Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

Candidate for practical post-quantum signature scheme

A new criterion of post-quantum security is used to design a practical signature scheme based on the computational complexity of the hidden discrete logarithm problem. A 4-dimensional finite non-commutative associative algebra is applied as algebraic support of the cryptoscheme. The criterion is formulated as computational intractability of the task of constructing a periodic function containing a period depending on the discrete logarithm value. To meet the criterion, the hidden commutative group possessing the 2-dimensional cyclicity is exploited in the developed signature scheme. The public-key elements are computed depending on two vectors that are generators of two different cyclic groups contained in the hidden group. When computing the public key two types of masking operations are used: i) possessing the property of mutual commutativity with the exponentiation operation and ii) being free of such property. The signature represents two integers and one vector S used as a multiplier in the verification equation. To prevent attacks using the value S as a fitting element the signature verification equation is doubled.

Mathieu–Zhao subspaces of vertex algebras

We introduce a notion of Mathieu–Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra [Formula: see text] and its subspace [Formula: see text] that contains [Formula: see text], [Formula: see text] is a Mathieu–Zhao subspace of [Formula: see text] if and only if the quotient space [Formula: see text] is a Mathieu–Zhao subspace of a commutative associative algebra [Formula: see text]. As a result, one can study the famous Jacobian conjecture in terms of Mathieu–Zhao subspaces of vertex algebras. In addition, for a [Formula: see text]-type vertex operator algebra [Formula: see text] that satisfies the [Formula: see text]-cofiniteness condition, we classify all Mathieu–Zhao subspaces [Formula: see text] that contain [Formula: see text].

Finite irreducible representations of map Lie conformal algebras

In this paper, we study the finite representation theory of the map Lie conformal algebra [Formula: see text], where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over [Formula: see text]. In particular, we give a complete classification of nontrivial finite irreducible conformal modules of [Formula: see text] provided A is finite-dimensional.

Constructive description of monogenic functions in a finite-dimensional commutative associative algebra

Let 𝔸

Centroids of Lie Supertriple Systems

We derive certain structural results concerning centroids of Lie supertriple systems. Centroids of the tensor product of a Lie supertriple system and a unital commutative associative algebra are studied. Furthermore, the centroid of a tensor product of a simple Lie supertriple system and a polynomial ring is partly determined.

The Centroid of a Lie Triple Algebra

General results on the centroids of Lie triple algebras are developed. Centroids of the tensor product of a Lie triple algebra and a unitary commutative associative algebra are studied. Furthermore, the centroid of the tensor product of a simple Lie triple algebra and a polynomial ring is completely determined.

Symmetric Symplectic Commutative Associative Algebras and Related Lie Algebras

A commutative associative algebra [Formula: see text] is called symmetric symplectic if it is endowed with both an associative non-degenerate symmetric bilinear form B and an invertible B-antisymmetric derivation D. We give a description of the commutative associative symmetric symplectic 𝕂-algebras by using the notion of T*-extension. Next, we introduce the notion of double extension of symmetric symplectic commutative associative algebras in order to give an inductive description of these algebras. Moreover, much information on the structure of symmetric commutative associative algebras is given in this paper.

The Second Cohomology of Current Algebras of General Lie Algebras

Let A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.