On Boolean Ideals and Varieties with Application to Algebraic Attacks

Finding the key of symmetric cipher takes computing common zero of polynomials, which de ne ideal and corresponding variety, usually considered over algebraically closed eld. The solution is the point of the variety over prime eld; it is de ned by a sum of the polynomial ideal and the eld ideal that de nes prime eld. Some authors use partitioning of this sum and reducing syzygies of polynomial ideal modulo eld ideal. We 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 eld ideal. Syzygies are reduced modulo this sum of ideals. Accuracy of de nition of the substitution ideal by short polynomials can be increased using a ne equivalence of ideals. This method decreases degree and length of syzygies and reduces complexity of Groebner basis computation.