Bijective proofs for Eulerian numbers in types B and D

Let $\left\langle{n\atop k}\right\rangle$, $\left\langle{B_n\atop k}\right\rangle$, and $\left\langle{D_n\atop k}\right\rangle$ be the Eulerian numbers in the types $A$, $B$, and $D$, respectively -- that is, the number of permutations of $n$ elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type $B$ descents, the number of even signed permutations (of $n$ elements) with $k$ type $D$ descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \left\langle{n\atop k}\right\rangle t^k$, $B_n(t) = \sum_{k = 0}^{n}\left\langle{B_n\atop k}\right\rangle t^k$, and $D_n(t) = \sum_{k = 0}^{n}\left\langle{D_n\atop k}\right\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2ntS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t)\ .$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.Comment: ALgebras, Graphs and Ordered Sets - August 26th to 28th 2020, Aug 2020, Nancy, Franc

General Diffusion Analysis: How to Find Optimal Permutations for Generalized Type-II Feistel Schemes

Type-II Generalized Feistel Schemes are one of the most popular versions of Generalized Feistel Schemes. Their round function consists in applying a classical Feistel transformation to p sub-blocks of two consecutive words and then shifting the k = 2p words cyclically. The low implementation costs it offers are balanced by a low diffusion, limiting its efficiency. Diffusion of such structures may however be improved by replacing the cyclic shift with a different permutation without any additional implementation cost. In this paper, we study ways to determine permutations with the fastest diffusion called optimal permutations. To do so, two ideas are used. First, we study the natural equivalence classes of permutations that preserve cryptographic properties; second, we use the representation of permutations as coloured trees. For both heuristic and historical reasons, we focus first on even-odd permutations, that is, those permutations for which images of even numbers are odd. We derive from their structure an upper bound on the number of their equivalence classes together with a strategy to perform exhaustive searches on classes. We performed those exhaustive searches for sizes k ≤ 24, while previous exhaustive searches on all permutations were limited to k ≤ 16. For sizes beyond the reach of this method, we use tree representations to find permutations with good intermediate diffusion properties. This heuristic leads to an optimal even-odd permutation for k = 26 and best-known results for sizes k = 64 and k = 128. Finally, we transpose these methods to all permutations. Using a new strategy to exhaust equivalence classes, we perform exhaustive searches on classes for sizes k ≤ 20 whose results confirmed the initial heuristic: there always exist optimal permutations that are even-odd and furthermore for k = 18 all optimal permutations are even-odd permutations

Permutation-invariant distance between atomic configurations

We present a permutation-invariant distance between atomic configurations, defined through a functional representation of atomic positions. This distance enables to directly compare different atomic environments with an arbitrary number of particles, without going through a space of reduced dimensionality (i.e. fingerprints) as an intermediate step. Moreover, this distance is naturally invariant through permutations of atoms, avoiding the time consuming associated minimization required by other common criteria (like the Root Mean Square Distance). Finally, the invariance through global rotations is accounted for by a minimization procedure in the space of rotations solved by Monte Carlo simulated annealing. A formal framework is also introduced, showing that the distance we propose verifies the property of a metric on the space of atomic configurations. Two examples of applications are proposed. The first one consists in evaluating faithfulness of some fingerprints (or descriptors), i.e. their capacity to represent the structural information of a configuration. The second application concerns structural analysis, where our distance proves to be efficient in discriminating different local structures and even classifying their degree of similarity

Triplicate functions

We define the class of triplicate functions as a generalization of 3-to-1 functions over $\mathbb {F}_{2^{n}}$ for even values of n. We investigate the properties and behavior of triplicate functions, and of 3-to-1 among triplicate functions, with particular attention to the conditions under which such functions can be APN. We compute the exact number of distinct differential sets of power APN functions and quadratic 3-to-1 functions; we show that, in this sense, quadratic 3-to-1 functions are a generalization of quadratic power APN functions for even dimensions, in the same way that quadratic APN permutations are generalizations of quadratic power APN functions for odd dimensions. We show that quadratic 3-to-1 APN functions cannot be CCZ-equivalent to permutations in the case of doubly-even dimensions. We compute a lower bound on the Hamming distance between any two quadratic 3-to-1 APN functions, and give an upper bound on the number of such functions over $\mathbb {F}_{2^{n}}$ for any even n. We survey all known infinite families of APN functions with respect to the presence of 3-to-1 functions among them, and conclude that for even n almost all of the known infinite families contain functions that are quadratic 3-to-1 or are EA-equivalent to quadratic 3-to-1 functions. We also give a simpler univariate representation in the case of singly-even dimensions of the family recently introduced by Göloglu than the ones currently available in the literature. We conduct a computational search for quadratic 3-to-1 functions in even dimensions n ≤ 12. We find six new APN instances for n = 10, and the first sporadic APN instance for n = 12 since 2006. We provide a list of all known 3-to-1 APN functions for n ≤ 12.publishedVersio