arXiv subjects

Pantelimon Stanica

Publications and source records attributed to Pantelimon Stanica.

At least 19 recordsLinked to original sources

arXiv2026

Affine and cyclic affine equivalence of rotation-symmetric S-boxes

A rotation-symmetric S-box on $n$ bits is a vectorial Boolean function that commutes with the right cyclic shift and is therefore determined by a single Boolean function. The map $χ$ used in Keccak is a well-known example. Ordinary affine equivalence allows arbitrary invertible affine transformations, which need not preserve rotation symmetry. We introduce \emph{cyclic affine equivalence}, in which the implementing transformations each have an invertible $k$-circulant linear part with $\gcd(k,n)=1$ and a translation vector fixed by the shift. For $n\ge3$, the two relations give the same classes of rotation-symmetric permutations exactly when $n$ is an odd prime for which $2$ is a primitive root. In every other dimension $n\ge3$, explicit constructions exhibit affine classes containing several cyclic affine classes. We derive exact class-size formulas, count cyclic affine classes for $3\le n\le9$, and classify a four-bit family of $χ$-siblings under both relations. Finally, we characterize when a permutation is affinely equivalent to a rotation-symmetric one and apply the criterion to the Ascon S-box.

math.CO↗

arXiv2026

Extraction meets Distillation - cryptanalytic techniques to recover parameters from a Deep Neural Network

We study fully connected neural-network reconstruction by combining cryptanalytic weight-direction extraction with learning from predicted labels. We adapt a ReLU extraction implementation to Leaky ReLU, compare sign-search methods, and estimate biases and output layers. We prove conditional bounds for direction recovery with approximate prefixes, preservation of local distance votes, prediction changes from aligned parameter errors, and sign selection with independent scoring data. We also specify exact row restoration and distinguish its guarantee from that of gradient masking. The experiments evaluate assisted extraction, with privileged access to gradients, hidden activations, and victim parameters, followed by label-based reconstruction. Across eight search seeds on the largest synthetic architecture, Leaky ReLU recovers 51--57 neurons in the deepest hidden layer, while ReLU recovers none. This advantage does not hold uniformly across architectures and layers. Removing victim-weight access from the sign walk gives less accurate signs, despite high final prediction agreement. Across 40 seeded runs on the two smallest synthetic victims, agreement on 5,000 inputs excluded from fitting and selection ranges from $0.9890$ to $0.9992$. On two CIFAR-10 victims, the reconstructed models achieve $56.40\%$ and $58.29\%$ agreement and exceed the distillation baseline on three functional metrics. We report label requests separately from privileged access and explain why prediction agreement does not identify network parameters.

cs.LG↗

arXiv2026

Reduced APN permutations and a disproof of the Galois-ring lifting conjecture

We construct infinite families of reduced almost perfect nonlinear (APN) permutations of $\mathbb{F}_{2^m}$ that admit coefficientwise permutation lifts to $\mathrm{GR}(2^k,m)$ for every $k>1$. These families disprove Conjecture~2 of Rønjom and Sandrib (Cryptogr. Commun., 2026), even for reduced polynomial representatives. For every odd $m\ge5$ with $3\nmid m$, we obtain explicit seven-term polynomials over $\mathbb{F}_2$ with formal derivative identically one. We derive them from a complete criterion for the reduced representative of $A(x)^{2^s+1}$ to have no critical point in $\mathbb{F}_{2^m}$, where $A$ is an invertible linearized polynomial and $\gcd(s,m)=1$. We then apply the cube case to construct a second family of degree $2^{m-1}+2$, whose nonconstant derivatives avoid zero by an explicit trace identity. A fixed parameter in $\mathbb{F}_{32}$ gives this family in every dimension $m=5r$ with $r$ odd, including a degree-$18$ example over $\mathbb{F}_{32}$. Both constructions remain in Gold linear equivalence classes, so liftability of reduced representatives is not invariant under invertible $\mathbb{F}_2$-linear input changes. We obtain the lifting conclusions from the standard Galois-ring permutation criterion. We also verify a degree-$24$ example supplied by Zhang and Haobo, prove a trace-adjoint criterion for reduced quadratic functions, and interpret the derivative condition through ramification at rational points.

math.NT↗

arXiv2026

Extended Differential Cryptanalysis of Kuznyechik

We study the input-side relation $F(cx\oplus a)\oplus F(x)=b$ as an extended differential for block-cipher analysis. Because multiplication occurs before the first S-box, fixing that S-box output leaves an ordinary XOR difference for subsequent binary linear layers and common key additions. For permutations, the extended and outer $c$-differential tables are related by inversion. For the Kuznyechik S-box, the exceptional inverse classes $\mathtt{02}/\mathtt{e1}$, $\mathtt{04}/\mathtt{91}$, and $\mathtt{03}/\mathtt{be}$ have exact extended differential uniformities $64$, $33$, and $21$. We connect them to the pseudo-exponential structure of Perrin and Udovenko through a cross-field transfer theorem. In the common polynomial basis, the disagreement rank between multiplication maps in the hidden and Kuznyechik fields equals the degree of the multiplier polynomial, yielding structural lower bounds $60$, $30$, and $16$. We also analyze the complete extended differential table: its normalization is doubly stochastic, its energy is an exact multiplicative correlation of ordinary DDT rows, and transferred hidden fibers give explicit Pearson and Shannon-information bounds. Whitening selects the effective row $a\oplus(1\oplus c)K$. For two-round standard Kuznyechik we derive the exact fixed-key channel $Q_{c,\ell}=W_cP_\ell W_1$ and a two-stage maximum-likelihood attack. With $c=\mathtt{04}$ it recovers the full $256$-bit master key with failure probability at most $0.01$, using $226{,}512$ chosen-plaintext pairs and at most $240{,}669$ encryption queries. Restricted product-model searches give gains of $5.2$, $4.6$, and about $1.6$ bits over the corresponding classical families at two, three, and four rounds. A separate nine-round no-pre-whitening Monte Carlo study is reported only as exploratory evidence because its final configurations followed preliminary screening.

cs.CR↗

arXiv2026

Infinite families of APN permutations in constrained trivariate classes over $\mathbb{F}_{2^m}$

We study trivariate permutation polynomials over $\mathbb{F}_{2^{m}}$ extending two APN permutation families of Li--Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over $\mathbb{F}_{2^m}^*$. For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where $a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in $\mathbb{F}_{2^m}^*$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least \[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \] values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li--Kaleyski family. For the second family \[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \] we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\neq 6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\neq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on $\mathbb{F}_{2^{3m}}$.

math.NT↗

arXiv2026

Permutation Polynomials Under Multiplicative-Additive Perturbations: Characterization via Difference Distribution Tables

We investigate permutation polynomials F over finite fields F_{p^n} whose generalized derivative maps x -> F(x + a) - cF(x) are themselves permutations for all nonzero shifts a. This property, termed perfect c-nonlinearity (PcN), represents optimal resistance to c-differential attacks - a concern highlighted by recent cryptanalysis of the Kuznyechik cipher variant. We provide the first characterization using the classical difference distribution table (DDT): F is PcN if and only if Delta_F(a,b) Delta_F(a,c^{-1}b) = 0 for all nonzero a,b. This enables verification in O(p^{2n}) time given a precomputed DDT, a significant improvement over the naive O(p^{3n}) approach. We prove a strict dichotomy for monomial permutations: the derivative F(x + alpha) - cF(x) is either a permutation for all nonzero shifts or for none, with the general case remaining open. For quadratic permutations, we provide explicit algebraic characterizations. We identify the first class of affine transformations preserving c-differential uniformity and derive tight nonlinearity bounds revealing fundamental incompatibility between PcN and APN properties. These results position perfect c-nonlinearity as a structurally distinct regime within permutation polynomial theory.

cs.IT↗

arXiv2026

Non-permutation phenomena in trivariate families over $\F_{2^m}$ and resolution of a conjecture

Constructing permutation polynomials over finite fields, particularly those with simple algebraic structure in multiple variables, is a fundamental problem with applications in cryptography and coding theory. Recently, Li and Kaleyski (IEEE Trans. Inf. Theory, 2024) generalized two sporadic quadratic APN permutations into infinite families of trivariate functions. Motivated by their work, we investigate conditions under which generalized trivariate functions fail to be permutations. We establish necessary conditions on coefficient parameters that prevent the permutation property, provide a complete computational classification for small field extensions, and prove general non-permutation results. As a key application of our algebraic geometry approach, we resolve the permutation part of a conjecture by Beierle, Carlet, Leander, and Perrin (Finite Fields Appl., 2022) regarding a related trivariate form. Specifically, we prove that for all odd characteristic-2 extension degrees $m \geq 23$, their function $C_u$ is not a permutation over $\mathbb{F}_{2^m}^3$ for any $u \in \mathbb{F}_{2^m}^*$, resolving the permutation part of their conjecture for sufficiently large fields.

math.NT↗

arXiv2026

On the Classification of Dillon's APN Hexanomials

We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family.

math.NT↗

arXiv2025

On the Codebook Design for NOMA Schemes from Bent Functions

Uplink grant-free non-orthogonal multiple access (NOMA) is a promising technology for massive connectivity with low latency and high energy efficiency. In code-domain NOMA schemes, the requirements boil down to the design of codebooks that contain a large number of spreading sequences with low peak-to-average power ratio (PAPR) while maintaining low coherence. When employing binary Golay sequences with guaranteed low PAPR in the design, the fundamental problem is to construct a large set of $n$-variable quadratic bent or near-bent functions in a particular form such that the difference of any two is bent for even $n$ or near-bent for odd $n$ to achieve optimally low coherence. In this work, we propose a theoretical construction of NOMA codebooks by applying a recursive approach to those particular quadratic bent functions in smaller dimensions. The proposed construction yields desired NOMA codebooks that contain $6\cdot N$ Golay sequences of length $N=2^{4m}$ for any positive integer $m$ and have the lowest possible coherence $1/\sqrt{N}$.

math.CO↗

arXiv2025

Extending Forrelation: Quantum Algorithms Related to Generalized Fourier-Correlation

In this paper, we study different cryptographically significant spectra of Boolean functions, including the Walsh-Hadamard, cross-correlation, and autocorrelation. The $2^k$-variation by Stanica [IEEE-IT 2016] is considered here with the formulation for any $m \in \mathbb{N}$. Given this, we present the most generalized version of the Deutsch-Jozsa algorithm, which extends the standard and previously extended versions, thereby encompassing them as special cases. Additionally, we generalize the Forrelation formulation by introducing the $m$-Forrelation and propose various quantum algorithms towards its estimation. In this regard, we explore different strategies in sampling these newly defined spectra using the proposed $m$-Forrelation algorithms and present a comparison of their corresponding success probabilities. Finally, we address the problem related to affine transformations of generalized bent functions and discuss quantum algorithms in identifying the shifts between two bent (or negabent) functions with certain modifications, and compare these with existing results.

quant-ph↗

arXiv2025

On APN functions in odd characteristic, the disproof of a conjecture and related problems

In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality $q$ is larger than~$9587$, then those functions will never be APN. Moreover, we discuss other connected families of functions, for potential APN functions, but we show that they are not good candidates for APNess if the underlying field is large, in spite of the fact that they though they are APN for small environments.

math.AG↗

arXiv2025

The revised boomerang connectivity tables and their connection to the Difference Distribution Table

It is well-known that functions over finite fields play a crucial role in designing substitution boxes (S-boxes) in modern block ciphers. In order to analyze the security of an S-box, recently, three new tables have been introduced: the Extended Boomerang Connectivity Table (EBCT), the Lower Boomerang Connectivity Table (LBCT), and the Upper Boomerang Connectivity Table (UBCT). In fact, these tables offer improved methods over the usual Boomerang Connectivity Table (BCT) for analyzing the security of S-boxes against boomerang-style attacks. Here, we put in context these new EBCT, LBCT, and UBCT concepts by connecting them to the DDT for a differentially $δ$-uniform function and also determine the EBCT, LBCT, and UBCT entries of three classes of differentially $4$-uniform power permutations, namely, Gold, Kasami and Bracken-Leander. We also determine the Double Boomerang Connectivity Table (DBCT) entries of the Gold function. As byproducts of our approach, we obtain some previously published results quite easily.

cs.CR↗

arXiv2025

The second-order zero differential spectra of some functions over finite fields

It was shown by Boukerrou et al.~[IACR Trans. Symmetric Cryptol. 1 (2020), 331--362] that the $F$-boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic) of perfect nonlinear functions is~$0$ on $\F_{p^n}$ ($p$ prime) and the one of almost perfect nonlinear functions on $\F_{2^n}$ is~$0$. It is natural to inquire what happens with APN or other low differential uniform functions in even and odd characteristics. Here, we explicitly determine the second-order zero differential spectra of several maps with low differential uniformity. In particular, we compute the second-order zero differential spectra for some almost perfect nonlinear (APN) functions over finite fields of odd characteristic, pushing further the study started in Boukerrou et al. and continued in Li et al.~[Cryptogr. Commun. 14.3 (2022), 653--662], and it turns out that our considered functions also have low second-order zero differential uniformity. Moreover, we study the second-order zero differential spectra of certain functions with low differential uniformity over finite fields of even characteristic. We connect this new concept to the sum-freedom and vanishing flats concepts and find some counts for the number of vanishing flats via our methods. We provide detailed analyses on several equations over finite fields that may have an interest outside of the scope of our paper.

cs.IT↗

arXiv2025

The differential properties of certain permutation polynomials over finite fields

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown by Anbar et al. (J. Comb. Des. 31(12):1-24, 2023). Additionally, in a recent manuscript by Pal et al. (Adv. Math. Communications, to appear), a very interesting connection between the $c$-differential uniformity and boomerang uniformity, when $c=-1$, was pointed out, showing that they are the same for an odd APN permutation, sparking yet more interest in the construction of functions with low $c$-differential uniformity. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. To solve the involved equations over finite fields, we use various number theoretical techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

math.CO↗

arXiv2024

A proof of a conjecture on permutation trinomials

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +αX^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$, from [A. Rai, R. Gupta, {\it Further results on a class of permutation trinomials}, Cryptogr. Commun. 15 (2023), 811--820].

math.NT↗

arXiv2024

A connection between the boomerang uniformity and the extended differential in odd characteristic and applications

This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $c$-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $(-1)$-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $(-1)$-differential uniformity. We next apply this result to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the composition of the inverse function with an order-$3$ cycle permutation, with differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is even; and $4$ otherwise. The second class is a family of binomials and we show that their differential uniformity equals~$4$. We next complete the open case of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for $p\geq 5$, and continued by Kölsch (2021), for $p=2$, $n\geq 5$, on the characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a permutation polynomial. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.

cs.IT↗

arXiv2024

Quantum Secure Protocols for Multiparty Computations

Secure multiparty computation (MPC) schemes allow two or more parties to conjointly compute a function on their private input sets while revealing nothing but the output. Existing state-of-the-art number-theoretic-based designs face the threat of attacks through quantum algorithms. In this context, we present secure MPC protocols that can withstand quantum attacks. We first present the design and analysis of an information-theoretic secure oblivious linear evaluation (OLE), namely ${\sf qOLE}$ in the quantum domain, and show that our ${\sf qOLE}$ is safe from external attacks. In addition, our scheme satisfies all the security requirements of a secure OLE. We further utilize ${\sf qOLE}$ as a building block to construct a quantum-safe multiparty private set intersection (MPSI) protocol.

quant-ph↗

arXiv2023

The second-order zero differential spectra of some APN and other maps over finite fields

The Feistel Boomerang Connectivity Table and the related notion of $F$-Boomerang uniformity (also known as the second-order zero differential uniformity) has been recently introduced by Boukerrou et al.~\cite{Bouk}. These tools shall provide a major impetus in the analysis of the security of the Feistel network-based ciphers. In the same paper, a characterization of almost perfect nonlinear functions (APN) over fields of even characteristic in terms of second-order zero differential uniformity was also given. Here, we find a sufficient condition for an odd or even function over fields of odd characteristic to be an APN function, in terms of second-order zero differential uniformity. Moreover, we compute the second-order zero differential spectra of several APN or other low differential uniform functions, and show that our considered functions also have low second-order zero differential uniformity, though it may vary widely, unlike the case for even characteristic when it is always zero.

cs.IT↗