Probabilistic Evaluation of the Exploration–Exploitation Balance during the Search, Using the Swap Operator, for Nonlinear Bijective S-Boxes, Resistant to Power Attacks

During the search for S-boxes resistant to Power Attacks, the S-box space has recently been divided into Hamming Weight classes, according to its theoretical resistance to these attacks using the metric variance of the confusion coefficient. This partition allows for reducing the size of the search space. The swap operator is frequently used when searching with a random selection of items to be exchanged. In this work, the theoretical probability of changing Hamming Weight class of the S-box is calculated when the swap operator is applied randomly in a permutation. The precision of these probabilities is confirmed experimentally. Its limit and a recursive formula are theoretically proved. It is shown that this operator changes classes with high probability, which favors the exploration of the Hamming Weight class of S-boxes space but dramatically reduces the exploitation within classes. These results are generalized, showing that the probability of moving within the same class is substantially reduced by applying two swaps. Based on these results, it is proposed to modify/improve the use of the swap operator, replacing its random application with the appropriate selection of the elements to be exchanged, which allows taking control of the balance between exploration and exploitation. The calculated probabilities show that the random application of the swap operator is inappropriate during the search for nonlinear S-boxes resistant to Power Attacks since the exploration may be inappropriate when the class is resistant to Differential Power Attack. It would be more convenient to search for nonlinear S-boxes within the class. This result provides new knowledge about the influence of this operator in the balance exploration–exploitation. It constitutes a valuable tool to improve the design of future algorithms for searching S-boxes with good cryptography properties. In a probabilistic way, our main theoretical result characterizes the influence of the swap operator in the exploration–exploitation balance during the search for S-boxes resistant to Power Attacks in the Hamming Weight class space. The main practical contribution consists of proposing modifications to the swap operator to control this balance better.

Coarse fundamental groups and box spaces

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

From the geometry of box spaces to the geometry and measured couplings of groups

In this paper, we prove that if two “box spaces” of two residually finite groups are coarsely equivalent, then the two groups are “uniform measured equivalent” (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a “uniform measured equivalent embedding” (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp.ã coarse embedding) between the box spaces gives rise to a coarse equivalence (resp.ã coarse embedding) between the groups. We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from [Formula: see text] cannot be coarsely embedded inside the expanders of [Formula: see text], where [Formula: see text] and [Formula: see text]. Moreover, we obtain a countable class of residually finite groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.

Dustproof cooling of the electrical box

In present are electrical boxes cooled by air through the intake hole on the bottom electrical box to the box space with electrotechnical elements and exhaust through the hole at the top to the surrounding by natural convection. This cooling method is effective but operate with the risk of contamination electrotechnical elements by dust sucking from surrounding air. The goal of this work is solution of the dustproof cooling of the electrical box by natural convection. The work deal with design of the device with the heat transfer by the phase change of the working fluid and experimental measuring its thermal performance at the cooling electrotechnical elements loaded by heat 1 200 W in the dustproof electrical box.

Full box spaces of free groups

In this paper we investigate full box spaces and coarse equivalences between them. We do this in two parts. In part one we compare the full box spaces of free groups on different numbers of generators. In particular, the full box space of a free group{F_{k}}is not coarsely equivalent to the full box space of a free group{F_{d}}, if{d\geq 8k+10}. In part two we compare{\Box_{f}\mathbb{Z}^{n}}to the full box spaces of 2-generated groups. In particular, we prove that the full box space of{\mathbb{Z}^{n}}is not coarsely equivalent to the full box space of any 2-generated group, if{n\geq 3}.

An elementary proof of a non-triviality of the E8subfactor planar algebra

In this paper, we show in a combinatorial way that the 0-box space of the E8subfactor planar algebra is 1-dimensional. In the proof, we improve on Bigelow's relations for the E8subfactor planar algebra and give an efficient algorithm to reduce any planar diagram to the empty diagram.