Suzuki 2-groups Research Articles

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

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

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.

CONSTRUCTION OF AN IMPROVED ENCRYPTION SCHEME ON GENERALIZED SUZUKI 2-GROUPS IN THE MST3 CRYPTOSYSTEM

This paper proposes a method for constructing an improved encryption scheme on generalized Suzuki 2-groups for the MST3 cryptosystem, which improves the security parameters of the original approach. The challenge of improving existing cryptosystem design approaches is driven by advances in building quantum computers with sufficient computing power to render many public-key cryptosystems insecure. In particular, this includes cryptosystems based on the factorization problem or the discrete logarithm problem, such as RSA and ECC. There have been several proposals in the past two decades for using non-commutative groups to create quantum-resistant cryptosystems. The unsolvable word problem is a promising area of research for building cryptosystems. It was formulated by Wagner and Magyarik and lies in the realm of permutation groups. Magliveras proposed logarithmic signatures, which are a special type of factorization that applies to finite groups. The latest version of this implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras demonstrated a transitive LS limit for the MST3 cryptosystem. Later, Svaba proposed the eMST3 cryptosystem with improved security parameters, achieved by adding a secret homomorphic cover. In 2018, T. van Trung proposed an MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted an extensive analysis of MST3 and noted that, since there are currently no publications on the quantum vulnerability of the algorithm, it can be considered a candidate for use in the post-quantum era. The first implementation of the cryptosystem on the generalized Suzuki 2-group does not provide encryption of the entire Suzuki 2-group and does not protect against attacks with sequential key recovery by the brute-force method. Further work has developed the idea of public cryptography using non-Abelian refinements of parameters. This paper proposes a method for constructing an encryption scheme on Suzuki 2-groups that improves the security parameters of the existing MST3 cryptosystem and solves its security problems.

DEVELOPMENTOFMETHODSFORTESTINGTHELIBRARYOFCRYPTOGRAPHICTRANSFORMATIONSONTHEEXAMPLEOFTHEMST3CRYPTOSYSTEMBASEDONGENERALIZEDSUZUKI2-GROUPS

The article proposes a methodology for testing a library of cryptographic transformations with the implementation of an improved encryption scheme on generalized Suzuki 2-groups in the MST3 cryptosystem. The need to improve existing methods of cryptosystem creation is driven by progress in quantum computer development, which possess sufficient computational power to compromise many existing public key cryptosystems. This is especially true for systems based on factorization and discrete logarithm, such as RSA and ECC. Over the last nearly 20 years, there have been proposals for using non-commutative groups to develop quantum-resistant cryptosystems. The unsolved word problem, formulated by Wagner and Magyarik, uses permutation groups and is a promising direction in cryptosystem development. Magliveras proposed logarithmic signatures, a special type of factorization applied to finite groups, and the latest version of this technology is known as MST3, based on the Suzuki group. The first implementation of the cryptosystem on the generalized Suzuki 2-group had limitations in encryption and protection against brute force attacks. Over the past years, many proposals have been made to improve the basic design. The research conducted by the authors expanded the possibilities of using public cryptography by refining parameters based on non-Abelian groups. The article demonstrates the methodology for conducting tests of the practical implementation of the library of cryptographic transformations with the implementation of an improved encryption scheme on Suzuki 2-groups, confirming its functionality.

TI subgroups and depth 3 subgroups in simple Suzuki groups

Abstract In this paper, we determine the TI subgroups of the simple Suzuki groups Sz ⁢ ( q ) \mathrm{Sz}(q) . More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively by Higman. These results were extended later by Goldschmidt, Shaw, Shult, Gross, Wilkens and Bryukhanova. As a corollary of our investigations, we get some interesting results for the Sylow 2-subgroups of Suzuki groups, as well. We relate this to an open problem on Suzuki 2-groups, and we ask a question concerning that. We also give some characterization of Suzuki groups.

Optimal Line Packings from Nonabelian Groups

We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.

Analysis of the implementation complexity of the cryptosystem on the Suzuki group

Implementations for cryptosystems of finite groups based on logarithmic signature and covering are considered. A logarithmic signature is exemplified by a permutation group with the asymmetry of encryption and decryption algorithms. Description of the improved cryptosystem MST3 in Suzuki 2-group with the order of the group q2 is given. The Suzuki 2-group use has a significant advantage in implementation, due to the large center and simple group operation. Cost estimates for encryption, decryption and comparison with the RSA algorithm are obtained.

Locally Finite Suzuki–Higman 2-Groups

We prove the following theorem. Let U be a locally finite Suzuki–Higman 2-group with respect to an automorphism group H. Then U and H are representable as the respective unions of ascending chains of finite subgroups U1 < U2 < . . . < U n < . . . and H1 < H2 < . . . < H n < . . ., in which case every subgroup U n is a Suzuki 2-group with respect to H n .

An IND-CCA2 Secure Public Key Cryptographic Protocol using Suzuki 2-Group

Objectives: The public key cryptographic protocol is one of the most important fields in computer security. These new public key cryptographic protocols provide high security as compare to past results in the same field. Methods/Statistical Analysis: Public key cryptographic is a protocol of transferring private info and data through open network communication, so only the receiver who has the secret key can read the encrypted messages which might be documents, phone conversations, images or other form of data. To implement privacy simply by encrypting the information intended to remain secret can be achieved by using methods of public key cryptography. Findings: In this study, we propose the new IND-CCA2 secure public key cryptographic protocol using the concept of integral coefficient ring polynomial based on Suzuki 2-group. We demonstrated the security of proposed public key cryptographic protocol in the adaptively chosen cipher text secure (IND-CCA2) in the random oracle model. Application/Improvements: We discussed the new strategy with change over an IND-CPA public key cryptographic protocol into an IND-CCA2 cryptographic protocol.

Public Key Cryptographic Technique Based On Suzuki 2-Group

Public key cryptography is one of the most important fields in computer security. . It is a method of transferring private information and data through open network communication, so only the receiver who has the secret key can read the encrypted messages which might be documents, phone conversations, images or other form of data. To implement privacy simply by encrypting the information intended to remain secret can be achieved by using methods of Public key cryptography. The information must be scrambled, so that other users will not be able to access the actual information. In this study, we propose new public key cryptographic technique based on Suzuki 2- group. We demonstrated the security of proposed public key cryptographic technique in the chosen plaintext attack in the random oracle model. Keywords:Public key cryptography, ring polynomial, Suzuki 2-group, chosen plaintext attack, random oracle.

Investigation of the cryptosystem MST3 based on a Suzuki 2-group

AbstractWe investigate a previously proposed attack against the public key cryptosystem MST

Bounding group orders by large character degrees: A question of Snyder

Abstract Let G be a nonabelian finite group and let d be an irreducible character degree of G. Then there is a positive integer e so that |G| = d(d + e). Snyder has shown that if e &gt; 1, then |G| is bounded by a function of e. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper we will show, for groups having a nontrivial, abelian normal subgroup, that |G| ≤ e 4 - e 3. Given that there are a number of solvable groups that meet this bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.

A GROUP WITH THREE REAL IRREDUCIBLE CHARACTERS: ANSWERING A QUESTION OF MORETÓ AND NAVARRO

In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moretó and Navarro who asked whether such a group exists.

A Characterization of the smallest Suzuki 2-group

Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.

On Suzuki 2-groups of finite Morley rank

On Suzuki 2-groups of finite Morley rank

Degrees of irreducible characters of the suzuki 2-groups

The degrees of the irreducible characters of the Suzuki 2-groupsA(m,ϕ) are described. Assume that the order of an automorphism ϕ of the fieldGF (2 m ) isk>1,G=A(m,ϕ), and cd (G) is the set of degrees of the irreducible characters ofG. Ifk is odd, then cd(G)={1, 2( m−m/k)/2 }, and ifk=2, then cd(G)={1, 2 m/2 }. Ifk is even andk≠2 then cd(G)={1,2 m/2 , 2 m/2−m/k }, and the groupG has (2 m −1)2 m/k /(2 m/k +1) characters of degree2 m/2 and (2m−1)22m/k/(2m/k+1) characters of degree 2m/2−m/k.

A condition on lengths of conjugacy classes and character degrees

We call this condition B-condition (B is due to E. Bannai). A few groups satisfying B-condition are known : abelian groups, Suzuki 2-groups A(n, <9)(See [3, VIII. 6. 7 Example and §7]), φ&, φ in [4], and groups isoclinic to them (For isoclinism, see [2]). In any case, they are nilpotent and their derived lengths are at most 2. In this paper, we shall construct a family of groups satisfying B-condition. Our groups are, in a sense, generalizations of Suzuki 2-groups. By our examples, we can say that

On finite groups all of whose elements of the same order are conjugate in their automorphism groups

The aim of this paper is to classify the finite groups G such that any two elements of the same order are conjugate in Aut( G). A new property of Suzuki 2-groups is also given.

Finite groups admitting the extensions of the automorphisms of a maximal elementary abelian 2-subgroup

1. INTRODUCTION AND STATEMENT OF THE THEOREM In [lo], Higman described the structure of a finite p-group admitting an automorphism which cyclically permutes the subgroups of order p. Such a group is either a direct sum of cyclic groups of the same order, a generalized quaternion group, or a Suzuki 2-group (defined in [lo]). Several authors have considered the structure of a finite p-group whose automorphism group acts transitively on the elements of order p of the group. (See [S] and the references cited in it.) Gaschiitz and Yen [S] have shown that if G is a group of odd order and, if for each prime divisor p of the order of G, the automorphism group of G acts transitively on the elements of G of order p, then G is a cyclic extension of a nilpotent Hall subgroup. Here, we consider a variation of the situations mentioned above. We say that a finite group G has the (extension) property B(p) for the prime p if the automorphisms of some elementary abelian p-subgroup E of G of the largest possible order can be obtained as the restrictions to E of the automorphisms of G. Our object here is to prove the following: THEOREM. Let G be a finite group possessing the property g(2), and let (G#)S E W,(G). (A) If O,(G) # (1) and O(G) = (1 ), then one of