The public key of the Oil-Vinegar scheme consists of a set of m quadratic equations in m + n variables over a finite field F_q. Kipnis and Shamir broke the balanced Oil-Vinegar scheme where d = n − m = 0 by finding equivalent keys of the cryptosytem. Later their method was extended by Kipnis et al to attack the unbalanced case where 0 < d < m and d is small with a complexity of O(q^{d−1} m^4). This method uses the matrices associated with the quadratic polynomials in the public key, which needs to be symmetric and invertible. In this paper, we give an optimized search method for Kipnis el al’s attack. Moreover, for the case that the finite field is of characteristic 2, we find the situation becomes very subtle, which, however, was totally neglected in the original work of Kipnis et al. We show that the Kipnis-Shamir method does not work if the field characteristic is 2 and d is a small odd number, and we fix the situation by proposing an alternative method and give an equivalent key recovery attack of complexity O(q^{d+1} m^4). We also prove an important experimental observation by Ding et al for the Kipnis-Shamir attack on balanced Oil-Vinegar schemes in characteristic 2.

A natural metric in 2-manifold surfaces is to use geodesic distance. If a 2-manifold surface is represented by a triangle mesh T , the geodesic metric on T can be computed exactly using computational geometry methods. Previous work for establishing the geodesic metric on T only supports using half-edge data structures; i.e., each edge e in T is split into two halves (he1, he2) and each half-edge corresponds to one of two faces incident to e. In this paper, we prove that the exact-geodesic structures on two halfedges of e can be merged into one structure associated with e. Four merits are achieved based on the properties which are studied in this paper: (1) Existing CAD systems that use edge-based data structures can directly add the geodesic distance function without changing the kernel to a half-edge data structure; (2) To find the geodesic path from inquiry points to the source, the MMP algorithm can be run in an onthe- fly fashion such that the inquiry points are covered by correct wedges; (3) The MMP algorithm is sped up by pruning unnecessary wedges during the wedge propagation process; (4) The storage of the MMP algorithm is reduced since fewer wedges need to be stored in an edge-based data structure. Experimental results show that when compared to the classic half-edge data structure, the edge-based implementation of the MMP algorithm reduces running time by 44% and storage by 29% on average.

Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii).

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