Action Of Quotient Research Articles

Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security

Substitution box (S-box) is a vital nonlinear component for the security of cryptographic schemes. In this paper, a new technique which involves coset diagrams for the action of a quotient of the modular group on the projective line over the finite field is proposed for construction of an S-box. It is constructed by selecting vertices of the coset diagram in a special manner. A useful transformation involving Fibonacci sequence is also used in selecting the vertices of the coset diagram. Finally, all the analyses to examine the security strength are performed. The outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

Renormalizations and Wandering Jordan Curves of Rational Maps

We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finite rational maps with wandering Jordan curves and prescribed renormalizations.

The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus

This elementary paper originated in the process of reading Katok and Spatzier's paper [3] concerning the first cohomology group of Z kand Rk-Anosov actions. It was proved in [3] that for the so-called standard Anosov actions any 1-cocycle (in C ~ and HSlder categories) with coefficients in R was cohomologous to a constant one. We will consider the .simplest class of actions, namely, the action of the discrete commutative group C SL (n, Z) by automorphisms of the n-toms T ~. We say that E is cohomologicalIy rigid if any ~ -cocyc l e : E ---* R is C~-cohomologous to a homomorphism ~ e Hom (E, R), i.e., Hi(Z, R) = H o m (~, R). It is proved in [3], in particular, that Z is cohomologically rigid provided that it is Anosov (i.e., contains an element a that has no eigenvalue of absolute value 1) and standard, i.e., contains a subgroup E' "-~ Z 2 that consists of ergodic elements (i.e., any nontrivial b E E' is ergodic on T~). Note that this result implies cohomological rigidity in the HSlder category [3], which is a special case of Katok and Schmidt's result [4] for mixing and expanding actions of Z k by automorphisms on a compact abetian group (see also [5]). On the other hand, higher cohomology groups for the Anosov ~ C SL (n, Z) were calculated in [2]. We will study more thoroughly the first cohomology group Hi(Z, R) of C~-cocycles with coefficients in R over the ergodic action of the commutative group Z C SL (n, Z) on T n by automorphisms. Our purpose here is to find a criterion for cohomological rigidity of ~. In fact, as Katok mentioned to the author, the result of [3] holds without the assumption that Z is Anosov. This requires involving the Lipschitz-type theorem of Veech [6] for ergodic automorphisms of a toms; we discuss this in Sec. 1. On the other hand, the cohomological rigidity easily implies that T" has no ~..-invariant proper subtorus T C T ~ such that the quotient action of ~ on T ~ / T is of rank I, i.e., contains a subgroup of finite index isomorphic to Z (see Sec. 1 for more details). Henceforth by a subtorus we mean a connected closed subgroup T C T~; thus T " / T ~_ T ~ .for some integer m and the quotient action of ~ lies in SL (m, Z). We will prove that the absence of rank-1 quotients is, in fact, sufficient for cohomological rigidity. The main result of the paper is as follows.

Aspects of (0,2) orbifolds and mirror symmetry

We study orbifolds of (0,2) models and their relation to (0,2) mirror symmetry. In the Landau-Ginzburg phase of a (0,2) model the superpotential features a whole bunch of discrete symmetries, which by quotient action lead to a variety of consistent (0,2) vacua. We study a few examples in very much detail. Furthermore, we comment on the application of (0,2) mirror symmetry to the calculation of Yukawa couplings in the space-time superpotential.

On orbifolds of (0, 2) models

We study orbifolds of (0, 2) models, including some cases with discrete torsion. Our emphasis is on models which have a Landau-Ginzburg realization, where we describe part of the massless spectrum by computing the elliptic genus for the orbifolded theory. Somewhat surprisingly, we find simple examples of (0, 2) mirror pairs that are related by a quotient action. We present a detailed description of a family of such pairs.