Detection and identification of cheaters in (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) secret sharing scheme

In a (t, n) secret sharing scheme, a secret s is divided into n shares and shared among a set of n shareholders by a mutually trusted dealer in such a way that any t or more than t shares will be able to reconstruct this secret; but fewer than t shares cannot know any information about the secret. When shareholders present their shares in the secret reconstruction phase, dishonest shareholder(s) (i.e. cheater(s)) can always exclusively derive the secret by presenting faked share(s) and thus the other honest shareholders get nothing but a faked secret. Cheater detection and identification are very important to achieve fair reconstruction of a secret. In this paper, we consider the situation that there are more than t shareholders participated in the secret reconstruction. Since there are more than t shares (i.e. it only requires t shares) for reconstructing the secret, the redundant shares can be used for cheater detection and identification. Our proposed scheme uses the shares generated by the dealer to reconstruct the secret and, at the same time, to detect and identify cheaters. We have included discussion on three attacks of cheaters and bounds of detectability and identifiability of our proposed scheme under these three attacks. Our proposed scheme is an extension of Shamir’s secret sharing scheme.      

Optimal (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) visual cryptographic schemes for general <Emphasis Type="Italic">k</Emphasis>

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L ₁-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.     

(<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-bi-Koszul algebras

The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

A Family of Binary (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">s</Emphasis>)-Nets of Strength 5

(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (m−k,m,s)₂-net is a family of ks vectors in F₂^m satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth et al. [5] recently constructed (2r−3,2r+2,2^r−1)₂-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2^r+1)₂-nets based on a family of binary linear codes of minimum distance 6.Communicated by: T. Helleseth     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Cubic and higher degree bounds for codes and (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-nets

The Plotkin bound and the quadratic bound for codes and (t, m, s)-nets can be obtained from the linear programming bound using certain linear and quadratic polynomials, respectively. We extend this approach by considering cubic and higher degree polynomials to find new explicit bounds as well as new non-existence results for codes and (t, m, s)-nets.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

(<Emphasis Type="Italic">q</Emphasis>,<Emphasis Type="Italic">t</Emphasis>)-Deformations of multivariate hook product formulae

We generalize multivariate hook product formulae for P-partitions. We use Macdonald symmetric functions to prove a (q,t)-deformation of Gansner’s hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a d-complete poset, we present a conjectural (q,t)-deformation of Peterson–Proctor’s hook product formula.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

Maximal regular subsemibands of finite order-preserving transformation semigroups <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)

Let O _n be the order-preserving transformation semigroup on X _n . For an arbitrary integer r such that 1≤r≤n−2, we completely describe the maximal regular subsemibands of the semigroup K(n,r)={α∈O _n :|im(α)|≤r}. We also formulate the cardinal number of such subsemigroups.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

On proper secrets, (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-bases and linear codes

This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir’s “k-out-of-n” threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k−1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme. In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k−1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k−1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce here and some well known objects from projective geometry.      

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.