Proof Theory of Skew Non-Commutative MILL

Lower Bounds for Non-Commutative Skew Circuits

Nisan (STOC 1991) exhibited a polynomial which is computable by linear-size non-commutative circuits but requires exponential-size non-commutative algebraic branching programs. Nisan’s hard polynomial is in fact computable by linear-size “skew circuits.” Skew circuits are circuits where every multiplication gate has the property that all but one of its children is an input variable or a scalar....

Non-commutative Bloch Theory

For differential operators which are invariant under the action of an abelian group Bloch theory is the preferred tool to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C-modules. It relates properties of C-algebras to spectral properties of module operators such as ...

Cryptanalysis of Cryptosystems Based on Non-commutative Skew Polynomials

We describe an attack on the family of Diffie-Hellman and El-Gamal like cryptosystems recently presented at PQ Crypto 2010. We show that the reference hard problem is not hard. 1 Description of the Cryptosystems Skew polynomials are polynomials with a particular noncommutative inner product. Let Fq denote the finite field with q elements, and p be the characteristic of the field. Automorphisms ...

Interactions between non-commutative algebraic geometry and skew PBW extensions

String theory and non-commutative gauge theory

I describe some recent developments concerning the role of non-commutative Yang– Mills theory in string theory.