Planar polynomial curves have rational offset curves, if they are either Pythagorean-hodograph (PH) or indirect Pythagorean-hodograph (iPH) curves. In this paper, we derive an algebraic and two geometric characterizations for planar quartic iPH curves. The characterizations are given in terms of quantities related to the Bézier control polygon of the curve, and naturally extend to quartic and cubic PH and quadratic iPH curves.

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in \cite{GHKR2}. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.

We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al [10], which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.

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We introduceM-tensors. This concept extends the concept ofM-matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M-tensors must be Ztensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M-tensor must be nonnegative. A symmetric M-tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an Mtensor is its smallest H+-eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M-tensor if and only if all its H+-eigenvalues are nonnegative. Some further spectral properties of M-tensors are given. We also introduce strong M-tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M-tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.