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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 , ⋯ , 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 ultimately 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. [ABSTRACT FROM AUTHOR]

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), 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(pn), 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 ℤm 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 evidence of this, we establish a correspondence between the two constructions in the case m = 2. [ABSTRACT FROM AUTHOR]

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 ⌈ n / 2 ⌉ into distinct parts. In the second case, an explicit bijection is provided. [ABSTRACT FROM AUTHOR]

The 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. [ABSTRACT FROM AUTHOR]

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 coincides with the sequence of the partial sums of the numbers of partitions into at least two distinct parts. [ABSTRACT FROM AUTHOR]

In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are obtained as the composition of different layers acting as a sequence of bijective transformations providing global increasing complexity. 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 recipe to avoid the imprimitivity attack, which exploits the invariance of some subspaces during the encryption. In the case of Lai–Massey schemes, where both Substitution Permutation Network and Feistel Network features are combined, the resistance against imprimitivity attacks has been a long-standing open problem. In this paper we consider a generalization of such a scheme and we prove its resistance against the imprimitivity attack. Our solution is obtained as a consequence of a more general result in which the problem of proving the primitivity of a generalized 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. We prove how this implies a reduction in the computational cost of invariant-subspace search. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationships between the elementary abelian regular subgroups and the Sylow 2-subgroups of their normalisers in the symmetric group Sym ( F 2 n) , in view of the interest that they have recently raised for their applications in symmetric cryptography. [ABSTRACT FROM AUTHOR]

In this paper we propose a tokenization algorithm of Reversible Hybrid type, as defined in PCI DSS guidelines for designing a tokenization solution, based on a block cipher with a secret key and (possibly public) additional input. We provide some formal proofs of security for it, which imply our algorithm satisfies the most significant security requirements described in PCI DSS tokenization guidelines. Finally, we give an instantiation with concrete cryptographic primitives and fixed length of the PAN, and we analyze its efficiency and security. [ABSTRACT FROM AUTHOR]

We study the relation among some security parameters for vectorial Boolean functions which prevent attacks on the related block cipher. We focus our study on a recently-introduced security criterion, called weak differential uniformity, which prevents the existence of an undetectable trapdoor based on imprimitive group action. We present some properties of functions with low weak differential uniformity, especially for the case of power functions and 4-bit S-Boxes. [ABSTRACT FROM AUTHOR]

The correctness in decrypting a ciphertext after some operations in the DGVH scheme depends heavily on the dimension of the secret key. In this paper we compute two bounds on the size of the secret key for the DGHV scheme to decrypt correctly a ciphertext after a fixed number of additions and a fixed number of multiplication. Moreover we improve the original bound on the dimension of the secret key for a general circuit. [ABSTRACT FROM AUTHOR]

Let ( Q, σ) be a symmetric quiver, where Q = ( Q, Q) is a finite quiver without oriented cycles and σ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form < , > on a representation V of Q. We shall call the representation orthogonal if < , > is symmetric, and symplectic if < , > is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c and, in the case when the matrix defining c is skew-symmetric, by the Pfaffians pf. In particular we give an explicit finite set of generators. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of digraphs Q, | Q|≤4 such that the associated Hecke-Kiselman monoids H are finite. In general, a necessary condition for H to be a finite monoid is that Q is acyclic and its Coxeter components are Dynkin diagrams. We show, by constructing examples, that such conditions are not sufficient. [ABSTRACT FROM AUTHOR]

A symmetric quiver ( Q, σ) is a finite quiver without oriented cycles Q = ( Q, Q) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if ( Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c and, when the matrix defining c is skew-symmetric, by the Pfaffians pf. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector. [ABSTRACT FROM AUTHOR]