At present, existing multi-receiver authentication (MRA) schemes can only handle situations where the capacities of all receivers in the schemes are the same. However, in reality, different receivers may need to have different storage capacities. In this paper, inspired by the secret sharing scheme based on the Chinese Remainder Theorem (CRT) for polynomial rings where each participant holds the share with different sizes, we propose three new constructions of unconditionally secure MRA schemes for multiple messages using the CRT for polynomial rings, including a$(k,n)$-threshold MRA scheme, a$(k,n,\omega)$-weighted threshold MRA scheme, and a$(\mathcal {Q},\mathcal {F})$-general MRA scheme. As far as we know, our proposed MRA schemes are the first MRA schemes with different storage capacities for different receivers and the first ones based on the CRT for polynomial rings. Moreover, the proposed schemes can be seen as extensions of the MRA scheme in Safavi-Naini and Wang. In particular, as for our$(\mathcal {Q},\mathcal {F})$-general MRA scheme, it has generally more communication complexity and much less computation complexity than the existing$(\mathcal {Q},\mathcal {F})$-general MRA scheme.

At present, existing multi-receiver authentication (MRA) schemes can only handle situations where the capabilities of all receivers are the same. However, in reality, different receivers may need to have different storage capabilities. In this paper, inspired by the secret sharing scheme based on the Chinese Remainder Theorem (CRT) for polynomial rings where distinct participants save shares with distinct sizes, we propose a new unconditionally secure MRA scheme for multiple messages by the same technique. As far as we know, our MRA scheme is the first MRA scheme with different storage capacities for different receivers and the first one based on the CRT for polynomial rings supporting general access structures. In contrast to the existing general MRA scheme, although our MRA scheme has more communication complexity, it has less computation complexity.

In this paper, for the first time, we propose a new explicit hierarchical threshold secret sharing (HTSS) scheme based on the Chinese Remainder Theorem (CRT) for polynomial rings, where the participant set is divided into disjoint subsets and the threshold of a superior subset is less than the threshold of an inferior subset. In addition, we present a rigorous security analysis to show that our HTSS scheme is both perfect and ideal. Moreover, a toy example of our HTSS scheme is given to enable readers to better understand our construction. By comparison, it appears that our scheme is the first CRT-based HTSS for polynomial rings and also the first ideal and perfect CRT-based HTSS scheme, which is easier to construct than its counterpart for integer rings, where different participants hold shares of different sizes. Besides, our HTSS can also distribute shares of the same size, similar to other HTSS.

Vectorial dual-bent functions have recently attracted some researchers’ interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions, and linear codes. In this paper, we further study vectorial dual-bent functions$F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$, where$2\leq m \leq \frac {n}{2}$, and$V_{n}^{(p)}$denotes an n-dimensional vector space over the prime field$\mathbb {F}_{p}$. For certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A), we present a more concise characterization in terms of partial difference sets than the one given in Wang et al. (2023), and give new characterizations in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices, respectively. When$p=2$, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Through the relationship between vectorial dual-bent functions and bent partitions, new characterizations of certain bent partitions in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices are obtained. For a vectorial dual-bent function$F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$with$F(0)=0, F(x)=F(-x)$, where$2\leq m \leq \frac {n}{2}$, we give a necessary and sufficient condition under which the preimage set partition of F induces an association scheme. By using two classes of vectorial dual-bent functions, more association schemes are obtained.

From the perspective of parity-check matrices, we present new constructions of$q$-ary maximum distance separable (MDS) array codes with multiple parities. Applying these constructions, some new types of MDS array codes with array numbers$m-\tau $can be derived, where${\mathrm{ gcd}}(m,q)=1$. Moreover, an explicit construction of binary MDS array codes is also presented. Compared to the existing MDS array codes, one important characteristic of these codes is that their available code lengths are much longer, which is suitable for large-scale storage systems. In some particular cases, the maximum code lengths of these codes and their extension can be up to$2^{m-\tau }$and$2^{m-\tau }+1$(or$2^{m-\tau }+2$), respectively. Moreover, to demonstrate the applicability of our constructed MDS array codes, we present an effective generic decoding method for the erased errors. In particular, when there are no more than three erasures occurring, a scheduled algorithm for the syndrome computation of our explicit construction is further proposed, whose computational complexity is asymptotically optimal. Furthermore, this algorithm can be directly applied to the encoding procedure of their extended form. The simulation shows that our new MDS array codes have better encoding and decoding performances than the corresponding extended RS codes coupled with different algorithms.

A verifiable multi‐secret sharing (VMSS) scheme allows distributors to share multiple secrets simultaneously and can detect fraud by both distributors and participants. After analysing the security of the VMSS schemes proposed by Dehkordi and Mashhadi in 2015, the authors point out that they could not detect the fraudulent behaviour of the dealer. By using the non‐homogeneous linear recursion and linear feedback shift rigister (LFSR) public key cryptosystem, they introduce two new VMSS schemes. The proposed schemes can not only overcome the defects mentioned above, but also have shorter private and public key lengths at the same level of security. Besides, the proposed schemes are dynamic.