Modular group acting on real quadratic fields

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 nite eld 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. e outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

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“…e fixed points of x and y, if they exist, are denoted by heavy dots. For more details about coset diagrams, we suggest reading [17][18][19].…”

Section: Preliminariesmentioning

confidence: 99%

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

Shahzad

Mushtaq

Razaq

2019

Security and Communication Networks

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“…In (6) , it was proved that the ambiguous numbers in the orbit γ G , γ ∈ Q * ( √ n) makes a G-circuit or simply circuit. Thus it becomes interesting to classify circuit.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of Icosahedral group and classification of G-circuits of length six

Bari¹

2020

IJST

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Objectives: To determine the exact number of equivalence classes of G-circuits of length q ≥ 2.Methods/Statistical Analysis: To classify G-orbits of Q(√ m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t. Applications/Improvements: We employ Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example.

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“…In fact, çarks can be thought of as Z-quotients of periodic rivers of Conway [5] or graphs dual to the coset diagrams of Mushtaq, [18]. As we shall see later in the paper, çarks provide a very nice reformulation of various concepts pertaining to indefinite binary quadratic forms, such as reduced forms and the reduction algorithm, ambiguous forms, reciprocal forms, the Markoff value of a form, etc.…”

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confidence: 99%

“…These graphs parametrize conjugacy classes of dihedral subgroups of the modular group. Çarks also provide a more conceptual way to understand the relation between coset diagrams and quadratic irrationalities and their properties as studied in [18] or in [16]. For us the importance of this correspondence between çarks and forms lies in that it suggests a concrete and clear way to consider modular graphs as arithmetic objects viz.…”

mentioning

confidence: 99%