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We study the last fall degrees of {\em semi-local} polynomial systems, andthe computational complexity of solving such systems for closed-point andrational-point solutions, where the systems are defined over a finite field. Asemi-local polynomial system specifies an algebraic set which is the image of aglobal linear transformation of a direct product of local affine algebraicsets. As a special but interesting case, polynomial systems that arise fromWeil restriction of algebraic sets in an affine space of low dimension aresemi-local. Such systems have received considerable attention due to theirapplication in cryptography. Our main results bound the last fall degree of asemi-local polynomial system in terms of the number of closed point solutions,and yield an efficient algorithm for finding all rational-point solutions whenthe prime characteristic of the finite field and the number of rationalsolutions are small. Our results on solving semi-local systems imply animprovement on a previously known polynomial-time attack on the HFE (HiddenField Equations) cryptosystems. The attacks implied in our results extend topublic key encryption functions which are based on semi-local systems whereeither the number of closed point solutions is small, or the characteristic ofthe field is small. It remains plausible to construct public key cryptosystemsbased on semi-local systems over a finite field of large prime characteristicwith exponential number of closed point solutions. Such a method is presentedin the paper, followed by further cryptanalysis involving the isomorphism ofpolynomials (IP) problem, as well as a concrete public key encryption schemewhich is secure against all the attacks discussed in this paper.

Last fall degree of semi-local polynomial systems