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Finding the key of symmetric cipher takes computing common zero of polynomials, which define ideal and corresponding variety, usually considered over algebraically closed field. The solution is the point of the variety over prime field; it is defined by a sum of the polynomial ideal and the field ideal that defines prime field. Some authors use partitioning of this sum and reducing syzygies of polynomial ideal modulo field ideal. They generalize this method and consider polynomial ideal as a sum of two ideals, one of them is given by short polynomials, and add this ideal to the field ideal. Syzygies are reduced modulo this sum of ideals.
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