Designs, Codes and Cryptography最新文献

Formally self-dual (FSD) codes and linear codes with small Euclidean (resp. Hermitian) hulls have recently attracted a lot of attention due to their theoretical and practical importance. However, there has been not much attention on FSD codes with small hulls. In this paper, we introduce two kinds of polynomial four Toeplitz codes and prove that they must be FSD. We characterize the linear complementary dual (LCD) properties and one-dimensional hull properties of such codes with respect to the Euclidean and Hermitian inner products. Using these characterizations, we find some improved binary, ternary Euclidean and quaternary Hermitian FSD LCD codes, as well as many non-equivalent ones that perform equally well with respect to best-known (FSD) LCD codes in the literature. Furthermore, some (near) maximum distance separable FSD codes with both one-dimensional Euclidean hull and one-dimensional Hermitian hull are also given as examples.

In this note we answer positively a question of Chris Godsil and Karen Meagher on the existence of a 2-design whose block graph has a non-canonical maximum clique without a design structure.

In this paper, we construct the first provably-secure isogeny-based (partially) blind signature scheme. While at a high level the scheme resembles the Schnorr blind signature, our work does not directly follow from that construction, since isogenies do not offer as rich an algebraic structure. Specifically, our protocol does not fit into the linear identification protocol abstraction introduced by Hauck, Kiltz, and Loss (EUROCYRPT’19), which was used to generically construct Schnorr-like blind signatures based on modules such as classical groups and lattices. Consequently, our scheme is provably secure in the random oracle model (ROM) against poly-logarithmically-many concurrent sessions assuming the subexponential hardness of the group action inverse problem. In more detail, our blind signature exploits the quadratic twist of an elliptic curve in an essential way to endow isogenies with a strictly richer structure than abstract group actions (but still more restrictive than modules). The basic scheme has public key size 128 B and signature size 8 KB under the CSIDH-512 parameter sets—these are the smallest among all provably secure post-quantum secure blind signatures. Relying on a new ring variant of the group action inverse problem ((textsf{rGAIP})), we can halve the signature size to 4 KB while increasing the public key size to 512 B. We provide preliminary cryptanalysis of ({textsf{rGAIP}} ) and show that for certain parameter settings, it is essentially as secure as the standard (textsf{GAIP}). Finally, we show a novel way to turn our blind signature into a partially blind signature, where we deviate from prior methods since they require hashing into the set of public keys while hiding the corresponding secret key—constructing such a hash function in the isogeny setting remains an open problem.

In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over (mathbb {F}_{2^r}). These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes.

In this paper, we introduce regular permutation inner products which contain the Euclidean inner product. And we generalize some properties of the Euclidean inner product to regular permutation inner products. As application, we construct a lot of cyclic codes with specific regular permutation hulls and also obtain the dimensions and distances of some BCH codes.

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of q-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square q, almost q/8 NMDS self-dual q-ary codes can be constructed.

Spence [9] constructed (left( frac{3^{d+1}(3^{d+1}-1)}{2}, frac{3^d(3^{d+1}+1)}{2}, frac{3^d(3^d+1)}{2}right) )-difference sets in groups (K times C_3^{d+1}) for d any positive integer and K any group of order (frac{3^{d+1}-1}{2}). Smith and Webster [8] have exhaustively studied the (d=1) case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in (A_4 times C_3) by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to (C_3^2). Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to (C_3^{d+1}) as long as (frac{3^{d+1}-1}{2}) is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that (frac{3^{d+1}-1}{2}) is a prime power. We conjecture that any group of order (frac{3^{d+1}(3^{d+1}-1)}{2}) with a normal subgroup isomorphic to (C_3^{d+1}) will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group (textrm{Aut}(mathcal {D})) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of (textrm{Aut}(mathcal {D})), uses (3, 3, 3, 1)-relative difference sets to describe the difference set.

In Pasalic et al. (IEEE Trans Inf Theory 69:2702–2712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235–249, 2018), two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from ({mathbb {F}}_2^n) to the cyclic group ({mathbb {Z}}_{2^k}). It is shown how to obtain nega-({mathbb {Z}}_{2^k})-bent functions from ({mathbb {Z}}_{2^k})-bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of ({mathbb {Z}}_8)-bent functions employing permutations with the (({mathcal {A}}_m)) property, and more generally we show that the inverse permutation gives rise to ({mathbb {Z}}_{2^k})-bent functions.

A 2-((v,k,lambda )) design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group G in such a way that its block set is contained in (or coincides with) the set of all zero-sum k-subsets of its point set. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG(_d(n,q)), which was known to be additive only for (q=2) or (d=n-1), is always established.

In this note we answer some questions on (text{ LCD }) group codes posed in de la Cruz and Willems (Des Codes Cryptogr 86:2065–2073, 2018) and (Vietnam J Math 51:721–729, 2023). Furthermore, over prime fields we determine completely the p-part of the divisor of an (text{ LCD }) group code. In addition we present a natural construction of nearly (text{ LCD }) codes.