arXiv subjects

Roberto Civino

Publications and source records attributed to Roberto Civino.

At least 19 recordsLinked to original sources

arXiv2026

A Geometric Characterization of Maximal Unrefinable Partitions via the Keith-Nath Transformation and Young Diagrams

We investigate the combinatorial structure of unrefinable partitions through their correspondence with numerical sets and Young diagrams. Building on the bijection introduced by Keith and Nath, we apply a general geometric criterion that links the unrefinability of a partition directly to the hook lengths of its associated Young diagram. This criterion provides a structural method for the characterization of any unrefinable partition. Using this general framework, we revisit the correspondence results between maximal unrefinable partitions and partitions into distinct parts, previously established using enumerative methods. We provide alternative and purely combinatorial proofs of these bijections, focusing on the rigid symmetry structures of the Young diagrams. In the triangular weight case, we show that the corresponding diagrams are quasi-symmetric, i.e. symmetric up to a single extra column. We extend this analysis to the nontriangular case, showing that the diagrams either exhibit this same quasi-symmetric structure or are perfectly self-conjugate, depending on the maximal part.

math.CO↗

arXiv2025

The number of maximal unrefinable partitions

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number.

math.CO↗

arXiv2025

Optimal s-boxes against alternative operations and linear propagation

Civino et al. (2019) have shown how some diffusion layers can expose a Substitution-Permutation Network to vulnerability from differential cryptanalysis when employing alternative operations coming from groups isomorphic to the translation group on the message space. In this study, we present a classification of diffusion layers that exhibit linearity with respect to certain parallel alternative operations, enabling the possibility of an alternative differential attack simultaneously targeting all the s-boxes within the block. Furthermore, we investigate the differential behaviour with respect to alternative operations for all classes of optimal 4-bit s-boxes, as defined by Leander and Poschmann (2007). Our examination reveals that certain classes contain weak permutations w.r.t. alternative differential attacks. Finally, we leverage these vulnerabilities to execute a series of experiments showing the effectiveness of the cryptanalysis performed with a parallel alternative operation compared to the classical one.

cs.CR↗

arXiv2025

Classification of small binary bibraces via bilinear maps

We classify small binary bibraces, using the correspondence with alternating algebras over the field F2, up to dimension eight, also determining their isomorphism classes. These finite-dimensional algebras, defined by an alternating bilinear multiplication and nilpotency of class two, can be represented by subspaces of skew-symmetric matrices, with classification corresponding to GL(m, F_2)-orbits under congruence. Our approach combines theoretical invariants, such as rank sequences and the identification of primitive algebras, with computational methods implemented in Magma. These results also count the number of possible alternative operations that can be used in differential cryptanalysis.

math.RA↗

arXiv2024

Binary bi-braces and applications to cryptography

In a XOR-based alternating block cipher the plaintext is masked by a sequence of layers each performing distinct actions: a highly nonlinear permutation, a linear transformation, and the bitwise key addition. When assessing resistance against classical differential attacks (where differences are computed with respect to XOR), the cryptanalysts must only take into account differential probabilities introduced by the nonlinear layer, this being the only one whose differential transitions are not deterministic. The temptation of computing differentials with respect to another difference operation runs into the difficulty of understanding how differentials propagate through the XOR-affine levels of the cipher. In this paper we introduce a special family of braces that enable the derivation of a set of differences whose interaction with every layer of an XOR-based alternating block cipher can be understood. We show that such braces can be described also in terms of alternating binary algebras of nilpotency class two. Additionally, we present a method to compute the automorphism group of these structures through an equivalence between bilinear maps. By doing so, we characterise the XOR-linear permutations for which the differential transitions with respect to the new difference are deterministic, facilitating an alternative differential attack.

math.GR↗

arXiv2024

Differential experiments using parallel alternative operations

The use of alternative operations in differential cryptanalysis, or alternative notions of differentials, are lately receiving increasing attention. Recently, Civino et al. managed to design a block cipher which is secure w.r.t. classical differential cryptanalysis performed using XOR-differentials, but weaker with respect to the attack based on an alternative difference operation acting on the first s-box of the block. We extend this result to parallel alternative operations, i.e. acting on each s-box of the block. First, we recall the mathematical framework needed to define and use such operations. After that, we perform some differential experiments against a toy cipher and compare the effectiveness of the attack w.r.t. the one that uses XOR-differentials.

cs.CR↗

arXiv2023

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n-2 terms in the chain. Moreover, they proved that the (n-1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m-1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As an evidence of this, we establish a correspondence between the two constructions in the case m=2.

math.RA↗

arXiv2023

A definitely periodic chain in the integral Lie ring of partitions

Given an integer $n$, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in $\{1,2,\dots, n-1\}$. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is definitely periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

math.CO↗

arXiv2023

Verification and generation of unrefinable partitions

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, being an unrefinable partition means that none of the parts can be written as the sum of smaller integers without introducing a repetition. We address the algorithmic aspects of unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given integer. We design two algorithms to solve the two mentioned problems and we discuss their complexity.

math.CO↗

arXiv2022

On the maximal part in unrefinable partitions of triangular numbers

A partition into distinct parts is refinable if one of its parts $a$ can be replaced by two different integers which do not belong to the partition and whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a $O(n^{1/2})$-upper bound for the largest part in an unrefinable partition of $n$, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if $n$ is even there exists only one maximal unrefinable partition of $n(n+1)/2$, and that if $n$ is odd the number of such partitions equals the number of partitions of $\lceil n/2\rceil$ into distinct parts. In the second case, an explicit bijection is provided.

math.CO↗

arXiv2022

On the primitivity of the AES-128 key-schedule

The key-scheduling algorithm in the AES is the component responsible for selecting from the master key the sequence of round keys to be xor-ed to the partially encrypted state at each iteration. We consider here the group $Γ$ generated by the action of the AES-128 key-scheduling operation, and we prove that the smallest group containing $Γ$ and all the translations of the message space is primitive. As a consequence, we obtain that no proper and non-trivial subspace can be invariant under its action.

math.GR↗

arXiv2021

Unrefinable partitions into distinct parts in a normalizer chain

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. Unrefinable partitions into distinct parts are those in which no part x can be replaced with integers whose sum is x obtaining a new partition into distinct parts. We prove here that the (n-1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition on the minimal excludant.

math.CO↗

arXiv2020

On properties of translation groups in the affine general linear group with applications to cryptography

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.

math.GR↗

arXiv2020

On the primitivity of Lai-Massey schemes

In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are often obtained as the composition of different layers providing confusion and diffusion. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the receipt to avoid the imprimitivity attack. In this paper a similar study is proposed on the subject of the Lai-Massey scheme, a framework which combines both Substitution Permutation Network and Feistel Network features. Its resistance to the imprimitivity attack is obtained as a consequence of a more general result in which the problem of proving the primitivity of the Lai-Massey scheme is reduced to the simpler one of proving the primitivity of the group generated by the round functions of a strictly related Substitution Permutation Network.

cs.CR↗

arXiv2020

Rigid commutators and a normalizer chain

The novel notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler's partition theorem.

math.GR↗

arXiv2020

A Chain of Normalizers in the Sylow $2$-subgroups of the symmetric group on $2^n$ letters

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.

math.GR↗

arXiv2020

An Authenticated Key Scheme over Elliptic Curves for Topological Networks

Nodes of sensor networks may be resource-constrained devices, often having a limited lifetime, making sensor networks remarkably dynamic environments. Managing a cryptographic protocol on such setups may require a disproportionate effort when it comes to update the secret parameters of new nodes that enter the network in place of dismantled sensors. For this reason, the designers of schemes for sensor network are always concerned with the need of scalable and adaptable solutions. In this work, we present a novel elliptic-curve based solution, derived from the previously released cryptographic protocol TAKS, which addresses this issue. We give a formal description of the scheme, built on a two-dimensional vector space over a prime field and over elliptic curves, where node topology is more relevant than node identity, allowing a dynamic handling of the network and reducing the cost of network updates. We also study some security concerns and their relation to the related discrete logarithm problem over elliptic curves.

cs.CR↗

arXiv2020

Some group-theoretical results on Feistel Networks in a long-key scenario

The study of the trapdoors that can be hidden in a block cipher is and has always been a high-interest topic in symmetric cryptography. In this paper we focus on Feistel-network-like ciphers in a classical long-key scenario and we investigate some conditions which make such a construction immune to the partition-based attack introduced recently by Bannier et al.

math.GR↗