RESUMEN
Maximum distance separable (MDS) matrices are often used in the linear layer of a block cipher due to their good diffusion property. A well-designed lightweight MDS matrix, especially an involutory one, can provide both security and performance benefits to the cipher. Finding the corresponding effective linear straight-line program (SLP) of the circuit of a linear layer is still a challenging problem. In this article, first, we propose a new heuristic algorithm called Superior Boyar-Peralta (SBP) in the computation of the minimum number of two-input Exclusive-OR (XOR) gates with the minimum circuit depth for the SLPs. Contrary to the existing global optimization methods supporting only two-input XOR gates, SBP heuristic algorithm provides the best global optimization solutions, especially for extracting low-latency circuits. Moreover, we give a new 4 × 4 involutory MDS matrix over F24, which requires only 41 XOR gates and depth 3 after applying SBP heuristic, whereas the previously best-known cost is 45 XOR gates with the same depth. In the second part of the article, for further optimization of the circuit area of linear layers with multiple-input XOR gates, we enhance the recently proposed BDKCI heuristic algorithm by incorporating circuit depth awareness, which limits the depth of the circuits created. By using the proposed circuit depth-bounded version of BDKCI, we present better circuit implementations of linear layers of block ciphers than those given in the literature. For instance, the given circuit for the AES MixColumn matrix only requires 44 XOR gates/depth 3/240.95 GE in the STM 130 nm (simply called ASIC4) library, while the previous best-known result is 55 XOR gates/depth 5/243.00 GE. Much better, our new 4 × 4 involutory MDS matrix requires only 19 XOR gates/depth3/79.75 GE in the STM 90 nm (simply called ASIC1) library, which is the lightest and superior to the state-of-the-art results.
RESUMEN
This article presents a new hybrid method (combining search based methods and direct construction methods) to generate all 4×4 involutory maximum distance separable (MDS) matrices over F2m. The proposed method reduces the search space complexity at the level of n, where n represents the number of all 4×4 invertible matrices over F2m to be searched for. Hence, this enables us to generate all 4×4 involutory MDS matrices over F23 and F24. After applying global optimization technique that supports higher Exclusive-OR (XOR) gates (e.g., XOR3, XOR4) to the generated matrices, to the best of our knowledge, we generate the lightest involutory/non-involutory MDS matrices known over F23, F24 and F28 in terms of XOR count. In this context, we present new 4×4 involutory MDS matrices over F23, F24 and F28, which can be implemented by 13 XOR operations with depth 5, 25 XOR operations with depth 5 and 42 XOR operations with depth 4, respectively. Finally, we denote a new property of Hadamard matrix, i.e., (involutory and MDS) Hadamard matrix form is, in fact, a representative matrix form that can be used to generate a small subset of all 2k×2k involutory MDS matrices, where k > 1. For k = 1, Hadamard matrix form can be used to generate all involutory MDS matrices.
RESUMEN
BACKGROUND: Despite using a variety of path-finding algorithms that use tracts, the most significant advancement in this study is considering the values of all brain areas by doing atlas-based segmentation for a more precise search. Our motivation comes from the literature's shortcomings in designing and implementing path-planning methods. Since planning paths with curvatures is a complex problem that requires considering many surgical and physiological constraints, most path-planning strategies focus on straight paths. There is also a lack of studies that focus on the complete structure of the brain with the tracks, veins, and segmented areas. Instrument dependence is another inadequacy of the methods proposed in the literature. AIMS: The aim of this study is to design a new surgical path planning framework that helps to plan the surgical path independently of the instrument, considers the entire structure of the brain, and allows curvilinear surgical paths. Thus, neurosurgeons can generate patient-specific possible optimal surgical pathways before the neurosurgical procedure. MATERIALS & METHODS: The proposed framework includes different path-finding algorithms (Dijkstra, A*, and their aggressive variants) that find optimal paths by taking the risk scores (surgeons assessed all the segmented regions, considering the extent of damage. In this evaluation, scores ranged from "0 to 10," with the most critical areas receiving a score of "10," while the least possible affected areas were assigned a score of "0") for sensitive brain areas into consideration. For the tract image processing the framework includes fractional anisotropy (FA), relative anisotropy (RA), spherical measure (SM), and linear measure (LM) methods. RESULTS: This is the first paper to handle tracts and atlas-based segmentation of the human brain altogether under a framework for surgical path planning. The framework has a dynamic structure that gives the flexibility to add different path-finding algorithms and generate different widths of surgical pathways. Moreover, surgeons can update the score table to guarantee minimally invasive surgery. The output file format of all the extracted surgical paths is NRRD, so it can be easily visualised, analysed, or processed over the third part software tools. DISCUSSION: In this study, we generated many possible surgical pathways then these pathways were evaluated by the surgeons the results were impressive because the framework could identify surgical pathways used in real-world surgery that correspond to the standard pathways such as anterior transsylvian, trans sulcal, transgyral, and sub-temporal. CONCLUSION: This study proposes a new surgical path planning framework for neurosurgery. Moreover, in the future by adding/adopting different parameters (such as operation time, and short and long-term complications after surgery) to the proposed framework, it would be possible to find new surgical pathways for difficult surgical conditions.
