（或积）表达式因子分解的新型算法，旨在优化AND计数。AND计数是决定经典电路实现可逆函数所需AND门数量、以及量子电路实现可逆函数所需Toffoli门数量的关键指标。我们开发的第一类算法基于图论方法：将布尔因子分解问题转化为双分图的双团覆盖问题，通过优化覆盖双分图所需双团的数量来减少因子个数，进而降低AND计数。第二类算法基于代数方法，源自多元霍纳法。我们将所提算法与EXORCISM-4、EPOEM2等现有主流方法在最多12变量的随机函数上进行了性能对比。实验表明：我们的多元霍纳法在速度上具有显著优势，而基于双团的方法则能实现最大程度的AND计数缩减——相较于EXORCISM-4，该方法的AND计数最多可降低5倍。