Groups Of Prime Order Research Articles

TOPAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS}$$\\end{document}2-pass key exchange with full perfect forward secrecy and optimal communication complexity

We present Transmission optimal protocol with active security (TOPAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS}$$\\end{document}), the first key agreement protocol with optimal communication complexity (message size and number of rounds) that provides security against fully active adversaries. The size of the protocol messages and the computational costs to generate them are comparable to the basic Diffie-Hellman protocol over elliptic curves (which is well-known to only provide security against passive adversaries). Session keys are indistinguishable from random keys—even under reflection and key compromise impersonation attacks. What makes TOPAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS}$$\\end{document}stand out is that it also features a security proof of full perfect forward secrecy (PFS), where the attacker can actively modify messages sent to or from the test-session. The proof of full PFS relies on two new extraction-based security assumptions. It is well-known that existing implicitly-authenticated 2-message protocols like HMQV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {HMQV}$$\\end{document}cannot achieve this strong form of (full) security against active attackers (Krawczyk, Crypto’05). This makes TOPAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS}$$\\end{document}the first key agreement protocol with full security against active attackers that works in prime-order groups while having optimal message size. We also present a variant of our protocol, TOPAS+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS+}$$\\end{document}, which, under the Strong Diffie-Hellman assumption, provides better computational efficiency in the key derivation phase. Finally, we present a third protocol termed FACTAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {FACTAS}$$\\end{document}(for factoring-based protocol with active security) which has the same strong security properties as TOPAS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS}$$\\end{document}and TOPAS+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {TOPAS+}$$\\end{document}but whose security is solely based on the factoring assumption in groups of composite order (except for the proof of full PFS).

Tightly secure (H)IBE in the random oracle model

We present an adaptively secure identity-based encryption (IBE) scheme with constant sized parameters and constant security loss in multi-instance multi-ciphertext (MIMC) setting in prime-order groups, in the random oracle model. We then further construct an adaptively secure hierarchical identity-based encryption (HIBE) scheme with shorter sized parameters and constant security loss in the MIMC setting in prime-order groups, in the random oracle model.At the core of our technique, we observe that the pseudorandomness property of the random oracle matches the conditions of Left Subgroup Indistinguishability (LS) and Right Subgroup Indistinguishability (RS) in dual system groups (DSG) well, so that we can transform the queried secret keys and the challenge ciphertexts for once. We construct our IBE scheme from Agrawal and Chase's work (CCS 2017) and construct our HIBE scheme from Boneh et al.'s work (Eurocrypt 2005). As a byproduct of our IBE scheme, we construct a new DSG instantiation from Chen et al.'s work (Eurocrypt 2015) under bi-MDDH assumption.

Verifiable Encryption from MPC-in-the-Head

Verifiable encryption (VE) is a protocol where one can provide assurance that an encrypted plaintext satisfies certain properties, or relations. It is an important building block in cryptography with many useful applications, such as key escrow, group signatures, optimistic fair exchange, and others. However, the majority of previous VE schemes are restricted to instantiation with specific public-key encryption schemes or relations. In this work, we propose a novel framework that realizes VE protocols using zero-knowledge proof systems based on the MPC-in-the-head paradigm (Ishai et al. STOC 2007). Our generic compiler can turn a large class of zero-knowledge proofs into secure VE protocols for any secure public-key encryption scheme with the undeniability property, a notion that essentially guarantees binding of encryption when used as a commitment scheme. Our framework is versatile: because the circuit proven by the MPC-in-the-head prover is decoupled from a complex encryption function, the work of the prover is focused on proving the encrypted data satisfies the relation, not the proof of plaintext knowledge. Hence, our approach allows for instantiation with various combinations of properties about the encrypted data and encryption functions. We then consider concrete applications, to demonstrate the efficiency of our framework, by first giving a new approach and implementation to verifiably encrypt discrete logarithms in any prime order group more efficiently than was previously known. Then we give the first practical verifiable encryption scheme for AES keys with post-quantum security, along with an implementation and benchmarks.

A Prime-Order Group with Complete Formulas from Even-Order Elliptic Curves

This paper describes a generic methodology for obtaining unified, and then complete formulas for a prime-order group abstraction homomorphic to a subgroup of an elliptic curve with even order. The method is applicable to any curve with even order, in finite fields of both even and odd characteristic; it is most efficient on curves with order equal to 2 modulo 4, dubbed "double-odd curves". In large characteristic fields, we obtain doubling formulas with cost as low as 1M + 5S, and the resulting group allows building schemes such as signatures that outperform existing fast solutions, e.g. Ed25519. In binary fields, the obtained formulas are not only complete but also faster than previously known incomplete formulas; we can sign and verify in as low as 18k and 27k cycles on x86 CPUs, respectively.

Non-Compatible Action Graph and Its Adjacency Matrix for The Non-abelian Tensor Product for Groups of Prime Power Order

This article focused on the notion of the non-abelian tensor product of groups of prime power order. Particularly, it presented new graph named as Non-compatible action graph and discussed some of its properties. Moreover, this graph concentrated on the case of non-compatible actions of the tensor product of two finite -groups. Furthermore, its adjacency matrix has been determined and discussed in detail. Moreover, the adjacency matrix has been denoted by A(G) and its inputs are 1 whenever there is adjacency and 0 otherwise.

On p -Group Isomorphism: Search-to-Decision, Counting-to-Decision, and Nilpotency Class Reductions via Tensors

In this article, we study some classical complexity-theoretic questions regarding Group Isomorphism ( GpI ). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneckcase for GpI , and work in the model of matrix groups over finite fields. Our main results are as follows: • Although search-to-decision and counting-to-decision reductions have been known for more than four decades for Graph Isomorphism , they had remained open for GpI , explicitly asked by Arvind and Torán ( EATCS Bull. , 2005). Extending methods from Tensor Isomorphism (TI) (Grochow and Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • Despite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI , there was no reduction to these groups from any larger class of groups. Again using methods from TI (ibid.), we show the first such reduction, namely from isomorphismtesting of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of Graph Isomorphism . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor and Lubotzky, Geom. Dedicata , 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with TI-completeness results (Grochow and Qiao, ibid.).

The structure of arbitrary Conze–Lesigne systems

Let Γ \Gamma be a countable abelian group. An (abstract) Γ \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ \Gamma - is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) \mathrm {Z}^2(\mathrm {X}) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ \Gamma , namely that they are the inverse limit of translational systems G n / Λ n G_n/\Lambda _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ n \Lambda _n . Results of this type were previously known when Γ \Gamma was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G .

Fixed point sets and orbit spaces of wedge of three spheres

Let X be a finite CW-complex having mod p cohomology isomorphic to a wedge of three spheres Sn∨Sm∨Sl,1≤n≤m≤l. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on X. We also classify the orbit spaces of free actions of G=Zp,p a prime or G=Sd,d=1 or 3, on X and derive the Borsuk-Ulam type results.

Comment on “Expressive Public-Key Encryption With Keyword Search: Generic Construction From KP-ABE and an Efficient Scheme Over Prime-Order Groups”

Comment on “Expressive Public-Key Encryption With Keyword Search: Generic Construction From KP-ABE and an Efficient Scheme Over Prime-Order Groups”

A note on the degree bounds of the invariant ring

<abstract><p>Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{F} $ be an algebraically closed field in characteristic $ p $. Let $ V $ be a direct sum of $ m $ non-trivial indecomposable $ G $-modules such that the norm polynomials of the simple $ H $-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.</p></abstract>

The construction of fuzzy subgroups of groups with units modulo 20 and 21

The number of fuzzy subgroups (NFSGs) of the groups of units U20 and U21 are determining in this study. The (NFSGs) of U20 was found based on diagram and using the lemma which state that the (NFSGs) of G is equal to the number of chains (NC) on the subgroups lattice of G. Moreover, we explain that there are only two fuzzy subgroups for a prime order group. Lagrange’s theorem is more important in this work, from Lagrange’s theorem, there is no nontrivial subgroup of U21 since the order of the group is prime. We prove that, if P1 (μ) = U21, then we get μ1(x) = θ1, ∀x ∈ U21. Thus we have one fuzzy subgroup of P1 (μ) = U21. If P1 (μ) = {1}, then we obtain μ2(x) = θ1, ∀x ∈ {1} and μ2 (x) = θ2,∀x∈U21\{1}. Thus we have 2 fuzzy subgroups of U21. We came up with the numbers 21 and 2 respectively.

Uniquely regular dessins with nilpotent automorphism groups of odd prime power order

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

Inner-Product Matchmaking Encryption: Bilateral Access Control and Beyond Equality

We present an inner-product matchmaking encryption (IP-ME) scheme achieving weak privacy and authenticity in prime-order groups under symmetric external Diffie–Hellman (SXDH) assumption in the standard model. We further present an IP-ME with Monotone Span Program Authenticity (IP-ME with MSP Auth) scheme, where the chosen sender policy is upgraded to MSP, and the scheme also achieves weak privacy and authenticity in prime-order groups under SXDH assumption in the standard model. Both of the schemes have more expressive functionalities than identity-based matchmaking encryption (IB-ME) scheme, and are simpler than Ateniese et al.’s modular ME scheme (Crypto’ 19). But our schemes only achieve a very limited flavor of security, which is reflected in the privacy.

Zero-knowledge proofs for set membership: efficient, succinct, modular

We consider the problem of proving in zero knowledge that an element of a public set satisfies a given property without disclosing the element, i.e., for some u, “u in S and P(u) holds”. This problem arises in many applications (anonymous cryptocurrencies, credentials or whitelists) where, for privacy or anonymity reasons, it is crucial to hide certain data while ensuring properties of such data. We design new modular and efficient constructions for this problem through new commit-and-prove zero-knowledge systems for set membership, i.e. schemes proving u in S for a value u that is in a public commitment c_u. We also extend our results to support non-membership proofs, i.e. proving u notin S. Being commit-and-prove, our solutions can act as plug-and-play modules in statements of the form “u in S and P(u) holds” by combining our set (non-)membership systems with any other commit-and-prove scheme for P(u). Also, they work with Pedersen commitments over prime order groups which makes them compatible with popular systems such as Bulletproofs or Groth16. We implemented our schemes as a software library, and tested experimentally their performance. Compared to previous work that achieves similar properties—the clever techniques combining zkSNARKs and Merkle Trees in Zcash—our solutions offer more flexibility, shorter public parameters and 3.7 times –30times faster proving time for a set of size 2^{64}.

Towards $3n-4$ in groups of prime order

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results towards the conjectured value $3|A|-4$ for which such an statement should hold..

Model structures on finite total orders

We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order [n], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of N_infty -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [n].

𝑝-group Galois covers of curves in characteristic 𝑝

We study cohomologies of a curve with an action of a finite p p -group over a field of characteristic p p . Assuming the existence of a certain “magical element” in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p p -group cover has a “magical element”. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.

A controllable delegation scheme for the subscription to multimedia platforms

With the development of the Internet, various multimedia platforms have sprung up like mushrooms after rain. Through subscription, people can enjoy the rich resources provided by the multimedia platform. To attract more subscribers, the multimedia platform will give users the ability to delegate access rights. Because in this way, the subscriber can share resources with his/her family or friends. However, the delegation ability should be controllable. Otherwise, there will be a risk of delegation abuse and damage to the interests of the multimedia platform. Therefore, we propose a new Key-Policy Attribute-Based Encryption (KP-ABE) scheme supporting key delegation ability, traceability, and single-layer delegation, which can work as a controllable delegation scheme for the subscription to multimedia platforms. Firstly, KP-ABE can realize access control to resources and protect the security of resources through encryption. Secondly, the key delegation ability can meet the requirement of users to share their access rights. Lastly, traceability and single-layer delegation can limit the key delegation ability of users, so as to resist key delegation abuse. In addition, the proposed scheme supports large universe and tree access structures and is established on prime-order groups, which makes it scalable, expressive, and efficient.

Class numbers, cyclic simple groups, and arithmetic

AbstractHere, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in a systematic way. To do this, we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms. Then, we classify optimal modules for the cyclic groups of prime order, in the special case of weight 2 and index 1, where class numbers of imaginary quadratic fields play an important role. Finally, we exhibit a connection between the classification we establish and the arithmetic geometry of imaginary quadratic twists of modular curves of prime level.

CI-property of [formula omitted] and [formula omitted] for digraphs

We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).