arXiv subjects

Pantelimon Stănică

Publications and source records attributed to Pantelimon Stănică.

9 recordsLinked to original sources

arXiv2026

Higher-order nonhomomorphicity under affine restrictions

We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\ge3$ and $0\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\ge3$. This proves the two threshold-function conjectures left open in~\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.

cs.IT↗

arXiv2026

Multi-target hyperbolic sieves and elliptic trace obstructions

Let $N=pq$ be a product of two primes, and let $\ell$ be an odd prime not dividing $N$. For $a\in\mathbb F_\ell^*$, the set \[ H_a(N;\ell)=\{ax+Nx^{-1}:x\in\mathbb F_\ell^*\} \] contains the residue of $ap+q$ modulo $\ell$. Since residues outside this set can be excluded when searching for that linear form, it has been applied as a sieve for linear combinations of $p$ and $q$ in recent deterministic integer factorization methods. Its size is $(\ell+χ(aN))/2$, where $χ$ is the Legendre symbol modulo $\ell$. We study unions of these sets, which give a common set of candidate residues for several linear forms and thus allow to sieve for multiple target forms at once. For distinct $a,b$, we obtain an exact formula for $|H_a(N;\ell)\cup H_b(N;\ell)|$ in terms of \[ K(A,B)=\sum_{z\in\mathbb F_\ell}χ((z^2-A)(z^2-B)), \qquad A=4aN,\quad B=4bN. \] The Hasse-Weil bound gives an error of $O(\sqrt\ell)$ from $3\ell/4$. For fixed $c\ne0,1$, we prove the elementary identity $K(A,cA)=-1-χ(A)t_c$, where $t_c=-1-K(1,c)$. Hence the union for a primitive-root pair $(a,b)=(γ,γ^2)$ has at most two sizes as $N$ varies. We give an exact condition on $t_γ$ for the bound $3\ell/4+1$ to hold for every nonzero $N$ modulo $\ell$, and compute the first and second moments. For $k$ distinct coefficients, we prove \[ \left|\left|\bigcup_{j=1}^k H_{a_j}(N;\ell)\right| -\ell(1-2^{-k})\right| \le (k-1+2^{-k})\sqrt\ell+k. \] We derive the corresponding bounds modulo products of distinct primes using the Chinese remainder theorem. We also determine the exact size of the image obtained by restricting $x$ to the nonzero $r$-th powers, which applies when searching for a divisor of the form $u^r$ in $N=u^rv$.

math.NT↗

arXiv2026

On the hull of linearized polynomial codes

Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}_{q^m}$. Our main tool is a unified Gram-matrix method. For image codes $\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$, with $Φ_{\boldsymbolα}=\sum_iα_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$, where $G(\boldsymbolα)$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{λ,μ}=\operatorname{im}(λx+μL(x))$, we obtain a quadratic Gram pencil $λ^2G_0+λμG_1+μ^2G_2$ whose determinant describes the LCD locus in $\mathbb{P}^1(\mathbb{F}_q)$. We also treat $\mathbb{F}_{q^m}$-linear rank-distance codes $\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}_{q^m}$ determines the hull dimension. For $L(X)=X^{q^k}$, with $d=\gcd(k,m)$, the resulting circulant Gram matrices yield a closed-form discriminant and a complete classification in three of the four bijectivity configurations over $\mathbb{P}^1(\mathbb{F}_{q^m})$. In the remaining case, the hull dimension equals $δ=\dim_{\mathbb{F}_q}(\operatorname{im}ϕ_{λ,μ}\cap\kerϕ_{λ,μ}^{\dagger})$, and the extremal condition $δ=d$ is characterized by an explicit trace-isotropy criterion. We conclude with an exact count of LCD and non-LCD points, showing that the LCD density tends to $1$ as $q\to\infty$, together with a worked example over $\mathbb{F}_{64}$ and a SageMath verification.

cs.IT↗

arXiv2026

Differential-linear profiles over finite fields of arbitrary characteristic

The (binary) differential-linear connectivity table (DLCT) measures the dependence between an input difference and a linear mask applied to the corresponding output difference. For vectorial Boolean functions, each DLCT entry is one half of an additive autocorrelation value. We extend this relation to functions over finite fields of arbitrary prime characteristic by introducing a level-resolved p-ary differential-linear profile. Its entries are the centered numbers of inputs for which a derivative component has each prescribed trace value in $\mathbb{F}_p$. The discrete Fourier transform of this profile is the family of additive autocorrelations obtained by multiplying the output mask by the nonzero elements of $\mathbb{F}_p$; when p=2, the usual binary identity is recovered. For a fixed input difference, we show that the profiles over all nonzero output masks determine the corresponding DDT row exactly, and we give an explicit inversion formula. We establish a second-moment identity: the total profile energy in one derivative direction is a constant multiple of the squared Euclidean distance between that DDT row and the balanced row. Thus this energy is determined by the full row differential spectrum, not by differential uniformity alone. It follows that all profiles in a direction vanish exactly when the derivative is balanced; for square maps in odd characteristic, this gives a characterization of planarity. As concrete odd-characteristic examples, we determine the complete profile of the monomial $x^{p^k+1}$ and derive an exact Kloosterman-sum formula for the inverse monomial. Finally, we determine the behavior of the profiles under equivalence. EA-equivalence reindexes the input and output masks and translates the trace level, whereas a general CCZ equivalence may mix several derivative directions. Nevertheless, for square maps the global nontrivial profile energy is CCZ-invariant.

cs.IT↗

arXiv2026

A Post-Quantum Secure Lattice-Based Forward-Secure Identity Based Encryption with Applications to Internet of Things Architecture

The rapid expansion of the Internet of Things (IoT) has led to an unprecedented scale of data exchange across heterogeneous and resource-constrained devices. Ensuring confidentiality and secure key management in such environments is challenging. Traditional public-key infrastructures require heavy certificate-handling overhead. Identity-Based Encryption (IBE) offers a lightweight alternative by deriving public keys directly from device identities, making it attractive for IoT deployments. However, IoT devices are highly vulnerable to side-channel and key-extraction attacks, motivating the need for Forward-Secure IBE(FS-IBE), where the compromise of a current secret key does not threaten past communications. Existing FS-IBE constructions based on classical hardness assumptions are not secure in the era of post-quantum, while the lattice-based (LWE-based) forward-secure scheme suffer from large key and ciphertext sizes, limiting their suitability for constrained IoT systems. Here, we propose a new lattice-based fs-IBE scheme in the ring setting, relying on the RLWE assumption to achieve post-quantum security and significant efficiency gains. Our design uses trapdoor delegation with a minimal-cover mechanism over a binary tree. It results in compact public parameters and efficient per-epoch key updates. Compared to prior LWE-based constructions, our scheme reduces public key, secret key, and ciphertext sizes, and thus, making it better suited for practical IoT environments.

cs.CR↗

arXiv2026

Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions

Generalized bent (gbent) functions from an $n$-variable Boolean space to $\mathbb{Z}_{2^k}$ are central in cryptography and sequence design. Instead of the usual binary decomposition, we introduce a $2^\ell$-adic representation, for $k=\ell r$, writing such functions as linear combinations of $r$ component functions valued in $\mathbb{Z}_{2^\ell}$. We prove a general result on overconstrained character sums over finite abelian groups: under a common-argument hypothesis, sequences with two-level Fourier magnitude spectra must be extremely sparse, with a conditional extension to multi-level spectra. As an application, we derive consequences for generalized plateaued functions under suitable assumptions. We then show that if $f:\mathbb{F}_2^n\to\mathbb{Z}_{2^k}$ is landscape, then under the $2^\ell$-adic decomposition every function in a certain affine space over $\mathbb{Z}_{2^\ell}$ is again landscape with the same Walsh magnitudes. This gives an unconditional necessity result, with no structural assumptions on $f$, together with a complete characterization using only a small subset of these maps. For generalized bent and generalized plateaued functions, sufficiency is also obtained from linear combinations of lower components under natural assumptions; a counterexample shows these assumptions are essential. Our method reduces verification for landscape functions from $2^{2^{k-1}}$ checks to fewer than $2^{k-\ell+1}+1$ conditions; for gbent functions this drops to a single basis function under the common-argument hypothesis, and for generalized plateaued functions, under additional assumptions, to $2^{k-\ell}$ checks. The $2^\ell$-adic framework also preserves key properties, including duality and differential uniformity.

math.NT↗

arXiv2026

On generalized Thabit numbers $(p+1)p^\mathfrak{a}-1$ in the $k$-Lucas sequence

Let $k\ge 2$ and $\{L_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$-Lucas numbers whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we solve the Diophantine equation $L_n^{(k)}=(p+1)p^\mathfrak{a}-1$, for a Mersenne or Fermat prime $p=2^{\ell}\pm 1$, and positive integers $n\ge 2$, $k\ge 2$, $\mathfrak{a}\ge 1$ and $\ell \ge 1$.

math.NT↗

arXiv2022

Bivariate functions with low $c$-differential uniformity

Starting with the multiplication of elements in $\mathbb{F}_{q}^2$ which is consistent with that over $\mathbb{F}_{q^2}$, where $q$ is a prime power, via some identification of the two environments, we investigate the $c$-differential uniformity for bivariate functions $F(x,y)=(G(x,y),H(x,y))$. By carefully choosing the functions $G(x,y)$ and $H(x,y)$, we present several constructions of bivariate functions with low $c$-differential uniformity. Many P$c$N and AP$c$N functions can be produced from our constructions.

cs.IT↗

arXiv2020

Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application

For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an "intermediate" concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it convey detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

cs.IT↗