A New Secure Encryption Scheme Based on Group Factorization Problem

Ye, Cong; Hong, Haibo; Shao, Jun; Han, Song; Lin, Jianhong; Zhao, Shuai

doi:10.1109/access.2019.2954672

Cited by 11 publications

(9 citation statements)

References 43 publications

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“…У 2009 р. Lempken та інші описали криптосистему з відкритим ключем MST3, засновану на логарифмічному підписі та 2-групі Сузукі [7]. Криптосистеми на неабелевих групах та їх властивості активно вивчались в [8,9]. У 2010 р. Сваба та інші [10] проаналізували всі відомі атаки на криптографію MST і створили більш безпечну криптосистему eMST3, включивши секретне гомоморфне накриття.…”

Section: вступunclassified

“…, ,..., ( , ) 17 , α 16 α 25 , α 17 , α 23 , α 27 α 13 , α 0 , α 28 , α 10 α 13 , α 0 , α 28 , α 10 α 30 ,α 2 ,α 17 ,α 2 α 6 , α 7 , α 30 , α 18 α 9 ,α 4 ,α 9 ,α 20 α 9 ,α 4 ,α 9 ,α 20 α 14 , α 28 , α 17 , α 22 α 26 , α 5 , α 16 , α 30 α 12 , α 15 , α 17 , α 6 α 12 , α 15 , α 17 , α 6 α 22 , α 30 , α 22 , α α 24 , α 29 , α 15 , α α 3 ,0,α 14 ,α 9 Зворотні елементи ( ) ( )…”

Section: практичні обчисленняunclassified

“…Таблиця 4 Обчислення обернених елементів ( ) ( ) 22 ,α 21 α 25 , α 7 , α 3 , α 15 α 13 , α 19 , α 7 , α 24 α 13 ,α 0 ,α 7 ,α 24 α 30 , α 2 , α 15 , α 21 α 6 , α 7 , α 28 , α 24 α 9 , α 4 , α 8 , α 25 α 9 ,α 4 ,α 9 ,α 25 α 14 , α 28 , α 17 , α 21 α 26 , α 5 , α 16 , α 13 α 12 , α 15 , α 17 , α 30 α 12 , α 15 , α 17 , α α 22 , α 30 , α 22 , α 16 α 24 , α 29 , α 15 , α 30 α 3 ,0,α 14 ,α 9 Аналогічно вибираємо випадкові 0( ) 1( ) ( ) , ,..., ( , )…”

Section: практичні обчисленняunclassified

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Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

Kotukh,

Khalimov,

Korobchinskyi

2023

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This article presents a new implementation of encryption based on MST, focused on generalized Suzuki 2-groups. The well-known MST cryptosystem, based on Suzuki groups, is constructed using a logarithmic signature at the center of the group, leading to a large array of logarithmic signatures. The proposed encryption is based on multi-parameter noncommutative groups, with a focus on the generalized multi-parameter Suzuki 2-group. This approach responds to the progress in the development of quantum computers, which may pose a threat to the security of many open cryptosystems, especially those based on factorization problems and discrete logarithms, such as RSA or ECC. The use of noncommutative groups to create quantum-resistant cryptosystems has been a known approach for the last two decades. The unsolvable word problem, proposed by Wagner and Magyarik, is used in the field of permutation groups and is key to the development of cryptosystems. Logarithmic signatures, introduced by Magliveras, represent a unique type of factorization suitable for finite groups. The latest version of such an implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras introduced the LS transitivity limit for the MST3 cryptosystem, and later Swaba proposed an improved version of the cryptosystem, eMST3. In 2018, T. van Trung suggested applying the MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted a deep analysis of MST3 and noted that due to the absence of publications on the quantum vulnerability of this algorithm, it can be considered a potential candidate for use in the post-quantum era. The main distinction of the new system is the use of homomorphic encryption to construct logarithmic signature coverings for all group parameters, which improves the secrecy of the cryptosystem, particularly against brute-force attacks.

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Section: вступunclassified

Section: практичні обчисленняunclassified

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Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

Kotukh,

Khalimov,

Korobchinskyi

2023

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“…Factoring a finite abelian group into subsets was first initiated by Hajós [8] to solve a famous geometry problem by H. Minkowski in 1941. In the past few decades, study of group factorization has received numerous research attentions (see [9,12]) and they found its applications in various field such as geometry of tiling, code theory, cryptography, graph theory and etc (see [5,6]). The current research has renew focused on factoring nonabelian groups into subsets (see [1,11]).…”

Section: Introductionmentioning

confidence: 99%

The Exhaustion Numbers of the Generalized Quaternion Groups

Chen¹,

Sin²

2023

MJMS

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Let G be a finite group and let T be a non-empty subset of G. For any positive integer k, let Tk={t1…tk∣t1,…,tk∈T}. The set T is called exhaustive if Tn=G for some positive integer n where the smallest positive integer n, if it exists, such that Tn=G is called the exhaustion number of T and is denoted by e(T). If Tk≠G for any positive integer k, then T is a non-exhaustive subset and we write e(T)=∞. In this paper, we investigate the exhaustion numbers of subsets of the generalized quaternion group Q2n=⟨x, y∣x2n−1=1, x2n−2=y2, yx=x2n−1−1y⟩ where n≥3. We show that Q2n has no exhaustive subsets of size 2 and that the smallest positive integer k such that any subset T⊆Q2n of size greater than or equal to k is exhaustive is 2n−1+1. We also show that for any integer k∈{3,…,2n}, there exists an exhaustive subset T of Q2n such that |T|=k .

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“…Більшість атак може буде імплементовано з використанням доступних обчислювачів. Водночас саме розуміння архітектури атаки стимулює спроби використання інших підходів до забезпечення потрібного рівня безпеки, на кшталт використання гомоморфного шифрування у якості додаткового елемента посилення конструкції[14], використання посилених логарифмічних підписів[15]. В останні роки, використовуючи обґрунтовані властивості неабелевих груп[16], результати робіт з пошуку кращих за характеристиками логарифмічних підписів[17], дослідникам вдалось запропонувати посилені конструкції Hard problems for non-abelian cryptography // 2021: Fifth International Scientific and Technical Conference "COMPUTER AND INFORMATION SYSTEMS AND TECHNOLOGIES", 2021, pp39-40, https://doi.org/10.30837/csitic52021232176 2.…”

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Cryptanalysis of the system based on word problems using logarithmic signatures

Kotukh¹,

Okhrimenko²,

Dyachenko³

et al. 2021

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Rapid development and advances of quantum computers are contributing to the development of public key cryptosystems based on mathematically complex or difficult problems, as the threat of using quantum algorithms to hack modern traditional cryptosystems is becoming much more real every day. It should be noted that the classical mathematically complex problems of factorization of integers and discrete logarithms are no longer considered complex for quantum calculations. Dozens of cryptosystems were considered and proposed on various complex problems of group theory in the 2000s. One of such complex problems is the problem of the word. One of the first implementations of the cryptosystem based on the word problem was proposed by Magliveras using logarithmic signatures for finite permutation groups and further proposed by Lempken et al. for asymmetric cryptography with random covers. The innovation of this idea is to extend the difficult problem of the word to a large number of groups. The article summarizes the known results of cryptanalysis of the basic structures of the cryptosystem and defines recommendations for ways to improve the cryptographic properties of structures and the use of non-commutative groups as basic structures.

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A New Secure Encryption Scheme Based on Group Factorization Problem

Cited by 11 publications

References 43 publications

Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

The Exhaustion Numbers of the Generalized Quaternion Groups

Cryptanalysis of the system based on word problems using logarithmic signatures

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