Introduction↩︎

Public-key cryptography, introduced in the mid-1970s, removed the longstanding requirement for a secure pre-shared channel to distribute secret keys by separating encryption and decryption into distinct, mathematically linked keys: a publicly disclosed encryption key and a privately held decryption key. This separation enabled any party to encrypt a message for a recipient without prior secret exchange and has become foundational to secure web protocols, electronic payments, digital signatures, and private messaging. Diffie and Hellman introduced the core idea of public-key exchange and motivated subsequent asymmetric constructions [1], which remains the conceptual basis for many discrete-log based schemes.

Early practical realizations of the paradigm include RSA, which bases security on integer factorization and demonstrated the feasibility of public-key encryption in practice [2]. Shortly thereafter, the ElGamal cryptosystem [3] offered an alternative construction rooted in the discrete logarithm problem, introducing probabilistic encryption—a feature that fundamentally distinguishes it from deterministic schemes. The probabilistic nature of ElGamal—where identical plaintexts encrypt to different ciphertexts with overwhelming probability—yields ciphertext indistinguishability under appropriate group choices and underpins many modern cryptographic constructions [4]. ElGamal’s algebraic pattern also directly inspired practical standards such as the Digital Signature Algorithm (DSA), which adapts ElGamal-style signatures for standardized use [5].

Motivated by the algebraic diversity of polyadic systems [6], this paper develops a ternary generalization of ElGamal-style public-key encryption. We present the algebraic preliminaries for nonderived ternary structures (special elements, ternary group rings, and a matrix ternarization mapping), formulate an abstract ternary encryption/decryption protocol that mirrors the three-step ElGamal flow (key generation, ephemeral encryption, and decryption), and validate the construction with concrete examples. The ternary setting replaces binary powers and inverses with \(\ell\)-ternary powers and querelements (the ternary analogue of inverses), drawing on the formalism of \(\left( m,n\right)\)-rings and fields [6]. To obtain practical instantiations we employ a matrix ternarization that maps binary rings and group rings to antidiagonal symbolic matrices closed under ternary multiplication, this construction is informed by prior work on group rings [7] and matrix realizations of \(n\)-ary algebras [8].

The remainder of the paper details the necessary ternary algebra, the matrix ternarization procedure that produces nonderived ternary unit groups, the ternary analogue of the ElGamal protocol with explicit formulas for encryption and decryption, and several numerical examples that illustrate correctness and structural properties relevant to security and implementation.

1 Ternary structures↩︎

We present the notation and general properties of ternary structures (see [6] for details). Let \(S^{\times3}\) be the \(3\)-fold Cartesian product of a non-empty set \(S\). Elements \((x_{1},x_{2},x_{3})\in S^{\times3}\) are called \(3\)-tuples \(({\mathrm{\small \upshape x}})\); an \(3\)-tuple of identical elements is denoted \((x^{3})\). A ternary operation is a mapping \(\mu^{\left[ 3\right] }:S^{\times3}\rightarrow S\), written \(\mu^{\left[ 3\right] }[{\mathrm{\small \upshape x}}]\). In general, a ternary structure \(\langle S\mid{\mu}_{i}^{\left[ 3\right] }\rangle\) consists of a set \(S\) closed under a family of the operations \({\mu }_{i}^{\left[ 3\right] }\).

1.1 Special elements in ternary groups and rings↩︎

The fundamental one-operation structure is the ternary magma \(\mathcal{M}^{\left[ 3\right] }=\langle S\mid\mu^{\left[ 3\right] }\rangle\). Adding axioms gives group-like structures: a ternary associative magma is a ternary semigroup \(\mathcal{S}^{\left[ 3\right] }=\langle S\mid\mu^{\left[ 3\right] }\mid assoc\rangle\). Ternary associativity means\[\mu^{\left[ 3\right] }\left[ \mu^{\left[ 3\right] }\left[ x_{1},x_{2},x_{3}\right] ,x_{4},x_{5}\right] =\mu^{\left[ 3\right] }\left[ x_{1},\mu^{\left[ 3\right] }[x_{2},x_{3},x_{4}],x_{5}\right] =\mu^{\left[ 3\right] }\left[ x_{1},x_{2},\mu^{\left[ 3\right] }\left[ x_{3},x_{4},x_{5}\right] \right]\] and so it is invariant under placing the inner multiplication in any of the \(2\) positions (giving \(2\) relations), allowing omission of parentheses. The iterated product is \[\mu^{\left[ 3\right] \circ\ell_{\mu}}[{\mathrm{\small \upshape x}}]=\overset{\ell_{\mu}}{\overbrace{\mu^{\left[ 3\right] }[\mu^{\left[ 3\right] }[\ldots \mu^{\left[ n\right] }[{\mathrm{\small \upshape x}}]]]}},\quad{\mathrm{\small \upshape x}}\in S^{2\ell_{\mu }+1}, \label{ml}\tag{1}\] where \(\ell_{\mu}\) is the number of composed ternary multiplications \(\mu^{\left[ 3\right] }\). If \(n=2\), then the iterated product (1 ) of arity \(\ell_{\mu}+1\) is called derived.

A key distinction from the binary case (\(n=2\)) is that the word length \(w\) in a composition of \(n\)-ary multiplications is quantized and depends of arity: multiplication is possible for \[w^{\text{admiss}}(n,\ell_{\mu})=\left( n-1\right) \ell_{\mu}+1, \label{wln}\tag{2}\] and for the ternary case\[w^{\text{admiss}}(\ell_{\mu})=2\ell_{\mu}+1 \label{wl}\tag{3}\] elements. This leads to two types of ternary operations: derived (iterated from binary/lower-arity operations) and nonderived; the latter are more interesting.

For \(x\in S\), the \(\ell_{\mu}\)-ternary power is \[x^{\langle\ell_{\mu}\rangle}=\mu^{\left[ 3\right] \circ\ell_{\mu}}[x^{2\ell_{\mu}+1}], \label{xl}\tag{4}\] which for \(n=2\) gives \(x^{\langle\ell_{\mu}\rangle}=x^{\ell_{\mu}+1}\), differing by one from the usual power. A ternary idempotent \(x_{id}\) satisfies \(x_{id}^{\langle\ell_{\mu}\rangle}=x_{id}\). A ternary zero \(z\) is uniquely defined by the \(3\) conditions\[{\mu}^{\left[ 3\right] }[z,x_{1},x_{2}]={\mu}^{\left[ 3\right] }[x_{1},z,x_{2}]={\mu}^{\left[ 3\right] }[x_{1},x_{2},z]=z,\;\;\;\;\;x_{1},x_{2},z\in S. \label{z}\tag{5}\] A ternary nilpotent \(x_{nil}\) obeys \[x_{nil}^{\langle\ell_{\mu}\rangle}=z. \label{xz}\tag{6}\] A neutral \(2\)-tuple \({\mathrm{\small \upshape e}}\) satisfies \({\mu}^{\left[ 3\right] }[x,{\mathrm{\small \upshape e}}]=x\) (typically non-unique). If all components of \({\mathrm{\small \upshape e}}\) are identical, \({\mathrm{\small \upshape e}}=e^{2}\), then the identity is defined by\[\mu^{\left[ 3\right] }[x,e^{2}]=\mu^{\left[ 3\right] }[x,e,x]=\mu^{\left[ 3\right] }[e^{2},x]=x \label{e}\tag{7}\] where \(e\) may appear in any 2 of the \(3\) positions. Obviously, that both \(z\) and \(e\) are idempotents. Some ternary structures lack idempotents, zero, or identity, or have multiple identities.

For ternary structures, invertibility is not tied to the identity but to the querelement \(\bar{x}=\bar{x}(x)\) (or \(\bar{x}=Q\left( x\right)\)), defined by the \(3\) relations\[\mu^{\left[ 3\right] }[x^{2},\bar{x}]=\mu^{\left[ 3\right] }[x,\bar {x},x]=\mu^{\left[ 3\right] }[x^{2},\bar{x}]=x\label{q}\tag{8}\] (holding for \(\bar{x}\) in each of the \(3\) positions). Such \(x\) is ternary invertible. If every element of an ternary semigroup \(\mathcal{S}_{3}\) is ternary invertible, then \(\mathcal{S}_{3}\) is an ternary group \(\mathsf{G}^{\left[ 3\right] }=\langle G\mid\mu^{\left[ 3\right] }\mid assoc\rangle\), where \(G\) is the underlying set. The identity and neutral element (necessary in binary groups) are not required for a ternary group \(\mathsf{G}^{\left[ 3\right] }\).

Structures with two operations are ring-like. A generic \((m,n)\)-ring, \(\mathcal{R}^{\left[ m,n\right] }=\langle R\mid\nu^{\left[ m\right] },\mu^{\left[ n\right] }\rangle\) consists of the underlying set \(R\) with an \(m\)-ary addition \(\nu^{\left[ m\right] }:R^{m}\rightarrow R\) and an \(n\)-ary multiplication \(\mu^{\left[ n\right] }:R^{n}\rightarrow R\), such that \(\langle R\mid\nu^{\left[ m\right] }\mid assoc\mid comm\rangle\) is an \(m\)-ary commutative group and \(\langle R\mid\mu^{\left[ n\right] }\mid assoc\rangle\) is an \(n\)-ary semigroup, linked by \(n\)-ary distributivity relations (for all positions of the sum in the [9]. The \((m,n)\)-ring \(\mathcal{R}^{\left[ m,n\right] }\) is ternary derived, if both operations are composed from ternary operations (1 ). For further details, see [6].

1.2 Ternary group rings↩︎

The binary group rings were studied in [10]–[12], while \(\left( m,n\right)\)-group rings were introduced and investigated in [7]. We here remind in brief definitions to fix notations.

Let us consider \(\left( m_{r},n_{r}\right)\)-ring \(\mathcal{R}^{\left[ m_{r},n_{r}\right] }\) and \(n_{g}\)-ary group \(\mathsf{G}^{\left[ n_{g}\right] }\), then the general \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\) is denoted by\[\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }=\mathcal{R}^{\left[ n_{r},m_{r}\right] }\left[ \mathsf{G}^{\left[ \mathsf{n}_{g}\right] }\right] ,\label{rm}\tag{9}\] and is the 3 set and 6 operation algebra-like structure\[\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }=\left\langle {\mathrm{\small \upshape R}},R,G\mid\boldsymbol{\nu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{m}_{r}\right] },\boldsymbol{\mu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{n}_{r}\right] },\boldsymbol{\rho}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{k}_{\rho}\right] }\mid\nu _{R}^{\left[ m_{r}\right] },\mu_{R}^{\left[ n_{r}\right] }\mid \mu_{\mathsf{G}}^{\left[ \mathsf{n}_{g}\right] }\right\rangle ,\label{rmn}\tag{10}\] where \(\mu_{\mathsf{G}}^{\left[ \mathsf{n}_{g}\right] }\) is \(\mathsf{n}_{g}\)-ary multiplication of the initial group \(\mathsf{G}^{\left[ n_{g}\right] }\) (with the underlying set \(G\)), \(\nu_{R}^{\left[ m_{r}\right] }\) and \(\mu_{R}^{\left[ n_{r}\right] }\) are \(m_{r}\)-ary addition and \(n_{r}\)-ary multiplication of the initial \(\left( m_{r},n_{r}\right)\)-ring \(\mathcal{R}^{\left[ n_{r},m_{r}\right] }\) (with the underlying set \(R\)), \(\boldsymbol{\nu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{m}_{r}\right] }\), \(\boldsymbol{\mu }_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{n}_{r}\right] }\) and \(\boldsymbol{\rho }_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{k}_{\rho}\right] }\) are \(\boldsymbol{\nu }_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{m}_{r}\right] }\)-ary addition, \(\boldsymbol{\mu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{n}_{r}\right] }\)-ary multiplication and \(\boldsymbol{k}_{\rho}\)-ary multiplication on scalar of the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\) (with the underlying set \({\mathrm{\small \upshape R}}\)). The \(\alpha\)th element \(\boldsymbol{r}\left( \alpha\right)\) of the group ring (or the element of its underlying set \({\mathrm{\small \upshape R}}\)) can be written as the formal sum (analog of the standard binary case \(\sum _{\mathsf{g}\in\mathsf{G}}r_{\mathsf{g}}\bullet\mathsf{g},\;\;\;r_{\mathsf{g}}\in R,\;\;\mathsf{g}\in G\))\[\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \alpha\right) =\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow {r}_{\overrightarrow{\mathsf{g}}}\left( \alpha\right) ,\overrightarrow {\mathsf{g}}\right) =\underset{i}{\boldsymbol{\Sigma}}\;r_{\mathsf{g}_{i}}\left( \alpha\right) \bullet\mathsf{g}_{i},\;\;\;\;r_{\mathsf{g}_{i}}\left( \alpha\right) \in R,\;\;\mathsf{g}_{i}\in G,\;\;\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \alpha\right) \in{\mathrm{\small \upshape R}},\label{ra}\tag{11}\] where \(r_{\mathsf{g}_{i}}\left( \alpha\right)\) are ring elements and \(\mathsf{g}_{i}\) are group elements. For conciseness, we denote the ordered families \(\overrightarrow{r}_{\overrightarrow{\mathsf{g}}}\left( \alpha\right) =\left( r_{\mathsf{g}_{1}}\left( \alpha\right) ,r_{\mathsf{g}_{2}}\left( \alpha\right) ,\ldots\right)\), \(\overrightarrow {\mathsf{g}}=\left( \mathsf{g}_{1},\mathsf{g}_{2},\ldots\right)\) as “formal vectors”. In case the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring (9 ) is finite, \(\alpha=1,2\ldots\left\vert {\mathrm{\small \upshape R}}\right\vert\).

In the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\), the \(\boldsymbol{m}_{r}\)-ary addition \(\boldsymbol{\nu}^{\left[ \boldsymbol{m}_{r}\right] }\) is defined by analogy with the binary case, \(\left( \sum_{\mathsf{g}\in\mathsf{G}}r_{\mathsf{g}}\bullet\mathsf{g}\right) +\left( \sum _{\mathsf{g}\in\mathsf{G}}r_{\mathsf{g}}^{\prime}\bullet\mathsf{g}\right) =\left( \sum_{\mathsf{g}\in\mathsf{G}}\left( r_{\mathsf{g}}+r_{\mathsf{g}}^{\prime}\right) \bullet\mathsf{g}\right)\), \(r_{\mathsf{g}}\in R\), \(\mathsf{g},\mathsf{h}\in G\), left-” componentwise”\[\begin{align} \boldsymbol{\nu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{m}_{r}\right] }\left[ \scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow{r}_{\overrightarrow{\mathsf{g}}}\left( \alpha_{1}\right) ,\overrightarrow{\mathsf{g}}\right) ,\ldots,\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow{r}_{\mathsf{\vec{g}}}\left( \alpha_{\boldsymbol{m}_{r}}\right) ,\overrightarrow{\mathsf{g}}\right) \right] & =\underset {i}{\boldsymbol{\Sigma}}\;\nu_{R}^{\left[ m_{r}\right] }\left[ r_{\mathsf{g}_{i}}\left( \alpha_{1}\right) ,\ldots,r_{\mathsf{g}_{i}}\left( \alpha_{m_{r}}\right) \right] \bullet\mathsf{g}_{i},\label{mm}\\ r_{\mathsf{g}_{i}}\left( \alpha_{1,\ldots,\boldsymbol{m}_{r}}\right) & \in R,\;\;\mathsf{g}_{i}\in G,\;\;\;\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow {r}_{\overrightarrow{\mathsf{g}}}\left( \alpha_{1,\ldots,\boldsymbol{m}_{r}}\right) ,\overrightarrow{\mathsf{g}}\right) \in{\mathrm{\small \upshape R}}.\nonumber \end{align}\tag{12}\]

It follows from (12 ), that the arity of addition in the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\) coincides with the arity of addition in the initial \(\left( m_{r},n_{r}\right)\)-ring \(\mathcal{R}^{\left[ n_{r},m_{r}\right] }\) that is\[\boldsymbol{m}_{r}=m_{r}.\label{mrm}\tag{13}\]

If the \(m_{r}\)-ary addition in the initial \(\left( m_{r},n_{r}\right)\)-ring \(\mathcal{R}^{\left[ m_{r},n_{r}\right] }\) is of higher ternary power \(\left( \nu_{R}^{\left[ 3\right] }\right) ^{\circ\ell_{m}}\), see (1 ), then instead of (13 ), we have the ” quantization”\[\boldsymbol{m}_{r}=2\ell_{m}+1, \label{mlm}\tag{14}\] which directly follows from the general form of admissible length of words (3 ).

We define the multiplication in the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\) (9 ) similarly to the binary case, \(\left( \sum_{\mathsf{g}\in G}r_{\mathsf{g}}\bullet\mathsf{g}\right) \left( \sum_{\mathsf{h}\in G}r_{\mathsf{h}}^{\prime}\bullet\mathsf{h}\right) =\left( \sum_{\mathsf{g}\in G}\sum_{\mathsf{h}\in G}\left( r_{\mathsf{g}}r_{\mathsf{h}}^{\prime}\right) \bullet\left( \mathsf{gh}\right) \right)\), \(r_{\mathsf{g}}\in R\), \(\mathsf{g},\mathsf{h}\in G\), by the both-sides “componentwise”convolution-like operation\[\begin{align} & \boldsymbol{\mu}_{{\mathrm{\small \upshape R}}}^{\left[ \boldsymbol{n}_{r}\right] }\left[ \scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow{r}_{\overrightarrow{\mathsf{g}}}\left( \alpha_{1}\right) ,\overrightarrow{\mathsf{g}}\right) ,\ldots,\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow{r}_{\overrightarrow{\mathsf{g}}}\left( \alpha _{\boldsymbol{n}_{r}}\right) ,\overrightarrow{\mathsf{g}}\right) \right] \nonumber\\ & =\underset{i_{1}}{\boldsymbol{\Sigma}}\;\ldots\;\underset{i_{n_{r}}}{\boldsymbol{\Sigma\;}}\underset{j_{1}}{\boldsymbol{\Sigma}}\;\ldots\;\underset {j_{\mathsf{n}_{g}}}{\boldsymbol{\Sigma}}\mu_{R}^{\left[ n_{r}\right] }\left[ r_{\mathsf{g}_{i_{1}}}\left( \alpha_{1}\right) ,\ldots,r_{\mathsf{g}_{i_{n_{r}}}}\left( \alpha_{n_{r}}\right) \right] \bullet\mu_{\mathsf{G}}^{\left[ \mathsf{n}_{g}\right] }\left[ \mathsf{g}_{j_{1}},\ldots ,\mathsf{g}_{j_{\mathsf{n}_{g}}}\right] ,\label{mng}\\ & r_{\mathsf{g}_{i}}\left( \alpha_{j}\right) \in R,\;\;\mathsf{g}_{i}\in G,\;\;\;\scalebox{1.15}{\boldsymbol{\mathrm{r}}}\left( \overrightarrow{r}_{\overrightarrow{\mathsf{g}}}\left( \alpha_{j}\right) ,\overrightarrow{\mathsf{g}}\right) \in{\mathrm{\small \upshape R}},\nonumber \end{align}\tag{15}\] if ternary power of all operations is \(1\), such that the arities of all multiplications defined in (15 ) coincide\[\boldsymbol{n}_{r}=n_{r}=\mathsf{n}_{g}.\label{nnn}\tag{16}\]

In case, the ternary powers of multiplications in the initial ternary ring \(\mu_{R}^{\left[ n_{r}\right] }\) and the \(\mathsf{n}_{g}\)-ary group \(\mu_{\mathsf{G}}^{\left[ \mathsf{n}_{g}\right] }\), denoted by \(\ell_{n}\) and \(\ell_{g}\), correspondingly, differ from \(1\), (see (4 )), then we have for the arity of multiplication \(\boldsymbol{n}_{r}\) in the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\), instead of (16 ),\[\boldsymbol{n}_{r}=2\ell_{n}+1=2\ell_{g}+1, \label{nln}\tag{17}\] which follows from the construction (15 ) and the ” quantization”condition (3 ).

It is well known (see, e.g., [6]) that the invertibility properties of polyadic structures are governed not by neutral elements (7 ), but by querelements (8 ). Let us consider the subset \(\emph{U}_{R}\subset R\) having the querelement \(\bar{r}\), so such elements form a multiplicative \({\mathrm{\small \upshape n}}_{r}\)-ary unit group \(\mathscr U^{\left[ n_{r}\right] }\left[ \mathcal{R}^{\left[ m_{r},n_{r}\right] }\right]\) ), while the \(n_{g}\)-ary group \(\mathsf{G}^{\left[ \mathsf{n}_{g}\right] }\) having (by its definition) the querelement \(\mathsf{\bar{g}}\) (8 ) for each \(\mathsf{g}\in\mathsf{G}\). Then in the group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\) the querelement \(\overline{\scalebox{1.15}{\boldsymbol{\mathrm{r}}}}\left( \alpha\right)\) for some elements is defined by\[\overline{\scalebox{1.15}{\boldsymbol{\mathrm{r}}}}\left( \alpha\right) =\underset{i}{\boldsymbol{\Sigma}}\;\bar{r}_{\mathsf{g}_{i}}\left( \alpha\right) \bullet\mathsf{\bar{g}}_{i},\;\;\;\;\;\;\;\bar{r}\in\emph{U},\;\;\mathsf{\bar{g}}_{i},\mathsf{g}_{i}\in G,\;\;\overline{\scalebox{1.15}{\boldsymbol{\mathrm{r}}}}\in{\mathrm{\small \upshape R}},\label{rq}\tag{18}\] and we call them multiplicatively invertible elements (sometimes called group units), denote their subset \(\emph{U}_{{\mathrm{\small \upshape R}}}\in{\mathrm{\small \upshape R}}\), which forms the \(\boldsymbol{n}_{r}\)-ary unit group \(\mathscr U^{\left[ \boldsymbol{n}_{r}\right] }\left( \scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\right)\) of the \(\left( \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right)\)-group ring \(\scalebox{1.15}{\boldsymbol{\mathrm{R}}}^{\left[ \boldsymbol{m}_{r},\boldsymbol{n}_{r}\right] }\).

The concept of ternary unit group is extremely important for cryptosystems, because the decoding procedure can be made only by group units which correspond to the binary multiplicatively invertible elements of (ternary) rings and group rings.

1.3 Matrix ternarization↩︎

The structure of binary rings is well known (see, e.g., [13]–[15]): a simple ring (having no nontrivial two-sided ideals) is isomorphic to the full matrix ring \(Mat^{full}\left( \mathcal{D}\right)\) over a division ring \(\mathcal{D}\) (each nonzero element has inverse, or field, if \(\mathcal{D}\) is commutative). For the simple \(\left( 2,n\right)\)-ring with binary addition and \(n\)-ary multiplication \(\mu^{\left[ n\right] }\) (as ordinary product of matrices) the structure of \(\mathcal{R}_{simple}^{\left[ 2,n\right] }\) was obtained in [16]. In the ternary case it is isomorphic to the antidiagonal \(2\times2\) matrices with blocks \(Mat^{full}\left( \mathcal{D}\right)\) of the form\[\boldsymbol{M}_{adiag}^{\left[ 3\right] }\left( \mathcal{D}\right) =\left( \begin{array} [c]{cc}0 & Mat^{full}\left( \mathcal{D}\right) \\ Mat^{full}\left( \mathcal{D}\right) & 0 \end{array} \right) . \label{mb}\tag{19}\] The matrices of the form (19 ) are not closed with respect to the binary product, but closed with respect to exactly ternary multiplication and binary addition, and therefore we call them ternary matrices. Note that the ternary multiplication \(\mu^{\left[ 3\right] }\) is nonderived, because it cannot be directly iterated (1 ) (without using additional homomorphisms, as in [17]–[19]) from binary product which is not defined (not closed) for the ternary matrices \(\boldsymbol{M}_{adiag}^{\left[ 3\right] }\left( \mathcal{D}\right)\).

Moreover, a finite-dimensional simple associative ternary algebra \(\mathcal{A}^{\left( 3\right) }\) over an algebraically closed field \(\mathcal{F}\) is isomorphic to the ternary matrix (19 ) \(\boldsymbol{M}_{adiag}^{\left[ 3\right] }\left( \mathcal{F}\right)\) [20].

Thus, going in the opposite way, we propose the matrix ternarization procedure for binary algebraic structures, firstly introduced in [8] for general case, which can be considered as immediate and nonderived ternary generalization of the known ordinary (binary) cryptosystems.

Let \(B\) be a binary structure, for instance, semigroup, group, ring, group ring. We defined the matrix ternarization as the mapping of the binary structure \(B\) to the symbolic matrix\[\boldsymbol{\Phi}_{\boldsymbol{M}}:B\rightarrow\widehat{\boldsymbol{M}}^{\left[ 3\right] }\left( B\right) , \label{f}\tag{20}\] where\[\widehat{\boldsymbol{M}}^{\left[ 3\right] }\left( B\right) \equiv \widehat{\boldsymbol{M}}_{adiag}^{\left[ 3\right] }\left( B\right) =\left( \begin{array} [c]{cc}0 & B\\ B & 0 \end{array} \right) . \label{mbb}\tag{21}\] In this way, the multiplicative arity \(n_{B}\) of \(B\) can be also \(3\), because matrices of the form (19 ) are closed with respect to ternary multiplication. The additive properties of \(B\) are transferred to the symbolic matrix addition without restrictions because of linearity.

The symbolic matrix construction of the ternary group ring from a binary group ring \(\mathcal{R}\left[ \mathsf{G}\right]\) is more nontrivial: we should use two symbolic matrices, one for the ring \(\mathcal{R}\) and another for the group \(\mathsf{G}\). If their sizes coincide, the resulting matrix will have the same size, and all the arities coincide. In this case \(\boldsymbol{\Phi }_{\boldsymbol{M}}\) (20 ) is a homomorphism-like mapping, because using (17 ) \[\widehat{\boldsymbol{M}}^{\left[ \boldsymbol{3}\right] }\left( \mathcal{R}\left[ \mathsf{G}\right] \right) =\widehat{\boldsymbol{M}}^{\left[ 3\right] }\left( \mathcal{R}\right) \left[ \widehat{\boldsymbol{M}}^{\left[ 3\right] }\left( \mathsf{G}\right) \right] , \label{mrg}\tag{22}\] which follows from the sum of formal products \(\sum_{\mathsf{g}\in\mathsf{G}}r_{\mathsf{g}}\bullet\mathsf{g}\) (11 ).

2 Public-key cryptosystems↩︎

Public-key cryptosystems revolutionized cryptography by eliminating the need for a shared secret key, instead utilizing a pair of mathematically linked keys—a public key for encryption and a private key for decryption. We first begin by introducing the foundational algorithm that made this paradigm possible, the ElGamal cryptosystem [3] which is based directly on Diffie-Hellman ideas [1], but extends them into a full asymmetric encryption scheme and a digital signature variant. Unlike RSA [2], which relies on the difficulty of integer factorization, ElGamal’s security is grounded in the computational infeasibility of the Discrete Logarithm Problem in cyclic groups [21], [22], most commonly implemented over prime fields or elliptic curves [23]. This probabilistic encryption mechanism produces distinct ciphertexts for the same plaintext each time it is run—a property that provides semantic security against chosen-plaintext attacks when used with appropriate group parameters [4]. The algorithm’s influence persists today in standards such as the Digital Signature Algorithm (DSA) [5], which is directly derived from ElGamal signatures [24], and in many privacy-preserving cryptographic protocols [25]. There are many extensions and improvements of the ElGamal cryptosystem, see, e.g., [26], [27], but all of them used only binary algebraic structures. Here we generalize the ElGamal cryptosystem to ternary algebraic structures (see, e.g., [6]), while further its extensions can be provided similarly to the binary case.

2.1 Binary cryptosystem↩︎

Let us formulate the abstract universal scheme of the binary cryptosystem (which is for cyclic binary groups indeed the ElGamal one [3]) in a unified way for binary groups \(\mathsf{G}\), rings \(\mathcal{R}\) and group rings \(\mathcal{R}[\mathsf{G}]\), and denote all them as \(\mathscr M\) with the underlying set \(M\). Their corresponding unit group is \(\mathscr U\left( \mathscr M\right)\) with the underlying set \(U_{M}\) of multiplicatively invertible elements. For \(u\in U_{M}\) and positive integer \(n\geq0\), \(u^{n}\) denotes \(\left( n-1\right)\)-fold composed binary multiplication (for cyclic unit groups using \(\operatorname{mod}\)), we call it \(n\)-power, and negative exponents are defined via the inverse \(u^{-1}\in U_{M}\).

The general idea of the binary cryptosystem is to transfer a secret message \(\textcolor{red}{\texttt{\upshape m}}\in M\) from Sender to Recipient by exchanging public keys and cipher texts. The private (secret) key \(\textcolor{red}{\texttt{\upshape x}}\) is known to Recipient only, who cannot send it openly to Sender, but Recipient can send openly some function \(h=f\left( \textcolor{red}{\texttt{\upshape x}}\right)\) which is difficult to solve (e.g., for cyclic groups this is the Discrete Logarithm Problem [21], [22]). Then Sender prepares as minimum two cipher texts (to make cancellation), such that the private key \(\textcolor{red}{\texttt{\upshape x}}\) dependence can be cancelled out to give the searched for secret message \(\textcolor{red}{\texttt{\upshape m}}\). The simplest choice for the function \(f\left( \textcolor{red}{\texttt{\upshape x}}\right)\) is the exponential function, because the ratio of two ciphers can be independent of \(\textcolor{red}{\texttt{\upshape x}}\).

The general algorithm of the binary cryptosystem consists of three steps:

1. Key generation by Recipient.

   • Recipient takes an invertible (group unit) \(u\in U_{M}\) which is a public parameter, since \(\mathscr M=\mathsf{G},\mathcal{R},\mathcal{R}[\mathsf{G}]\) is known to all.

   • Using his own private integer key \(\textcolor{red}{\texttt{\upshape x}}\), Recipient prepares the public key using \(\textcolor{red}{\texttt{\upshape x}}\)-powers\[h=u^{\textcolor{red}{\texttt{\upshape x}}},\;\;\;\;\;\;\;\;h,u\in U_{M} \label{h}\tag{23}\] and sends the pair \(\left( u,h\right)\) to Sender.

2. Encryption by Sender.

   • Having the public key \(h\) and the secret message \(\textcolor{red}{\texttt{\upshape m}}\), for the first glance Sender can multiply them, but this will not hide \(\textcolor{red}{\texttt{\upshape m}}\), the function should be more complicated.

   • Sender chooses an additional parameter really to hide \(\textcolor{red}{\texttt{\upshape m}}\), a random element (or ephemeral key), integer \(\textcolor{red}{\texttt{\upshape k}}\), that is not known to Recipient and exists locally, just for the purposes of randomly encrypting a single secret message \(\textcolor{red}{\texttt{\upshape m}}\).

   • Then Sender creates the following two-dimensional cipher text by using \(\textcolor{red}{\texttt{\upshape k}}\)-powers\[\begin{align} c_{1} & =u^{\textcolor{red}{\texttt{\upshape k}}},\tag{24}\\ c_{2} & =\textcolor{red}{\texttt{\upshape m}}\cdot h^{\textcolor{red}{\texttt{\upshape k}}}, \tag{25} \end{align}\] where \(\left( \cdot\right)\) is multiplication in the unit group \(\mathscr U\left( \mathscr M\right)\). Different random choices of \(\textcolor{red}{\texttt{\upshape k}}\) give different two-dimensional ciphers \(\left( c_{1},c_{2}\right)\) which correspond to the same secret message \(\textcolor{red}{\texttt{\upshape m}}\).

   • Sender sends the pair \(\left( c_{1},c_{2}\right)\) to Recipient (but not the random element \(\textcolor{red}{\texttt{\upshape k}}\)).

3. Decryption by Recipient.

   • Recipient checks that the both parameters of two-dimension cipher (24 )–(25 ) are in the unit group \(\left( c_{1},c_{2}\right) \in U_{M}\times U_{M}\) .

   • Then Recipient uses his private key \(\textcolor{red}{\texttt{\upshape x}}\) and creates the \(\textcolor{red}{\texttt{\upshape x}}\)-power of the first cipher component \(\left( c_{1}\right) ^{\textcolor{red}{\texttt{\upshape x}}}\), and check that it is in \(U_{M}\) as well. The main idea of the procedure is the \(\textcolor{red}{\texttt{\upshape x}}\)-power (\(\textcolor{red}{\texttt{\upshape x}}\) is known to Recipient only) of the public cipher is derived through the public parameter \(h\), because, see (23 ) and (24 ),\[\left( c_{1}\right) ^{\textcolor{red}{\texttt{\upshape x}}}=\left( u^{\textcolor{red}{\texttt{\upshape k}}}\right) ^{\textcolor{red}{\texttt{\upshape x}}}=\left( u^{\textcolor{red}{\texttt{\upshape x}}}\right) ^{\textcolor{red}{\texttt{\upshape k}}}=h^{\textcolor{red}{\texttt{\upshape k}}}. \label{cu}\tag{26}\]

   • The fraction \(\textcolor{red}{\texttt{\upshape m}}^{\prime}=c_{2}\left( c_{1}\right) ^{-\textcolor{red}{\texttt{\upshape x}}}\) (which exists, because \(c_{1}\in U_{M}\)) immediately gives the sought for secret message \(\textcolor{red}{\texttt{\upshape m}}\). Indeed, using (23 ) and (24 )–(25 ), but without knowledge of the random element \(\textcolor{red}{\texttt{\upshape k}}\), Recipient obtains\[\textcolor{red}{\texttt{\upshape m}}^{\prime}=c_{2}\left( c_{1}\right) ^{-\textcolor{red}{\texttt{\upshape x}}}=\textcolor{red}{\texttt{\upshape m}}\cdot h^{\textcolor{red}{\texttt{\upshape k}}}\cdot\left( u^{\textcolor{red}{\texttt{\upshape k}}}\right) ^{-\textcolor{red}{\texttt{\upshape x}}}=\textcolor{red}{\texttt{\upshape m}}\cdot\left( u^{\textcolor{red}{\texttt{\upshape x}}}\right) ^{\textcolor{red}{\texttt{\upshape k}}}\cdot\left( u^{\textcolor{red}{\texttt{\upshape k}}}\right) ^{-\textcolor{red}{\texttt{\upshape x}}}\equiv\textcolor{red}{\texttt{\upshape m}}. \label{cc}\tag{27}\]

     Note, that Attacker, the passive eavesdropper, can see all messages sent over the public channel—the public parameters \(u\), \(\mathscr U\left( \mathscr M\right)\), the Recipient’s public key \(h\), and the Sender’s ciphertext \(\left( c_{1},c_{2}\right)\)—but cannot decrypt and get \(\textcolor{red}{\texttt{\upshape m}}\), because Attacker does not know the Recipient’s private key \(\textcolor{red}{\texttt{\upshape x}}\). Trying to solve the public key definition (23 ) (to get the private key \(\textcolor{red}{\texttt{\upshape x}}\)) is the nontrivial Discrete Logarithm Problem [21], [22] and is related to the Diffie-Hellman problem [1].

In the classical binary cryptosystem algorithms, the unit group \(\mathscr U\left( \mathscr M\right)\) is assumed cyclic for several reasons: 1) Uniform message space. Every group element can be represented as \(u^{n}\) for some exponent \(n\). 2) Hardness of the Discrete Logarithm Problem [21], [22], such that the best known algorithms assume knowledge of the group order and its structure, while cyclic groups of prime order are optimal. 3) During key generation, any element of group can be a public key \(h=u^{x}\).

However, these are cryptographic conveniences, not algebraic necessities. Moreover, the main relation (27 ) of the above algorithm demands \(\left( u^{\textcolor{red}{\texttt{\upshape x}}}\right) ^{\textcolor{red}{\texttt{\upshape k}}}\cdot\left( u^{\textcolor{red}{\texttt{\upshape k}}}\right) ^{-\textcolor{red}{\texttt{\upshape x}}}=u^{\textcolor{red}{\texttt{\upshape xk}}}\cdot u^{-\textcolor{red}{\texttt{\upshape kx}}}=1\), \(u\in U_{M}\). This relation holds valid as for noncyclic groups, and even for nonabelian (noncommutative) groups, since in both cases integer powers \(r,s\in \mathbb{N}_{0}\) of the same element commute \(u^{r}{}^{s}=u^{sr}\). The only algebraic requirement is that \(u\) belongs to the unit group \(U_{M}\) (so that inverses \(u^{-1}\in U_{M}\) exist) and that multiplication in the unit group \(\mathscr U\left( \mathscr M\right)\) is associative (for more complicated nonassociative cryptosystem algorithms, see, e.g., [28]).

2.2 Ternary cryptosystem↩︎

Now we propose a fundamentally novel approach to cryptosystems, which is based on the new structures, ternary algebraic structures, see, e.g., [6]. Here we present the abstract universal scheme of the ternary cryptosystem generalizing the cyclic binary one [3], [29] in a unified way for ternary groups \(\mathsf{G}^{\left[ 3\right] }\), rings with ternary product \(\mathcal{R}^{\left[ m,3\right] }\) and group rings \(\mathcal{R}^{\left[ m,3\right] }[\mathsf{G}^{\left[ 3\right] }]\), and denote all them as \(\mathscr M^{\left[ 3\right] }\) with the underlying set \(M\). Their corresponding unit group is \(\mathscr U\left( \mathscr M^{\left[ 3\right] }\right)\) with the underlying set \(U_{M}\) of multiplicatively ternary invertible elements, such that each element has its querelement (8 ). For \(u\in U_{M}\) and positive integer \(\ell\geq0\), \(u^{2\ell+1}\) denotes \(\ell\)-fold composed ternary multiplication (for cyclic unit groups using \(\operatorname{mod}\)), we call it \(\ell\)-ternary power (4 ). For numbers we omit the symbol of product \(\mu\). The negative exponents of binary case correspond now to querelements \(u^{-1}\rightarrow Q\left( u\right) \in U_{M}\) (8 ).

If we use the matrix ternarization, \(\mathscr U\left( \mathscr B\right) \rightarrow\widehat{\mathscr U}\left( \mathscr M^{\left[ 3\right] }\right)\) and for the underlying set \(U_{M}\rightarrow\widehat{U_{M}}\) (20 ), to obtain nonderived ternary groups, the underlying set of the unit ternary group \(U_{M}\) consists of the symbolic matrices of the form (21 ), such that each element of the matrix ternary unit group \(\widehat{\mathscr U}\left( \mathscr M^{\left[ 3\right] }\right)\) is effectively a pair \(\widehat {u}=\left( u_{\left( 1\right) },u_{\left( 2\right) }\right)\), because\[\widehat{u}=\left( \begin{array} [c]{cc}0 & u_{\left( 1\right) }\\ u_{\left( 2\right) } & 0 \end{array} \right) . \label{u}\tag{29}\]

The goal of the ternary cryptosystem remains the same: to send from Sender to Recipient the message \(\textcolor{red}{\texttt{\upshape m}}\) securely by exchanging public keys and cipher texts only. The difference is in more complicates structures of products, because they can be only of admissible length (3 ). This make much more difficult to attack and find solutions without knowledge of the private key which is now \(\ell_{\textcolor{red}{\texttt{\upshape x}}}\)-ternary power.

The ternary cryptosystem algorithm consists of three steps:

1. Key generation by Recipient.

   • Recipient takes an invertible, i.e. having querelement (8 ) (group unit) \(\widehat{u}\in\widehat{U}_{M}\) which is a public parameter (pair), since \(\mathscr M=\mathsf{G}^{\left[ 3\right] },\mathcal{R}^{\left[ m,3\right] },\mathcal{R}^{\left[ m,3\right] }[\mathsf{G}^{\left[ 3\right] }]\) is public, i.e. known to all.

   • Recipient uses the private secret integer key \(\ell _{\textcolor{red}{\texttt{\upshape x}}}\) and prepares the public key using \(\ell_{\textcolor{red}{\texttt{\upshape x}}}\)-ternary powers (below we omit the product symbol \(\mu\))\[\widehat{h}=\mu^{\left[ 3\right] \circ\ell_{\textcolor{red}{\texttt{\upshape x}}}}\left[ \widehat {u}^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right] \equiv\left[ \widehat{u}^{2\ell _{\textcolor{red}{\texttt{\upshape x}}}+1}\right] _{\times},\;\;\;\;\;\;\;\;\widehat{h},\widehat{u}\in\widehat{U}_{M}. \label{hh}\tag{30}\] where \(\left[ \;\right] _{\times}\) is ternary multiplication (and its ternary powers) in the matrix ternary unit group \(\widehat{\mathscr U}\left( \mathscr M\right)\).

   • Recipient sends openly the public pair \(\left( \widehat {u},\widehat{h}\right)\) (effectively 4 parameters) to Sender.

2. Encryption by Sender.

   • Sender chooses a secret message \(\textcolor{red}{\texttt{\upshape m}}\). To hide \(\textcolor{red}{\texttt{\upshape m}}\) he introduce an additional parameter, a random element (or ephemeral key), integer \(\ell_{\textcolor{red}{\texttt{\upshape k}}}\), that is not known to Recipient and exists locally, just for the purposes of randomly encrypting a single secret message \(\textcolor{red}{\texttt{\upshape m}}\).

   • Then Sender creates the following two-dimensional cipher text by using \(\ell_{\textcolor{red}{\texttt{\upshape k}}}\)-ternary powers\[\begin{align} \widehat{c}_{1} & =\left[ \widehat{u}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times},\tag{31}\\ \widehat{c}_{2} & =\left[ \widehat{\textcolor{red}{\texttt{\upshape m}}},\widehat{h}^{2\ell _{\textcolor{red}{\texttt{\upshape k}}}+1},\widehat{h}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times}, \tag{32} \end{align}\] Different random choices of \(\ell_{\textcolor{red}{\texttt{\upshape k}}}\) give different two-dimensional ciphers \(\left( \widehat{c}_{1},\widehat{c}_{2}\right)\) which correspond to the same secret message \(\textcolor{red}{\texttt{\upshape m}}\).

   • Sender sends the pair \(\left( \widehat{c}_{1},\widehat{c}_{2}\right)\) to Recipient (without the random element \(\ell_{\textcolor{red}{\texttt{\upshape k}}}\)).

3. Decryption by Recipient.

   • Recipient checks that the both parameters of two-dimension cipher (24 )–(25 ) are in the ternary unit group \(\left( \widehat{c}_{1},\widehat{c}_{2}\right) \in\widehat{U}_{M}\times\widehat {U}_{M}\) .

   • Then Recipient uses his private key \(\ell_{\textcolor{red}{\texttt{\upshape x}}}\) and creates the \(\ell_{\textcolor{red}{\texttt{\upshape x}}}\)-ternary power of the first cipher component \(\left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\), and check that it is in \(\widehat{U}_{M}\) as well. Then, he obtains\[\left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}=\left[ \left( \left[ \widehat{u}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right] _{\times}=\left[ \left( \left[ \widehat {u}^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right] _{\times}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times}=\left[ \widehat{h}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times}, \label{ch}\tag{33}\] which is the main idea of the ternary procedure: the \(\ell_{\textcolor{red}{\texttt{\upshape x}}}\)-ternary power of the first cipher is expressed through the public \(\widehat{h}\), as in (26 ).

   • In the matrix ternary unit group \(\widehat{\mathscr U}\left( \mathscr M\right)\) the role of inverse plays the querelement (8 ). We compute, using (33 )\[Q\left( \left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right) =\left[ \widehat{h}^{-\left( 2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1\right) }\right] _{\times}, \label{qc}\tag{34}\] where on the r.h.s. there are usual matrix inverses.

   • Now we construct the ternary analog of the fraction \(c_{2}c_{1}^{-\textcolor{red}{\texttt{\upshape x}}}\) (27 ) as\[\widehat{\textcolor{red}{\texttt{\upshape m}}}^{\prime}=\left[ \widehat{c}_{2},Q\left( \left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right) ,Q\left( \left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right) \right] _{\times}, \label{c21}\tag{35}\] which should give the secret message \(\widehat{\textcolor{red}{\texttt{\upshape m}}}\). Indeed, even without knowledge of the random element \(\ell_{\textcolor{red}{\texttt{\upshape k}}}\), using (30 ), (34 ), (35 ) and (31 )–(32 ), Recipient derives\[\begin{align} \widehat{\textcolor{red}{\texttt{\upshape m}}}^{\prime} & =\left[ \widehat{c}_{2},Q\left( \left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right) ,Q\left( \left( \widehat{c}_{1}\right) ^{2\ell_{\textcolor{red}{\texttt{\upshape x}}}+1}\right) \right] _{\times }\nonumber\\ & =\left[ \left[ \widehat{\textcolor{red}{\texttt{\upshape m}}},\widehat{h}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1},\widehat{h}^{2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1}\right] _{\times},\left[ \widehat {h}^{-\left( 2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1\right) }\right] _{\times},\left[ \widehat{h}^{-\left( 2\ell_{\textcolor{red}{\texttt{\upshape k}}}+1\right) }\right] _{\times}\right] _{\times}=\widehat{\textcolor{red}{\texttt{\upshape m}}}. \label{m1} \end{align}\tag{36}\]

The first difference from the binary case is the nontrivial ternary formulas, which can be not known to Attacker. Second, one matrix ternary encryption/decryption procedure can transfer double information, because the plaintext (secret messages) consists of pairs \(\widehat{\textcolor{red}{\texttt{\upshape m}}}=\left( \textcolor{red}{\texttt{\upshape m}}_{\left( 1\right) },\textcolor{red}{\texttt{\upshape m}}_{\left( 2\right) }\right)\), see (29 ).

If we do not use the matrix ternarization (20 ), all the above formulas hold valid for the ternary unit group \(\mathscr U\left( \mathscr M^{\left[ 3\right] }\right)\), where \(\mathscr M^{\left[ 3\right] }=\mathsf{G}^{\left[ 3\right] },\mathcal{R}^{\left[ m,3\right] },\mathcal{R}^{\left[ m,3\right] }[\mathsf{G}^{\left[ 3\right] }]\).

3 Examples of ternary cryptosystem↩︎

Here we first consider a ternary cryptosystem over fractions without matrix ternarization, then matrix ternarization of the finite binary group ring \(\mathbb{Z}_{5}[C_{3}]\), and finally over the finite \(\left( 6,3\right)\)-ring with \(5\) elements \(\mathbb{Z}_{5}^{\left[ 6,3\right] }\).

Now we consider the finite matrix ternary group ring, when the ring part is matrix ternarization of the finite \(\left( m,3\right)\)-ring, but not a binary ring, as in Example 3. Recall, that the finite \(\left( m,n\right)\)-ring is the set of secondary congruence classes of infinite \(\left( m,n\right)\)-ring \(\mathbb{Z}^{\left[ a,b\right] }(m,n)\) being set of representatives of the congruence class \(\left[ a\right] _{b}\), \(0\leq a\leq b-1\), \(b\in\mathbb{N}\). That is the finite \(\left( m,n\right)\)-ring \(\mathbb{Z}_{q}^{\left[ a,b\right] }(m,n)\) having \(q\) elements (of order \(q\)) is defined by \(\mathbb{Z}_{q}^{\left[ a,b\right] }(m,n)=\mathbb{Z}^{\left[ a,b\right] }(m,n)\diagup\left( bq\right) \mathbb{Z}\). The corresponding exotic \(\left( m,n\right)\)-fields have unusual properties, and can be zeroless, zeroless-nonunital or have several units, and it is even possible for all elements to be units, also there exist non-isomorphic finite fields of the same arity shape and order. None of the above is possible in the binary case (for further details, see [6], [30]).

Conclusions↩︎

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 representative family.

The familiar ElGamal three-step pattern—publish a public key derived from a private exponent, use ephemeral randomness to produce a two-part ciphertext, and cancel the randomness at decryption—admits a direct analogue in ternary algebraic settings. Replacing binary powers and inverses with \(\ell\)-ternary powers and querelements yields a coherent encryption/decryption algebra in which the decryption step recovers the plaintext by applying querelement operations to the first ciphertext component and combining the result with the second component.

Matrix ternarization provides a practical instantiation: mapping binary rings and group rings to antidiagonal symbolic matrices produces nonderived ternary unit groups closed under ternary multiplication while preserving additive linearity. This construction increases per-element information density (paired/plaintext vectors) and introduces additional algebraic structure that an attacker must model, which can complicate straightforward adaptations of binary discrete-log attacks.

Concrete numerical examples in the paper demonstrate correctness and illustrate important structural features such as admissible word-length quantization and cycle behaviour of ternary powers; these properties influence message-space design and parameter selection. Remaining work includes formalizing hardness assumptions for ternary analogues of discrete logarithm and Diffie-Hellman problems, developing efficient and secure implementations (including side-channel mitigations), and performing comparative performance and cryptanalytic studies versus classical and post-quantum schemes.

Overall, the ternary framework is a viable algebraic generalization that merits further cryptanalytic and implementation research before any practical deployment.

References↩︎