Browsing by Author "Kusmus, Omer"

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A Characterization of the Unit Group in Z[T X C2]
(Korean Mathematical Soc, 2016) Bilgin, Tevfik; Kusmus, Omer; Low, Richard M.
Describing the group of units U(ZG) of the integral group ring SG, for a finite group G, is a classical and open problem. In this note, we show that U-1(Z[T x C-2]) similar or equal to [F-97 x F-5] x [T x C-2], where T = < a, b : a(6) = 1, a(3) = b(2), ba = a(5)b > and F-97, F-97 are free groups of ranks 97 and 5, respectively.
Değişmeli Grup Halkalarında G-nilpotent Birimsel Elemanların Direkt Çarpım Gruplarına Bir Genellemesi
(2023) Hanoymak, Turgut; Kusmus, Omer
V(RG), bir R halkası üzerindeki bir G grubunun RG grup halkasının normalleştirilmiş birim grubunu göstersin. Değişmeli bir grup halkasındaki G-nilpotent birimsel kavramı (Danchev, 2012)'de tanımlanmıştır. Bu çalışmada da, bir G×H direkt çarpım grubunun değişmeli grup halkasında normallenmiş birimsel elemanlar grubunun sadece G×H-nilpotent birimsel elemanlardan oluşabilmesi için bazı gerek ve yeter şartlar verilmiştir. Ayrıca özel olarak G×C_3 ve G×C_4 gruplarına dair bazı sonuçlar sunulmuştur ki burada C_3 ve C_4 sırasıyla 3 ve 4 mertebeli devirli gruplardır. Bu bağlamda, makale (Danchev, 2012)’deki sonuçları genişletir diyebiliriz. Sonunda, gelecek çalışma için açık problem sunulmuştur.
A Glance at Blockchain Technology and Cryptocurrencies as an Application
(2022) Hanoymak, Turgut; Kusmus, Omer
Blockchain technology, which includes cryptocurrencies such as Bitcoin, Ethereum,…etc [1,2] which has been evaluated as an investment tool by many people all over the world in recent years, needs to be examined in details, both mathematically and conceptually [8,9]. In fact, it can be said that blockchain technology, which is characterized as an accounting system and database based on distributed ledgers in its most basic form, is extremely secure in terms of copying data or attacking. For this reason, we can say that technology has a more effective security mechanism than any central state-of-the-art authoritative system used today. However, as it is almost impossible to bring all of the security, speed and cost parameters to their full extend in a system at the same time, as in any cryptosystem, the security parameter from the distributed ledger structure in blockchain technology adversely affects the speed and cost parameters. In this article, we discuss the cryptographic working principles of cryptocurrencies, which is an application field of blockchain technology, together with blockchain technology and the features and structures of the blocks contained.
İki-devirli Bir Grubun İntegral Grup Halkasındaki Burulmalı Birimsel Elemanlar Üzerine
(2020) Kusmus, Omer
𝐺 bir grup olsun. ℤ𝐺 integral grup halkasındaki herhangi iki birimsel elemanın, ℚ𝐺 grup cebrindeki birimseller bakımından eşlenik olması durumunda rasyonel eşlenik olarak ifade edildiklerini anımsayalım. Zassenhaus, bir konjektür olarak sunmuştur ki ℤ𝐺’deki herhangi bir sonlu mertebeli birimsel eleman, 𝐺 grubunun bir elemanı ile rasyonel eşleniktir. Bu, Zassenhaus’un ilk konjektürü olarak bilinir [4]. Biz bu konjektürü makale boyunca ZC1 ile göstereceğiz. ZC1, çözülebilir ve meta-devirli grupların bazı sınıfları için çözülmüştür. Bunun yanı sıra, biri görebilir ki metabelyen gruplarda bazı aksine örnekler vardır. Bu makalede temel amaç, 𝑇3 = ⟨𝑎, 𝑏: 𝑎 6 = 1, 𝑎 3 = 𝑏 2 , 𝑏𝑎𝑏 −1 = 𝑎 −1⟩ iki-devirli grubunun ℤ𝑇3 integral grup halkasındaki burulmalı birimsel elemanların yapısını, ikinci dereceden bir kompleks indirgenemez güvenilir temsil kullanarak karakterize etmektir. Birinci Zassenhaus konjektürü (ZC1) ile göstereceğiz ki ℤ𝑇3 integral grup halkasındaki aşikar olmayan burulmalı birimsel elemanlar 3, 4 veya 6 mertebeli ve bunların her biri üç serbest parametre cinsinden ifade edilebilir.
A New Multi-Party Key Exchange Protocol and Symmetric Key Encryption Scheme Over Non-Commutative Group Rings
(2019) Hanoymak, Turgut; Kusmus, Omer
The importance of secure communication over an insecure channel has increased day by day in almost all applicationssuch as commercial purposes, money transactions, military and sanitary services. Therefore, many encryption algorithms basedon various types of algebraic structures have become more considerable because of the underlying mathematically hard problemssuch as integer factorization, discrete logarithm, conjugacy search problem in group theory, finding the inverse of a given unit ingroup rings. Key exchange protocols also have monumental significance for generating shared keys between parties by exchangingcryptographic keys to allow a secure communication.In this paper, we first briefly mention about the basics of group rings, the fundamental properties of units, Diffie-Hellman type keyexchange protocol then we generalize this to a multi-party type key exchange protocol using units in a given group ring and finallywe propose a symmetric key encryption scheme over a non-commutative group ring which is different from the encryption schemein [1] by illustrating a concrete example. We also give a security analysis of the proposed protocol and comparisons with [1] and[8].
A Novel Public-Key Encryption Scheme Based on Bass Cyclic Units in Integral Group Rings
(Taylor & Francis Ltd, 2022) Kusmus, Omer; Hanoymak, Turgut
In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non- commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.
On the Units of Integral Group Ring of Cn X C6
(Luhansk Taras Shevchenko Natl Univ, 2015) Kusmus, Omer
There are many kind of open problems with varying difficulty on units in a given integral group ring. In this note, we characterize the unit group of the integral group ring of C-n x C-6 where C-n = < a : a(n) = 1 > and C-6 = < x : x(6) = 1 >. We show that u(1)(Z[C-n x C-6]) can be expressed in terms of its 4 subgroups. Furthermore, forms of units in these subgroups are described by the unit group u(1)(ZC(n)). Notations mostly follow [11].
On Torsion Units in Integral Group Ring of a Type of Cyclic-By Groups
(Badih Ghusayni, 2025) Kusmus, Omer
In this paper, using a complex representation of degree 2 of the group U12 which is given in [5] and belongs to the class of cyclic-byabelian groups [4], we investigate some parametric characteristics of normalized torsion units in integral group ring of U12 relying on the fact that all cyclic-by-abelian groups satisfy the first of Zassenhaus conjectures [4].
Trivial Units in Commutative Group Rings of G X Cn
(Luhansk Taras Shevchenko Natl Univ, 2024) Kusmus, Omer
It is known that if the unit group of an integral group ring ZG is trivial, then the unit group of Z(G x C-2) is trivial as well [3]. The aim of this study is twofold: firstly, to identify rings R that are D-adapted for the direct product D = G x H of abelian groups G and H, such that the unit group of the ring R(G x H) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring R and the direct factors of D to satisfy the property that the normalized unit group V (RD) is trivial in the case where D is one of the groups G x C-3, G x K-4 or G x C-4, where G is an arbitrary finite abelian group, C-n denotes a cyclic group of order n and K-4 is Klein 4-group. Hence, the study extends the related result in [18].
Unit Group of Integral Group Ring Z(G X C 3 )
(Univ Miskolc inst Math, 2024) Kusmus, Omer
Presenting an explicit descryption of unit group in the integral group ring of a given non-abelian group is a classical and open problem. Let S3 be a symmetric group of order 6 and C3 be a cyclic group of order 3. In this study, we firstly explore the commensurability in unit group of integral group ring Z(S3 xC3) by showing the existence of a subgroup as (F55 & rtimes; F3) & rtimes; (S & lowast;3 xC2) where F rho denotes a free group of rank rho. Later, we introduce an explicit structure of the unit group in Z(S3 x C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 where R = Z[w] is the complex integral domain since w is the primitive 3rd root of unity. At the end, we give a general method that determines the structure of the unit group of Z(GxC3) for an arbitrary group G depends on torsion-free normal complement V (G) of Gin U(Z(G xC3)) in an implicit form. As a consequence, a conjecture which is found in [21] is solved.
Z(S_3×c_3) İntegral Grup Halkasındaki Birimsel Elemanlar Grubu Üzerine
(2024) Denizler, Ismail Hakki; Low, Richard; Kusmus, Omer
Verilen bir sonlu grubun integral grup halkasındaki birimsel elemanların grubunu belirlemek çoğu grup için meşhur ve klasik bir açık problemdir. S_3 ve C_3 sırasıyla 6 mertebeli simetrik grup ve 3 mertebeli bir devirli grup olsun. Bu makalede, iki dereceli bir kompleks temsile göre S_3×C_3 direkt çarpım grubunun Z(S_3×C_3) integral grup halkasının birimsel elemanlarının yapısı verilmektedir. Sonuç olarak, kompleks bir tamlık bölgesi üzerinde tanımlı grup halkalarına ilişkin (Low, 2008)’ de sunulan konjektürün bir kısmı, temsil teorisi kullanılarak çözülmüştür.