permutation representation
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2019 ◽  
Vol 44 (2) ◽  
pp. 159-180
Author(s):  
Muhammad Sarwar Saeed ◽  
Muhammad Ashiq ◽  
Tariq Alraqad ◽  
Tahir Imran

2018 ◽  
Vol 28 (4) ◽  
pp. 231-247
Author(s):  
Boris A. Pogorelov ◽  
Marina A. Pudovkina

Abstract 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).


2018 ◽  
Vol 98 (3) ◽  
pp. 434-438
Author(s):  
ROBERT CHAMBERLAIN

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.


10.37236/5541 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Jamshid Moori ◽  
B. D. Rodrigues

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.


2012 ◽  
Vol 05 (04) ◽  
pp. 1250055
Author(s):  
Qaiser Mushtaq ◽  
Sadef Baig

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〉.


10.37236/2404 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Joe Sawada ◽  
Roy Li

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.


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