Stiefel–Whitney invariants for bilinear forms

Cohomological invariants associated to symmetric bilinear forms

This paper is an outgrowth of the notes for a lecture given at the Mathematics Department of Queen’s University in December of 1988. I would like to take this opportunity to thank the members of that department for their hospitality during my visit. I would also like to thank Bruno Kahn for reading early versions of this manuscript and making several helpful remarks. The basic idea of the lectu...

Invariants of Binary Bilinear Forms Modulo Two Larry

Bilinear Forms

The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study algebraic and geometric notions related to bilinear forms (especially the concept of orthogonality in all its manifestations: orthogonal vectors, orthogonal subspaces, and orthogonal bases). Section 1 defines a bilinear form on a vector space a...

Practical Functional Encryption for Bilinear Forms

We present a practically efficient functional encryption scheme for the class of functionalities that can be expressed via bilinear forms over the integers. Bilinear forms are a general class of quadratic functions that includes, for instance, multivariate quadratic polynomials. Our realization works over asymmetric bilinear groups and is surprisingly simple, efficient and easy to implement. Fo...

Variational principles for symmetric bilinear forms

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant’s minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general f...