Since the first appearance in Fridrich’s design, the usage of permutation-diffusion structure for designing digital image cryptosystem has been receiving increasing research attention in the field of chaos-based cryptography. Recently, a novel chaotic Image Cipher using one round Modified Permutation-Diffusion pattern (ICMPD) was proposed. Unlike traditional permutation-diffusion structure, the...

The quantum permutation group of the set Xn = {1, . . . , n} corresponds to the Hopf algebra Aaut(Xn). This is an algebra constructed with generators and relations, known to be isomorphic to C(Sn) for n ≤ 3, and to be infinite dimensional for n ≥ 4. In this paper we find an explicit representation of the algebra Aaut(Xn), related to Clifford algebras. For n = 4 the representation is faithful in...

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the ...

The random permutation is the Fräıssé limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all struct...

Bichain graphs form a bipartite analog of split permutation graphs, also known as split graphs of Dilworth number 2. Unlike graphs of Dilworth number 1 that enjoy many nice properties, split permutation graphs are substantially more complex. To better understand the global structure of split permutation graphs, in the present paper we study their bipartite analog. We show that bichain graphs ad...

We define a statistic on the graph of commutation classes permutation symmetric group which is used to show that these graphs are equipped with ranked poset structure, minimum and maximum. This characterization also allows us compute diameter for any permutation, from results longest fully commutative permutations recovered.

This talk describes a clean and unified way to enumerate all juxtaposition classes of the form “Av(abc) next to Av(xy)”, where abc is a permutation of length three and xy is a permutation of length two. The main tools are Dyck paths, decorated by sequences of points, and context free grammars, used afterwards to enumerate these decorated Dyck paths. Juxtapositions are a simple special case of p...

The existence of Almost Perfect Non-linear (APN) permutations operating on an even number of bits has been a long standing open question until Dillon et al., who work for the NSA, provided an example on 6 bits in 2009. In this paper, we apply methods intended to reverse-engineer S-Boxes with unknown structure to this permutation and find a simple decomposition relying on the cube function over ...

We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to f...