Some trace class commutators of trace zero

Cryptography on Trace-Zero Varieties

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of som...

An Attack on a Trace-Zero Cryptosystem

It was recently proposed in the literature that the discretelogarithm problem (DLP) in trace-zero groups of genus 2 curves withrespect to constant field extensions of degree 3 is a fast and secure al-ternative to the well established cryptographic primitive of the DLP in elliptic curves over prime fields. We present a novel attack on this primitive. We show that the DLP inthe tr...

Pairing with Supersingular Trace Zero Varieties Revisited

A Trace Zero Variety is a specific subgroup of the group of the divisor classes on a hyperelliptic curve C/Fq, which are rational over a small degree extension Fqr of the definition field. Trace Zero Varieties (TZV) are interesting for cryptographic applications since they enjoy properties that can be exploited to achieve fast arithmetic and group construction. Furthermore, supersingular TZV al...

On Some Matrix Trace Inequalities

A is further called positive definite, symbolized A > 0, if the strict inequality in 1.1 holds for all nonzero x ∈ C. An equivalent condition forA ∈ Mn to be positive definite is thatA is Hermitian and all eigenvalues of A are positive real numbers. Given a positive semidefinite matrix A and p > 0, A denotes the unique positive semidefinite pth power of A. Let A and B be two Hermitian matrices ...