arXiv subjects

Stanislav Semenov

Publications and source records attributed to Stanislav Semenov.

17 recordsLinked to original sources

arXiv2025

One-way multilinear functions of the second order with linear shifts

We introduce and analyze a novel class of binary operations on finite-dimensional vector spaces over a field K, defined by second-order multilinear expressions with linear shifts. These operations generate polynomials whose degree increases linearly with each iterated application, while the number of distinct monomials grows combinatorially. We demonstrate that, despite being non-associative and non-commutative in general, these operations exhibit power associativity and internal commutativity when iterated on a single vector. This ensures that exponentiation a^n is well-defined and unambiguous. Crucially, the absence of a closed-form expression for a^n suggests a one-way property: computing a^n from a and n is efficient, while recovering n from a^n (the Discrete Iteration Problem) appears computationally hard. We propose a Diffie-Hellman-like key exchange protocol based on this principle, introducing the Algebraic Diffie-Hellman Problem (ADHP) as an underlying assumption of security. In addition to the algebraic foundations, we empirically investigate the orbit structure of these operations over finite fields, observing frequent emergence of long cycles and highly regular behavior across parameter sets. Motivated by these dynamics, we further propose a pseudorandom number generation (PRNG) strategy based on multi-element multiplication patterns. This approach empirically achieves near-maximal cycle lengths and excellent statistical uniformity, highlighting the potential of these operations for cryptographic and combinatorial applications.

math.GM↗

arXiv2025

Stratified Algebra

We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their implications for algebraic dynamics, and present concrete matrix-based models that satisfy different subsets of these axioms. Both associative and bracket-sensitive constructions are considered, with an emphasis on stratum-breaking propagation and permutation symmetry. This framework proposes a paradigm shift in the way algebraic structures are conceived: instead of enforcing uniform global rules, it introduces stratified layers with context-dependent interactions. Such a rethinking of algebraic organization allows for the modeling of systems where local consistency coexists with global asymmetry, non-associativity, and semantic transitions.

math.GM↗

arXiv2025

Invariant-Based Cryptography: Toward a General Framework

We develop a generalized framework for invariant-based cryptography by extending the use of structural identities as core cryptographic mechanisms. Starting from a previously introduced scheme where a secret is encoded via a four-point algebraic invariant over masked functional values, we broaden the approach to include multiple classes of invariant constructions. In particular, we present new symmetric schemes based on shifted polynomial roots and functional equations constrained by symmetric algebraic conditions, such as discriminants and multilinear identities. These examples illustrate how algebraic invariants -- rather than one-way functions -- can enforce structural consistency and unforgeability. We analyze the cryptographic utility of such invariants in terms of recoverability, integrity binding, and resistance to forgery, and show that these constructions achieve security levels comparable to the original oscillatory model. This work establishes a foundation for invariant-based design as a versatile and compact alternative in symmetric cryptographic protocols.

cs.CR↗

arXiv2025

Invariant Bridges Between Four Successive Points: A New Tool for Data Coding

We introduce a simple yet powerful invariant relation connecting four successive terms of a class of exponentially decaying alternating functions. Specifically, for the sequence defined by f(n) = ((1/2)^n + (-1)^n) / n, we prove that the combination [(n-2)f(n-2) + (n-3)f(n-3)] / [n f(n) + (n-1)f(n-1)] is universally equal to 4 for all integers n >= 4. This invariant bridge across four points opens new possibilities for predictive coding, data compression, and error detection. We demonstrate how the relation can be used to reconstruct missing data, verify data integrity, and reduce redundancy in data streams with minimal computational overhead. The simplicity and universality of this invariant make it a promising tool for a wide range of applications in information theory and coding systems.

math.GM↗

arXiv2025

Invariant-Based Cryptography

We propose a new symmetric cryptographic scheme based on functional invariants defined over discrete oscillatory functions with hidden parameters. The scheme encodes a secret integer through a four-point algebraic identity preserved under controlled parameterization. Security arises not from algebraic inversion but from structural coherence: the transmitted values satisfy an invariant that is computationally hard to forge or invert without knowledge of the shared secret. We develop the full analytic and modular framework, prove exact identities, define index-recovery procedures, and analyze security assumptions, including oscillator construction, hash binding, and invertibility conditions. The result is a compact, self-verifying mechanism suitable for secure authentication, parameter exchange, and lightweight communication protocols.

cs.CR↗

arXiv2025

Smooth Integer Encoding via Integral Balance

We introduce a novel method for encoding integers using smooth real-valued functions whose integral properties implicitly reflect discrete quantities. In contrast to classical representations, where the integer appears as an explicit parameter, our approach encodes the number N in the set of natural numbers through the cumulative balance of a smooth function f_N(t), constructed from localized Gaussian bumps with alternating and decaying coefficients. The total integral I(N) converges to zero as N tends to infinity, and the integer can be recovered as the minimal point of near-cancellation. This method enables continuous and differentiable representations of discrete states, supports recovery through spline-based or analytical inversion, and extends naturally to multidimensional tuples (N1, N2, ...). We analyze the structure and convergence of the encoding series, demonstrate numerical construction of the integral map I(N), and develop procedures for integer recovery via numerical inversion. The resulting framework opens a path toward embedding discrete logic within continuous optimization pipelines, machine learning architectures, and smooth symbolic computation.

math.OC↗

arXiv2025

Smooth Approximations of the Rounding Function

We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions centered at each integer, and (2) normalized weighted sums of sigmoid derivatives representing local densities. The first method approximates the step-like behavior of rounding through differences of shifted sigmoids, while the second method achieves smooth interpolation between integers via density-based weighting. Both methods converge pointwise to the classical rounding function as the sharpness parameter k tends to infinity, and allow controlled trade-offs between smoothness and approximation accuracy. We demonstrate that by restricting the summation to a small set of nearest integers, the computational cost remains low without sacrificing precision. These constructions provide fully differentiable alternatives to hard rounding, which are valuable in contexts where gradient-based methods are essential.

cs.LG↗

arXiv2025

Wave Arithmetic: A Smooth Integral Representation of Number Theory

We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this formulation, each number arises as the accumulated amplitude of a structured waveform -- an interference pattern encoded by carefully designed kernels. Arithmetic operations such as addition, multiplication, and exponentiation are realized as geometric and tensorial constructions over multidimensional integration domains. Rational numbers emerge through amplitude scaling, and negative values through sign inversion, preserving all classical arithmetic identities within a continuous and differentiable structure. This representation embeds number theory into the realm of smooth analysis, enabling new interpretations of primality, factorization, and divisibility as geometric and spectral phenomena. Beyond technical formulation, Wave Arithmetic proposes a paradigm shift: numbers as the collapsed states of harmonic processes -- analytic resonances rather than atomic symbols.

math.GM↗

arXiv2025

A Smooth Analytical Approximation of the Prime Characteristic Function

We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m < n, and to remain close to 1 otherwise. We prove that P(n) approaches 1 for prime n and P(n) is less than 1 for composite n, under appropriate limits of the smoothing parameters. The construction is fully differentiable and admits both asymptotic and finite approximations, offering a continuous surrogate for primality that is compatible with analytical, numerical, and optimization methods. We compare our approach with classical number-theoretic techniques, explore its computational aspects, and suggest potential applications in spectral analysis, machine learning, and probabilistic models of primes.

math.GM↗

arXiv2025

On the Nature of Fractal Numbers and the Classical Continuum Hypothesis (CH)

We propose a reinterpretation of the continuum grounded in the stratified structure of definability rather than classical cardinality. In this framework, a real number is not an abstract point on the number line, but an object expressible at some level Fn of a formal hierarchy. We introduce the notion of "fractal numbers" -- entities defined not within a fixed set-theoretic universe, but through layered expressibility across constructive systems. This reconceptualizes irrationality as a relative property, depending on definability depth, and replaces the binary dichotomy between countable and uncountable sets with a gradated spectrum of definability classes. We show that the classical Continuum Hypothesis loses its force in this context: between aleph_0 and c lies not a single cardinal jump, but a stratified sequence of definitional stages, each forming a countable yet irreducible approximation to the continuum. We argue that the real line should not be seen as a completed totality but as an evolving architecture of formal expressibility. We conclude with a discussion of rational invariants, the relativity of irrationality, and the emergence of a fractal metric for definitional density.

math.GM↗

arXiv2025

Fractal Origin of the Continuum: A Hypothesis on Process-Relative Definability

We propose a new constructive model of the real continuum based on the notion of fractal definability. Rather than assuming the continuum as a completed uncountable totality, we view it as the cumulative result of a vast space of stratified formal systems, each defining a countable layer of real numbers via constructive means. The union of all such definable layers across all admissible chains yields a set of continuum cardinality, yet no single system or definability path suffices to capture it in full. This leads to the Fractal Origin Hypothesis: the apparent uncountability of the real line arises not from actual infinity, but from the meta-theoretical continuity of definability itself. Our framework models the continuum as a process-relative totality, grounded in syntax and layered formal growth. We develop this idea through a formal analysis of definability hierarchies and show that the resulting universe of constructible reals is countable-by-construction (that is, each element is definable within some finite syntactic system, but no single procedure enumerates all of them uniformly) yet inaccessible to any uniform enumeration. The continuum, in this view, is not a static set but a stratified semantic horizon.

math.GM↗

arXiv2025

Axiomatic Foundations of Fractal Analysis and Fractal Number Theory

We develop an axiomatic framework for fractal analysis and fractal number theory grounded in hierarchies of definability. Central to this approach is a sequence of formal systems F_n, each corresponding to a definability level S_n contained in R of constructively accessible mathematical objects. This structure refines classical analysis by replacing uncountable global constructs with countable, syntactically constrained approximations. The axioms formalize: - A hierarchy of definability levels S_n, indexed by syntactic and ordinal complexity; - Fractal topologies and the induced notions of continuity, compactness, and differentiability; - Layered integration and differentiation with explicit convergence and definability bounds; - Arithmetic and function spaces over the stratified continuum R_{S_n}, which is a subset of R. This framework synthesizes constructive mathematics, proof-theoretic stratification, and fractal geometric intuition into a unified, finitistically structured model. Key results include the definability-based classification of real numbers (e.g., algebraic, computable, Liouville), a stratified fundamental theorem of calculus with syntactic error bounds, and compatibility with base systems such as RCA_0 and ACA_0. The framework enables constructive approximation and syntactic regularization of classical analysis, with applications to proof assistants, computable mathematics, and foundational studies of the continuum.

math.GM↗

arXiv2025

Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability

This paper develops a technical and practical reinterpretation of the real interval [a,b] under the paradigm of fractal countability. Instead of assuming the continuum as a completed uncountable totality, we model [a,b] as a layered structure of constructively definable points, indexed by a hierarchy of formal systems. We reformulate classical notions from real analysis -- continuity, measure, differentiation, and integration -- in terms of stratified definability levels S_n, thereby grounding the analytic apparatus in syntactic accessibility rather than ontological postulation. The result is a framework for fractal analysis, in which mathematical operations are relativized to layers of expressibility, enabling new insights into approximation, computability, and formal verification.

cs.LO↗

arXiv2025

Fractal Countability as a Constructive Alternative to the Power Set of N: A Meta-Formal Approach to Stratified Definability

Classical set theory constructs the continuum via the power set P(N), thereby postulating an uncountable totality. However, constructive and computability-based approaches reveal that no formal system with countable syntax can generate all subsets of N, nor can it capture the real line in full. In this paper, we propose fractal countability as a constructive alternative to the power set. Rather than treating countability as an absolute cardinal notion, we redefine it as a stratified, process-relative closure over definable subsets, generated by a sequence of conservative extensions to a base formal system. This yields a structured, internally growing hierarchy of constructive definability that remains within the countable realm but approximates the expressive richness of the continuum. We compare fractally countable sets to classical countability and the hyperarithmetical hierarchy, and interpret the continuum not as a completed object, but as a layered definitional horizon. This framework provides a constructive reinterpretation of power set-like operations without invoking non-effective principles.

math.GM↗

arXiv2025

Fractal Boundaries of Constructivity: A Meta-Theoretical Critique of Countability and Continuum

All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques -- effectively generates only countable fragments within a closed formal system. We formalize this limitation as the "fractal boundary of constructivity", the asymptotic limit of all constructive extensions under syntactically enumerable rules. A central theorem establishes the impossibility of fully capturing the structure of the continuum within any such system. We further introduce the concept of "fractal countability", a process-relative refinement of countability based on layered constructive closure. This provides a framework for analyzing definability beyond classical recursion without invoking uncountable totalities. We interpret the continuum not as an object constructively realizable, but as a horizon of formal expressibility.

math.GM↗

arXiv2025

Constructive Limits of Cantor's Diagonal Method: Countability, Enumerability, and the Impossibility of Exhausting the Continuum

Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application -- including generalizations with computable permutations and infinite hierarchies of diagonal extensions -- the resulting set remains countable. Thus, the method demonstrates the incompleteness of countable coverage but is unable to generate an uncountable set. This highlights its limitations as a constructive tool and reveals the boundary between constructive enumerability and the completeness of the continuum.

math.GM↗

arXiv2016

Novel Feature Extraction, Selection and Fusion for Effective Malware Family Classification

Modern malware is designed with mutation characteristics, namely polymorphism and metamorphism, which causes an enormous growth in the number of variants of malware samples. Categorization of malware samples on the basis of their behaviors is essential for the computer security community, because they receive huge number of malware everyday, and the signature extraction process is usually based on malicious parts characterizing malware families. Microsoft released a malware classification challenge in 2015 with a huge dataset of near 0.5 terabytes of data, containing more than 20K malware samples. The analysis of this dataset inspired the development of a novel paradigm that is effective in categorizing malware variants into their actual family groups. This paradigm is presented and discussed in the present paper, where emphasis has been given to the phases related to the extraction, and selection of a set of novel features for the effective representation of malware samples. Features can be grouped according to different characteristics of malware behavior, and their fusion is performed according to a per-class weighting paradigm. The proposed method achieved a very high accuracy ($\approx$ 0.998) on the Microsoft Malware Challenge dataset.

cs.CR↗