On groups containing the additive group of the residue ring or the vector space

Groups which are most frequently used as key addition groups in iterative block ciphers include the regular permutation representation $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ of the group of vector key addition, the regular permutation representation $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ of the additive group of the residue ring, and the regular permutation representation $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n} + 1}^ \odot \end{array}$ of the multiplicative group of a prime field (in the case where 2n + 1 is a prime number). In this work we consider the extension of the group Gn generated by $\begin{array}{} \displaystyle V_{n}^{+} \end{array}$ and $\begin{array}{} \displaystyle \mathbb{Z}_{{2^n}}^{+} \end{array}$ by means of transformations and groups which naturally arise in cryptographic applications. Examples of such transformations and groups are the groups $\begin{array}{} \displaystyle \mathbb{Z}_{{2^d}}^{+} \times V_{n - d}^{+} ~\text{and}~ V_{n - d}^{+}\times \mathbb{Z}_{{2^d}}^{+} \end{array}$ and pseudoinversion over the field GF(2n) or over the Galois ring GR(2md, 2m).

MINIMAL EXCEPTIONAL -GROUPS

For a finite group $G$, denote by $\unicode[STIX]{x1D707}(G)$ the degree of a minimal permutation representation of $G$. We call $G$ exceptional if there is a normal subgroup $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$. To complete the work of Easdown and Praeger [‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc.38(2) (1988), 207–220], for all primes $p\geq 3$, we describe an exceptional group of order $p^{5}$ and prove that no exceptional group of order $p^{4}$ exists.

Some Self-Orthogonal Codes Related to Higman's Geometry

We examine some self-orthogonal codes constructed from a rank-5 primitive permutation representation of degree 1100 of the sporadic simple group ${\rm HS}$ of Higman-Sims. We show that ${\rm Aut}(C) = {\rm HS}{:}2$, where $C$ is a code of dimension 21 associated with Higman's geometry.

A GRAPHICAL TECHNIQUE TO FIND THE PERMUTATION REPRESENTATION OF THE TRIANGLE GROUPS △(2, 3, k)

In this paper we have developed a graphical technique by which a suitably created fragment of a coset diagram for the action of PSL(2, ℤ) on projective lines over Galois fields can be used to obtain a complete coset diagram depicting finite homomorphic images of groups Δ(2, 3, k) = 〈x, y : x2 = y3 = (xy)k = 1〉.

Stamp Foldings, Semi-Meanders, and Open Meanders: Fast Generation Algorithms

By considering a permutation representation for stamp-foldings and semi-meanders we construct tree-like data structures that will allow us to generate these objects in constant amortized time. Additionally, by maintaining the wind-factor and applying an additional optimization, the algorithm for semi-meanders can be modified to produce the fastest known algorithm to generate open meanders.