Browsing by Author "Hanoymak, Turgut"

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Another Approach to Factoring by Continued Fractions
(Association of Mathematicians (MATDER), 2025) Hanoymak, Turgut; Kayak, Cihan
The problem of prime factorization is particularly important in fields such as cryptography, where it plays a crucial role, especially in the security of public key cryptosystems like RSA. There are numerous factorization algorithms that have been developed over time, each with varying levels of complexity. These algorithms have played a crucial role in fields like mathematics and cryptography, where prime factorization remains a key challenge. In this study, the continued fraction method one of the factorization methods, is examined. To highlight the importance of the continued fraction factorization method, a brief mention is made of RSA's vulnerability to attacks, such as Weiner's attack, which exploits small private keys. Our approach aims to enhance the efficiency of factorization by integrating this method with relevant theorems by giving concrete examples with detailed tables.
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.
Fundamental Structure of Shor’s Quantum Algorithm for Factoring Integers
(2019) Chehrazi, Akram; Hanoymak, Turgut
One of the most well known mathematically hard problems in number theory is the integer factorizationproblem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography,RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespreadand up to date no classical algorithm having polynomial running time for the factorization of large numbers isknown. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we mention aboutthe fundamentals of Shor’s quantum algorithm illustrating a concrete example.
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.
Ind-Cca Secure Encryption Based on a Zheng-Seberry Scheme
(Elsevier Science Bv, 2014) Ak, Murat; Hanoymak, Turgut; Selcuk, Ali Aydin
In 1993, Zheng and Seberry proposed three methods for strengthening public key cryptosystems. These methods aimed to obtain schemes that are secure against adaptively chosen ciphertext attacks. One method was improving security by using digital signatures. Zheng and Seberry gave an example scheme that employs this method. However, they were not able to prove 1ND-CCA security of their cryptosystem. In this paper, we modify this cryptosystem by employing a Schnorr signature scheme and prove this new scheme to be IND-CCA secure in the random oracle model. (C) 2013 Elsevier B.V. All rights reserved.
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 Prime Factorization Algorithms and Their Comparisons
(2025) Kayak, Cihan; Hanoymak, Turgut
Bilgisayarların evler ve işletmelerde yaygınlaşması ve internetin hızla büyümesiyle birlikte, güvenli elektronik iletişim önemli bir konu haline gelmiştir. Elektronik bilgiyi korumak için en çok kullanılan açık anahtarlı sistemlerden biri olan RSA, (Rivest,1978) büyük bir tam sayının asal çarpanlarına ayrılmasının hesaplama açısından zor olduğu gerçeğine dayanır. Eğer uygun (polinom zamanlı bir algoritma bulunabilirse) bir süre içinde herhangi büyük bir tam sayıyı asal çarpanlarına ayırabilen etkili bir algoritma geliştirilirse, RSA kripto sistemi kırılır. Bu tezde Fermat çarpanlara ayırma algoritması, Euler çarpanlara ayırma algoritması, Quadratik Sieve çarpanlara ayırma algoritması, Pollard rho çarpanlara ayırma algoritması, Pollard çarpanlara ayırma algoritması ve sürekli kesirler gibi verilen bir sayının asal olmadığını kabul edilerek asal çarpanlarına ayırma metotları detaylı bir şekilde incelenecek ilgili teoremler ispatlarıyla verilecek bu metotlar birbirileriyle kıyaslanarak örneklendirilecektir.
On Prime Number Test Algorithms and Applications in Cryptology
(2020) Ağçakaya, Erol; Hanoymak, Turgut
Bu tez çalışması beş ana bölümden oluşmaktadır. Birinci bölümde asal sayı testleri hakkında günümüze kadar olan çalışmalar ile ilgili bilgi verilmiş ve asal sayı testlerinin önemi vurgulanmıştır. İkinci bölümde asal sayı test algoritmaları hakkında kaynak taraması yapılmıştır. Üçüncü bölümde çalışmamız boyunca kullanabileceğimiz bilgiler ve asal sayılarla ilgili temel özellikler ile kuadratik rezidülerle ilgili temel tanım ve teoremlere yer verilmiştir. Bu bilgiler özellikle Slovay-Strassen testinin uygulanması için gereklidir. Dördüncü bölümde Fermat, Euler, Miller-Rabin ve Slovay-Strassen olasılıksal (probabilistic) ve AKS kesin (deterministic) asallık testlerine geniş yer verilmiş ve bazı açık anahtarlı şifreleme algoritmalarından kısaca bahsedilip somut örnekler verilmiştir. Son bölümde ise tezin değerlendirildiği tartışma ve sonuç kısmına yer verilmiştir.