For each integer $m\ge 2$, every Boolean function f can be expressed as a unique multilinear polynomial modulo m, and the degree of this multilinear polynomial is called its modulo m degree. In this paper we investigate the modulo degree complexity of total Boolean functions initiated by Parikshit Gopalan et al. [9], in which they asked the following question: whether the degree complexity of a Boolean function is polynomially related with its modulo m degree. For m be a power of primes, it is already known that the module m degree can be arbitrarily smaller compare to the degree complexity (see Section 2 for details). When m has at least two distinct prime factors, the question remains open. Towards this question, our results include: (1) we obtain some nontrivial equivalent forms of this question; (2) we affirm this question for some special classes of functions; (3) we prove a no-go theorem, explaining why this problem is difficult to attack from the computational complexity point of view; (4) we show a super-linear separation between the degree complexity and the modulo m degree.

对于每个整数 $米\ge 2$，每个布尔函数f都可以表示为一个唯一的多元多项式模m，这个多元多项式的次数称为其模次数。在本文中，我们研究了由Parikshit Gopalan等人发起的总布尔函数的模数复杂度。[9]，其中他们提出以下问题：是否布尔函数的程度复杂多项式以其模相关米程度。对于m是素数的幂，已经知道，与度数复杂度相比，m度数的模块可以任意小（有关详细信息，请参见第2节）。当m有至少两个不同的主要因素，这个问题仍然悬而未决。针对这个问题，我们的结果包括：（1）我们获得了该问题的一些平凡形式。（2）对于某些特殊的功能类别，我们肯定这个问题；（3）我们证明了一个不通过定理，从计算复杂性的角度解释了为什么这个问题难以解决；（4）我们展示了度复杂度和模m度之间的超线性分离。