Jakob Jonsson

Generalized triangulations and diagonal-free subsets of stack polyominoes

For n≥3, let Ωn be the set of line segments between vertices in a convex n-gon. For j≥1, a j-crossing is a set of j distinet and mutually intersecting line segments from Ωn such that all 2j endpoints are distinct. For k≥1, let Δn,k be the simplicial complex of subsets of Ωn not containing any (k + 1)-crossing. For example, Δn,k has one maximal set for each triangulation of the n-gon, Dress, Koolen, and Moulton were able to prove that all maximal sets in Δn,k have the same number k(2n - 2k - 1) of line segments. We demonstrate that the number of such maximal sets in counted by a k × k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We gerneralize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.

Cryptanalysis of the NTRU Signature Scheme (NSS) from Eurocrypt 2001

In 1996, a new cryptosystem called NTRU was introduced, related to the hardness of finding short vectors in specific lattices. At Eurocrypt 2001, the NTRU Signature Scheme (NSS), a signature scheme apparently related to the same hard problem, was proposed. In this paper, we show that the problem on which NSS relies is much easier than anticipated, and we describe an attack that allows efficient forgery of a signature on any message. Additionally, we demonstrate that a transcript of signatures leaks information about the secret key: using a correlation attack, it is possible to recover the key from a few tens of thousands of signatures. The attacks apply to the recently proposed parameter sets NSS251-3-SHA1-1, NSS347-3-SHA1-1, and NSS503-3-SHA1-1 in [2]. Following the attacks, NTRU researchers have investigated enhanced encoding/verification methods in [11].

Funkspiel schemes: an alternative to conventional tamper resistance

We investigate a simple method of fraud management for secure devices that may serve as an alternative or complement to conventional hardware-based tamper resistance. Under normal operating conditions in our scheme, a secure device includes an authentication code in its communications, e.g., in the digital signatures it issues. This code may be verified by a fraud management center under a pre-determined key σ. When the device detects an attempted break-in, it modifies σ. This results in a change to the authentication codes issued by the device such that the fraud management center can detect the apparent break-in. Hence, in contrast to the case with typical tamper-resistance schemes, the deployer of our proposed scheme seeks to trace break-ins, rather than prevent them. In reference to the wartime practice of physically capturing and subverting underground radio transmitters – a practice analogous to the capture and use of secret information on secure devices – we denote this idea by the German term funkspiel, meaning “radio game.” One challenge in constructing a funkspiel scheme is to ensure that an attacker privy to the authentication codes of the secure device both before and after the break-in, as well as the secrets of the device following the break-in, cannot detect the alteration to σ. Additional challenges ∗Some of this work was done while visiting RSA Laboratories.

On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs

Using techniques from Robin Formans discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Δn3 of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of (n - 3). (n - 2)!/2 spheres of dimension 2n - 4, thereby verifying a conjecture by Babson, Bjorner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex Γn of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Δn2 of not 2- connected graphs on n vertices. The homotopy type of Δn2 has been determined, independently, by the five authors listed above and by Turchin. While Γn and Δn2 are homotopy equivalent for small values on n, they are nonequivalent for n = 10.

Certain Homology Cycles of the Independence Complex of Grids

Let G be an infinite graph such that the automorphism group of G contains a subgroup K≅ℤd with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r⋅|G/I|−1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩ℚ={r∈ℚ:a≤r≤b}. For example, for the square grid, we obtain the interval

Five-torsion in the homology of the matching complex on 14 vertices

[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}

More Torsion in the Homology of the Matching Complex

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Securing RSA-KEM via the AES

AbstractJ.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex

3-torsion in the Homology of Complexes of Graphs of Bounded Degree

\mathsf {M}_{14}

Simplicial Complexes of Graphs and Hypergraphs with a Bounded Covering Number

on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case n=14 is exceptional; for all other n, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of