arXiv subjects

Steven Duplij

Publications and source records attributed to Steven Duplij.

At least 19 recordsLinked to original sources

arXiv2026

Ternary public-key cryptosystem

Public-key cryptosystems eliminate the requirement for pre-shared secret keys by enabling encryption with a publicly disclosed key and decryption with a corresponding private key. In this article we generalize the public-key cryptosystems to ternary algebraic structures, with particular attention to ElGamal as a representative family. We introduce the necessary algebraic background for nonderived ternary structures, including special elements, ternary group rings, and a matrix ternarization procedure that maps binary rings and group rings to antidiagonal symbolic matrices closed under ternary multiplication. Building on these foundations, we formulate a ternary analogue of the ElGamal three-step protocol (key generation, ephemeral encryption, and decryption via querelements) and derive explicit ternary power and querelement formulas that enable correct decryption. Concrete instantiations and numerical examples over a ternary fraction field, a matrix-ternarized finite group ring, and a finite $(6,3)$-ring (field) validate the construction and illustrate admissible word-length quantization and cycle behaviour of ternary powers. The ternary framework highlights two practical advantages: richer algebraic structure (querelements replace binary inverses) that increases algebraic complexity for attackers, and higher information density (matrix ternarization transfers paired/plaintext vectors). Formal hardness assumptions, optimized parameter choices, and comprehensive security and performance analyses remain necessary future work.

cs.CR↗

arXiv2026

Multiary gradings

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, the First Isomorphism Theorem for graded polyadic algebras and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility.

math.RA↗

arXiv2025

Cryptographic transformations over polyadic rings

This article introduces a novel cryptographic paradigm based on nonderived polyadic algebraic structures. Traditional cryptosystems rely on binary operations within groups, rings, or fields, whose well-understood properties can be exploited in cryptanalysis. To overcome these vulnerabilities, we propose a shift to polyadic rings, which generalize classical rings by allowing operations of higher arity: an $m$-ary addition and an $n$-ary multiplication. The foundation of our approach is the construction of polyadic integers -- congruence classes of ordinary integers endowed with such $m$-ary and $n$-ary operations. A key innovation is the parameter-to-arity mapping $\Phi(a,b)=(m,n)$, which links the parameters $(a,b)$ defining a congruence class to the specific arities required for algebraic closure. This mapping is mathematically intricate: it is non-injective, non-surjective, and multivalued. This complex, non-unique relationship forms the core of the proposed cryptosystem's security. We present two concrete encryption procedures that leverage this structure by encoding plaintext within the parameters of polyadic rings and transmitting information via polyadically quantized analog signals. In one method, plaintext is linked to the additive arity $m_{i}$ and secured using the summation of such signals; in the other, it is linked to a ring parameter $a_{i}$ and secured using their multiplication. In both cases, the "quantized" nature of polyadic operations generates systems of equations that are straightforward for a legitimate recipient with the correct key but exceptionally difficult for an attacker without it. The resulting framework promises a substantial increase in cryptographic security. This work establishes the theoretical foundation for this new class of encryption schemes and highlights their potential for constructing robust, next-generation cryptographic protocols.

cs.CR↗

arXiv2025

Higher power polyadic group rings

This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings $\mathcal{R}[\mathsf{G}]$. We construct the fundamental operations of these structures, defining the $\mathbf{m}_{r}$-ary addition and $\mathbf{n}_{r} $-ary multiplication for a polyadic group ring $\mathrm{R}^{[\mathbf{m} _{r},\mathbf{n}_{r}]}=\mathcal{R}^{[m_{r},n_{r}]}[\mathsf{G}^{[n_{g}]}]$ built from a nonderived $(m_{r},n_{r})$-ring and a nonderived $n_{g}$-ary group. A central result is the derivation of the "quantization" conditions that interrelate these arities, governed by the arity freedom principle, which also extends to operations with higher polyadic powers. We establish key algebraic properties, including conditions for total associativity and the existence of a zero element and identity. The concepts of the polyadic augmentation map and augmentation ideal are generalized, providing a bridge to the classical theory. The framework is illustrated with explicit examples, solidifying the theoretical constructions. This work establishes a new foundation in ring theory with potential applications in cryptography and coding theory, as evidenced by recent schemes utilizing polyadic structures.

math.RA↗

arXiv2025

Polyadic encryption

A novel original procedure of encryption/decryption based on the polyadic algebraic structures and on signal processing methods is proposed. First, we use signals with integer amplitudes to send information. Then we use polyadic techniques to transfer the plaintext into series of special integers. The receiver restores the plaintext using special rules and systems of equations.

cs.CR↗

arXiv2025

Positional numeral systems over polyadic rings

We construct positional numeral systems that work natively over nonderived polyadic $\left( m,n\right) $-rings whose addition takes $m$ arguments and multiplication takes $n$. In such rings, the length of an admissible additive word and a multiplicative tower are not arbitrary (as in the binary case), but "quantized". Our main contributions are the following. Existence: every commutative $\left( m,n\right) $-ring admits a base-$p$ place-value expansion that respects the word length constraint in terms of numbers of operation compositions $\ell_{mult}=\ell_{add}(m-1)+1$. Lower bound: the minimum number of digits is greater than or equal to the arity of addition $m$. Representability gap: for $m,n\geq3$ only a proper subset of ring elements possess finite expansions, characterized by congruence-class arity shape invariants $I^{(m)}$ and $J^{(n)}$. Mixed-base "polyadic clocks": allowing a different base at each position enlarges the design space quadratically in the digit count. Catalogues: explicit tables for the integer rings $\mathbb{Z}_{4,3}$ and $\mathbb{Z}_{6,5}$ illustrate how ordinary integers lift to distinct polyadic variables. These results lay the groundwork for faster arity-aware arithmetic, exotic coding schemes, and hardware that exploits operations beyond the binary pair.

math.NT↗

arXiv2024

Polyadic supersymmetry

We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the $n$-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the $N$-extended and multigraded SQM. While constructing the corresponding supersymmetry as an $n$-ary Lie superalgebra ($n$ is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of $2\leq m<n$ and a related series of $m$-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity $m$ we obtain a tower of higher order (as differential operators) even Hamiltonians, while for $m$ odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.

hep-th↗

arXiv2024

Polyadic sigma matrices

We generalize $\sigma$-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived $n$-ary version of $SU\left( 2\right) $ using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary $\Sigma$-matrices introduced here play a role similar to ordinary matrix units, and their sums are full $\Sigma$-matrices which can be treated as a polyadic analog of $\sigma$-matrices. The presentation of $n$-ary $SU\left( 2\right) $ in terms of full $\Sigma$-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted $\sigma$-matrices with multipliers in cyclic groups of order $4q$ ($q>4$), and for the polyadic case we construct the correspondent finite $n$-ary semigroup of phase-shifted elementary $\Sigma$-matrices of order $4q\left( n-1\right) +1$, and the finite $n$-ary group of phase-shifted full $\Sigma$-matrices of order $4q$. Finally, we introduce the finite $n$-ary group of heterogeneous full $\mathit{\Sigma}^{het}$-matrices of order $\left( 4q\left( n-1\right) \right) ^{4}$. Some examples of the lowest arities are presented.

math.GR↗

arXiv2023

Hyperpolyadic structures

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a $n$-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative.

math.RA↗

arXiv2023

Polyander visualization of quantum walks

We investigate quantum walks which play an important role in the modelling of many phenomena. The detailed and thorough description is given to the discrete quantum walks on a line, where the total quantum state consists of quantum states of the walker and the coin. In addition to the standard walker probability distribution, we introduce the coin probability distribution which gives more complete quantum walk description and novel visualization in terms of the so called polyanders (analogs of trianders in DNA visualization). The methods of final states computation and the Fourier transform are presented for the Hadamard quantum walk.

quant-ph↗

arXiv2023

On superqubits

We first reconsider the mathematical background of superqubit theory and describe important peculiarities of superspaces and supermatrices which are usually out of attention. Then we study states in super Hilbert spaces using super-bra/super-ket formalism in details. The qubit (qudit) and superqubit (superqudit) are defined as linear spans in the corresponding Hilbert subspaces. A new kind of superqubit carring the odd parity is introduced. The multi-superqubit states are studied, and the superconcurrence which distinguishes separable states is proposed.

quant-ph↗

arXiv2022

Polyadic rings of $p$-adic integers

In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.

math.RA↗

arXiv2022

Extensions of special 3-fields

We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.

math.RA↗

arXiv2022

Polyadization of algebraic structures

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures ("double" decomposition of two kinds). We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.

math.RA↗

arXiv2022

Polyadic analog of Grothendieck group

We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.

math.RA↗

arXiv2022

Polyadic analogs of direct product

We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be \textquotedblleft entangled\textquotedblright\ such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields. Many illustrative concrete examples are presented.

math.RA↗

arXiv2022

Gauge gravity vacuum in constraintless Clairaut-type formalism

The gauged Lorentz theory with torsion has been argued to have an effective theory whose non-trivial background is responsible for background gravitational curvature if torsion is treated as a quantum-mechanical variable against a background of constant curvature. We use the CDG decomposition to argue that such a background can be found without including torsion. Adapting our previously published Clairaut-based treatment of QCD, we go on to study the implications for second quantisation.

hep-th↗

arXiv2022

Obscure qubits and membership amplitudes

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the "product" obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

quant-ph↗