On the last fall degree of zero-dimensional Weil descent systems. (July 2018) - Huang, Ming-Deh A.; Kosters, Michiel; Yang, Yun; Yeo, Sze Ling
Abstract: In this article we will discuss a mostly theoretical framework for solving zero-dimensional polynomial systems. Complexity bounds are obtained for solving such systems using a new parameter, called the last fall degree, which does not depend on the choice of a monomial order. The method is similar to certain MutantXL algorithms, but our abstract formulation has advantages. For example, we can prove that the cryptographic systems multi-HFE and HFE are insecure. More generally, let k be a finite field of cardinality q n and let k ′ be the subfield of cardinality q . Let F ⊂ k [ X 0, …, X m − 1 ] be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of F from k to k ′, which depends on q, m, the last fall degree of F, the degree of F and the number of solutions of F, but not on n . This shows that such Weil descent systems can be solved efficiently if n grows and the other parameters are fixed. In particular, one can apply these results to show a weakness in the cryptographic protocols HFE and multi-HFE.