数学最新文献

We study the hardness of the Syndrome Decoding problem, the base of most code-based cryptographic schemes, such as Classic McEliece, in the presence of side-channel information. We use ChipWhisperer equipment to perform a template attack on Classic McEliece running on an ARM Cortex-M4, and accurately classify the Hamming weights of consecutive 32-bit blocks of the secret error vector (textbf{e}in {{mathbb {F}}}_2^n). With these weights at hand, we optimize Information Set Decoding algorithms. Technically, we demonstrate how to speed up information set decoding via a dimension reduction, additional parity-check equations, and an improved information set search, all derived from the Hamming-weight information. Consequently, using our template attack, we can practically recover an error vector (textbf{e}in {{mathbb {F}}}_2^n) in dimension (n=2197) in a matter of seconds. Without side-channel information, such an instance has a complexity of around 88 bit. We also estimate how our template attack affects the security of the proposed McEliece parameter sets. Roughly speaking, even an error-prone leak of our Hamming weight information leads for (n=3488) to a security drop of 89 bits.

Given integers (n> k > 0), and a set of integers (L subset [0, k-1]), an L-system is a family of sets (mathcal {F}subset left( {begin{array}{c}[n] kend{array}}right) ) such that (|F cap F'| in L) for distinct (F, F'in mathcal {F}). L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent to finding the independence number of the graph G(n, k, L). The Lovász number (vartheta (G)) is a semidefinite programming approximation of the independence number (alpha ) of a graph G. In this paper, we determine the leading order term of (vartheta (G(n, k, L))) of any generalized Johnson graph with k and L fixed and (nrightarrow infty ). As an application of this theorem, we give an explicit construction of a graph G on n vertices with a large gap between the Lovász number and the Shannon capacity c(G). Specifically, we prove that for any (epsilon > 0), for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio (vartheta (G)/c(G) = Omega (n^{1-epsilon })), which improves on all known constructions. The graph G a fortiori also has ratio (vartheta (G)/alpha (G) = Omega (n^{1-epsilon })), which greatly improves on the best known explicit construction.

Wolbachia, an intracellular bacterium, is well-known for inducing cytoplasmic incompatibility, which has become a promising and environmentally sustainable strategy for controlling pest populations. The strain wStri, specifically identified in Nilaparvata lugens (brown planthopper), has shown potential for such biocontrol applications. In this study, we develop a comprehensive discrete mathematical model to analyze the dynamics of wStri spread in a mixed population of wStri-infected, wLug-infected, and uninfected Nilaparvata lugens under both constant and periodically varying environmental conditions. Under a constant environment, the model identifies the critical threshold necessary for the successful establishment of wStri within the population. Our analysis reveals that the model exhibits a strong Allee effect, where a population must exceed a certain critical density-the Allee threshold-for the wStri strain to persist and spread. Below this threshold, the wStri strain is likely to be eliminated, failing in pest control efforts. When the environment varies periodically, the model transforms into a non-autonomous periodic discrete model, introducing additional complexity. In this scenario, we derive sufficient conditions that ensure the composition of finitely many Allee maps continues to function as an Allee map. Furthermore, we prove that a unique periodic orbit exists within such a periodic environment. This orbit is characterized as unstable and acts as a threshold, determining whether wStri will establish itself in the population or die out over time. The findings from this model provide critical insights into the conditions under which wStri can be effectively used to control Nilaparvata lugens, particularly in environments that are not constant but fluctuate periodically. These insights have significant implications for the practical deployment of Wolbachia-based biocontrol methods in pest management strategies.