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We describe$T$-equivariant Schubert calculus on$G$($k$,$n$),$T$being an$n$-dimensional torus, through derivations on the exterior algebra of a free$A$-module of rank$n$, where$A$is the$T$-equivariant cohomology of a point. In particular,$T$-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian,$Pure Appl. Math. Quart.$$2$(2006), 699–717).

Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.

The Hidden Field Equations (HFE) Cryptosystem as proposed by Patarin is one of the best known and most studied multivariate schemes. While the security of the basic scheme appeared to be very weak, the HFEv- variant seems to be a good candidate for digital signature schemes on the basis of multivariate polynomials. However, the currently existing scheme of this type, the QUARTZ signature scheme, is hardly used in practice because of its poor efficiency. In this paper we analyze recent results from Ding and Yang about the degree of regularity of HFEv- systems and derive from them design principles for signature schemes of the HFEv- type. Based on these results we propose the new HFEv- based signature scheme Gui, which is more than 100 times faster than QUARTZ and therefore highly comparable with classical signature schemes such as RSA and ECDSA.

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