Designs, Codes and Cryptography最新文献

This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let (mathbb {F}_q) denote the finite field with q elements, and let G be a finite Abelian group of order n. The paper investigates Abelian codes defined as ideals of the group algebra (mathbb {F}_qG) with coefficients in (mathbb {F}_q). Specifically, it delves into Abelian hull dimensional codes in the group algebra (mathbb {F}_qG), where G is a finite Abelian group of order n with (gcd (n,q)=1). Specifically, we first examine general hull Abelian codes and then narrow its focus to Abelian one-dimensional hull codes. Next, we focus on Abelian one-dimensional hull codes and present some necessary and sufficient conditions for characterizing them. Consequently, we generalize a recent result on Abelian codes and show that no binary or ternary Abelian codes with one-dimensional hulls exist. Furthermore, we construct Abelian codes with one-dimensional hulls by generating idempotents, derive optimal ones with one-dimensional hulls, and establish several existing results of Abelian codes with one-dimensional hulls. Finally, we develop enumeration results through a simple formula that counts Abelian codes with one-dimensional hulls in (mathbb {F}_qG). These achievements exploit the rich algebraic structure of those Abelian codes and enhance and increase our knowledge of them by considering their hull dimensions, reducing the gap between their interests and our understanding of them.

Quasi-complementary sequence sets (QCSSs) are important in modern communication systems as they are capable of supporting more users, which is desired in applications like MC-CDMA nowadays. Although several constructions of aperiodic QCSSs have been proposed in the literature, the known optimal aperiodic QCSSs have limited length or have large alphabet. In this paper, based on extended Boolean functions, we present two constructions of aperiodic QCSSs with parameters ((q(p_0-1),q,q-t,q)) and ((q^m(p_0-1),q^m,q^m-t,q^m)), where (qge 3) is an odd integer, (p_0) is the minimum prime factor of q. The proposed constructions can generate asymptotically optimal or near-optimal aperiodic QCSSs with new parameters.

By using the notion of a d-embedding (Gamma ) of a (canonical) subgeometry (Sigma ) and of exterior sets with respect to the h-secant variety (Omega _{h}({mathcal {A}})) of a subset ({mathcal {A}}), ( 0 le h le n-1), in the finite projective space ({textrm{PG}}(n-1,q^n)), (n ge 3), in this article we construct a class of non-linear (n, n, q; d)-MRD codes for any ( 2 le d le n-1). A code of this class ({mathcal {C}}_{sigma ,T}), where (1in T subseteq {mathbb {F}}_q^*) and (sigma ) is a generator of (textrm{Gal}({mathbb {F}}_{q^n}|{mathbb {F}}_q)), arises from a cone of ({textrm{PG}}(n-1,q^n)) with vertex an ((n-d-2))-dimensional subspace over a maximum exterior set ({mathcal {E}}) with respect to (Omega _{d-2}(Gamma )). We prove that the codes introduced in Cossidente et al (Des Codes Cryptogr 79:597–609, 2016), Donati and Durante (Des Codes Cryptogr 86:1175–1184, 2018), Durante and Siciliano (Electron J Comb, 2017) are suitable punctured ones of ({mathcal {C}}_{sigma ,T}) and we solve completely the inequivalence issue for this class showing that ({mathcal {C}}_{sigma ,T}) is neither equivalent nor adjointly equivalent to the non-linear MRD codes ({mathcal {C}}_{n,k,sigma ,I}), (I subseteq {mathbb {F}}_q), obtained in Otal and Özbudak (Finite Fields Appl 50:293–303, 2018).

Recently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on the prime field ({mathbb {F}}_p) for some large prime p. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives is new, a rigorous cryptanalysis of such primitives is yet to be done. In this paper, we investigate arithmetization-oriented APN functions. More precisely, we investigate APN permutations in the CCZ-classes of known families of APN power functions over the prime field ({mathbb {F}}_p). Moreover, we present a class of binomial permutation having differential uniformity at most 5 defined via the quadratic character over finite fields of odd characteristic. Computationally it is confirmed that the latter family contains new APN permutations for some small parameters. We conjecture it to contain an infinite subfamily of APN permutations.

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural q-analog of a subcube partition of the hypercube (which can be seen as a polar space with (q=1)). Our main result proves that a generalized ovoid of k-spaces in polar spaces of large rank does not exist.

We consider the problem of error correction in a network where the errors can occur only on a proper subset of the network edges. For a generalization of the so-called Diamond Network we consider lower and upper bounds for the network’s (1-shot) capacity for fixed alphabet sizes.

Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for (t > 1) were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength (t=2), resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over (mathbb {F}_2) and (mathbb {F}_3).

Complex orthogonal designs (CODs) have been used to construct space-time block codes. Its real analog, real orthogonal designs, or equivalently, sum of squares composition formula, have a long history in mathematics. Driven by some practical considerations, Adams et al. (IEEE Trans Info Theory, 57(4):2254–2262, 2011) introduced the definition of balanced complex orthogonal designs (BCODs). The code rate of BCODs is 1/2, and their minimum decoding delay is proven to be (2^m), where 2m is the number of columns. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters ([2^m, 2m, 2^{m-1}]), and moreover, they are all equivalent.

A (v, k, t) packing of size b is a system of b subsets (blocks) of a v-element underlying set such that each block has k elements and every t-set is contained in at most one block. P(v, k, t) stands for the maximum possible b. A packing is called abundant if (b> v). We give new estimates for P(v, k, t) around the critical range, slightly improving the Johnson bound and asymptotically determine the minimum (v=v_0(k,t)) when abundant packings exist. For a graph G and a positive integer c, let (chi _ell (G,c)) be the minimum value of k such that one can properly color the vertices of G from any assignment of lists L(v) such that (|L(v)|=k) for all (vin V(G)) and (|L(u)cap L(v)|le c) for all (uvin E(G)). Kratochvíl, Tuza and Voigt in 1998 asked to determine (lim _{nrightarrow infty } chi _ell (K_n,c)/sqrt{cn}) (if it exists). Using our bound on (v_0(k,t)), we prove that the limit exists and equals 1. Given c, we find the exact value of (chi _ell (K_n,c)) for infinitely many n.

In this paper, we introduce Bandersnatch, a new elliptic curve built over the BLS12-381 scalar field. The curve is equipped with an efficient endomorphism, allowing a fast scalar multiplication algorithm. Our benchmark shows that the multiplication is 42% faster, 21% reduction in terms of circuit size in the form of rank 1 constraint systems (R1CS), and 10% reduction in terms of Plonk circuit, compared to another curve, called Jubjub, having similar properties. Many zero-knowledge proof systems that rely on the Jubjub curve can benefit from our result.