A novel very simple method for finding roots of polynomials over finite fields has been proposed. The essence of the proposed method is to search the roots via nested cycles over the subgroups of the multiplicative group of the Galois field. The modified Chien search is actually used in the inner cycles, but the internal polynomials are small. The word "modulus" was used because the search is doing on subsets like alpha^(a+bi), where a,b=const. In addition, modulo division of polynomials is actively used. The algorithm is applicable not for all Galois fields, but for selective ones, starting from GF(2^8). The algorithm has an advantage for large polynomials. The number of operations is significant for small polynomials, but it grows very slowly with the degree of the polynomial. When the polynomial is large or very large, the proposed method can be 10-100 times faster than Chien search.

翻译：提出了一种新颖且极为简单的有限域多项式根查找方法。该方法的核心在于通过伽罗瓦域乘法群子群上的嵌套循环进行根搜索。内层循环实际采用改进的Chien搜索，但内部多项式的规模较小。由于搜索在形如alpha^(a+bi)的子集上进行（其中a,b为常数），故使用"模数"一词。此外，算法还大量运用多项式的模除运算。该算法并非适用于所有伽罗瓦域，而是适用于从GF(2^8)开始的特定域。对于高次多项式具有显著优势，虽然小规模多项式的运算量较大，但其增长速率随多项式次数增加而极为缓慢。当多项式规模较大或极大时，本方法的运算速度可比传统Chien搜索快10-100倍。