On the Affine Equivalence and Nonlinearity Preserving Bijective Mappings

A Toolbox for Cryptanalysis: Linear and Affine Equivalence Algorithms

This paper presents two algorithms for solving the linear and the affine equivalence problem for arbitrary permutations (S-boxes). For a pair of n×n-bit permutations the complexity of the linear equivalence algorithm (LE) is O(n2). The affine equivalence algorithm (AE) has complexity O(n2). The algorithms are efficient and allow to study linear and affine equivalences for bijective S-boxes of a...

Convergence of an Implicit Iteration for Affine Mappings in Normed and Banach Spaces

Linear Maps Preserving Invertibility or Spectral Radius on Some $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra which has a faithful state. If $varphi:Arightarrow A$ is a unital linear map which is bijective and invertibility preserving or surjective and spectral radius preserving, then $varphi$ is a Jordan isomorphism. Also, we discuss other types of linear preserver maps on $A$.

Adjacency Preserving Mappings of Rectangular Matrices

Let D be a division ring and let m,n be integers ≥ 2. Let Mm×n(D) be the space of m × n matrices. In the fundamental theorem of the geometry of rectangular matrices all bijective mappings φ of Mm×n(D) are determined such that both φ and φ−1 preserve adjacency. We show that if a bijective map φ of Mm×n(D) preserves the adjacency then also φ −1 preserves the adjacency. Thus the supposition that φ...

Some Observations on Dirac Measure-Preserving Transformations and their Results

Dirac measure is an important measure in many related branches to mathematics. The current paper characterizes measure-preserving transformations between two Dirac measure spaces or a Dirac measure space and a probability measure space. Also, it studies isomorphic Dirac measure spaces, equivalence Dirac measure algebras, and conjugate of Dirac measure spaces. The equivalence classes of a Dirac ...