Designs, Codes and Cryptography最新文献

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.

Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the (2^ell ) binary sequences of length (ell ) and investigate the distribution of the demerit factors of these sequences. Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the pth central moment of the demerit factor for binary sequences of length (ell ). These techniques prove that for (pge 2) and (ell ge 4), all the central moments are strictly positive. For any given p, one may use the technique to obtain an exact formula for the pth central moment of the demerit factor as a function of the length (ell ). Jedwab’s formula for variance is confirmed by our technique with a short calculation, and we go beyond previous results by also deriving an exact formula for the skewness. A computer-assisted application of our method also obtains exact formulas for the kurtosis, which we report here, as well as the fifth central moment.

A storage code is an assignment of symbols to the vertices of a connected graph G(V, E) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in G. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on ({mathbb Z}) and ({mathbb Z}^2) (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.