Decomposition of Permutations in a Finite Field

We describe a method to decompose any power permutation, as a sequence of power permutations of lower algebraic degree. As a result we obtain decompositions of the inversion in $\mathrm{GF}(2^n)$ for small $n$ from $3$ up to $16$, as well as for the APN functions, when $n=5$. More precisely, we find decompositions into quadratic power permutations for any $n$ not multiple of $4$ and decompositions into cubic power permutations for $n$ multiple of $4$. Finally, we use the Theorem of Carlitz to prove that for $3 \leq n \leq 16$ any $n$-bit permutation can be decomposed in quadratic and cubic permutations

Redei Actions On Finite Fields And Multiplication Map In Cyclic Group

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We describe the functional graph of the multiplication-by-n map in a cycle group and use this to obtain the structure of the functional graph associated with a Redei function over a nonbinary finite field F-q. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Redei functions. We also extend characterizations of Redei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Redei permutations and we give an algorithm to construct Redei permutations with prescribed length cycles in a geometric progression.29314861503Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)NSERC of CanadaFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2012/10600-2, 2014/04096-5

Algebraic properties of generalized Rijndael-like ciphers

We provide conditions under which the set of Rijndael functions considered as permutations of the state space and based on operations of the finite field \GF (p^k) ($p\geq 2$ a prime number) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the AES (Rijndael block cipher with specific block sizes) in case AES became practically insecure. In Sparr and Wernsdorf (2008), R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field \GF (2^k) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field \GF (p^k) ($p\geq 2$) is equal to the symmetric group or the alternating group on the state space.Comment: 22 pages; Prelim0

Protected gates for topological quantum field theories

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.Comment: 50 pages, many figures, v3: updated to match published versio

M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update