Designs, Codes and Cryptography最新文献

The quantum security of message authentication codes (MACs) has been gaining increasing attention in recent years, particularly with regard to proving the quantum security of classical MACs, which has emerged as a significant area of interest. In this work, we present two variants of classical MACs: QPMAC, a quantum-secure parallel version of PMAC, and QCBCMAC, a quantum-secure variant of CBCMAC and NMAC that supports variable-length input. We demonstrate that QPMAC is a parallel quantum-secure MAC, with an inverse relationship between its degree of parallelism and its level of quantum security. On the other hand, QCBCMAC provides quantum security for variable-length inputs. To achieve an optimal balance between parallelism and quantum security, we propose QPCBC, a hybrid construction that combines the strengths of QPMAC and QCBCMAC. We also provide an instantiation of QPCBC using tweakable block ciphers.

Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in Özbudak and Pelen (J Algebr Comb 56:635–658, 2022), Özbudak and Pelen constructed infinite families of symmetric association schemes of classes 5 and 6 by using ternary non-weakly regular bent functions. They also stated that “constructing 2p-class association schemes from p-ary non-weakly regular bent functions is an interesting problem", where (p>3) is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes 2p, ((2p+1)) and (frac{3p+1}{2}) for any odd prime p. Fusing those association schemes, we obtain t-class symmetric association schemes, where (t=4,5,6,7). In addition, we give the sufficient and necessary conditions for the partitions P, D, T, U and V (defined in this paper) to induce symmetric association schemes.

An H-decomposition of a graph (Gamma ) is a partition of its edge set into subgraphs isomorphic to H. A transitive decomposition is a special kind of H-decomposition that is highly symmetrical in the sense that the subgraphs (copies of H) are preserved and transitively permuted by a group of automorphisms of (Gamma ). This paper concerns transitive H-decompositions of the graph (K_n Box K_n) where H is a path. When n is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are considerably large compared to the number of vertices. Our main result supports well-known Gallai’s conjecture and an extended version of Ringel’s conjecture.

For each odd prime power q, we present two rational functions (f(X)in mathbb {F}_q(X)) which have the unusual property that, for every odd n, the function induced by f(X) on (mathbb {F}_{q^n}setminus mathbb {F}_q) is ((q-1))-to-1.

For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of synchronising and separating are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of spreading permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an AB-Lemma) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

Nonlinear feedback shift registers (NFSRs) are used in many recent stream ciphers as their main building blocks. One security criterion for the design of a stream cipher is to assure its used NFSR has a long period. As the period of a Fibonacci NFSR is equal to its largest cycle length, a common way to get a maximum-period Fibonacci NFSR is to join the cycles of an original Fibonacci NFSR into a maximum cycle. Nevertheless, so far only the maximum-period Fibonacci NFSRs with stage numbers no greater than 33 have been found. Considering that Galois NFSRs may have higher implementation efficiency than Fibonacci NFSRs, this paper first generalizes the cycle joining method for Fibonacci NFSRs to Galois NFSRs and establishes some conditions for maximum-period Galois NFSRs. It then reveals the cycle structure of some cascade connections of two Fibonacci NFSRs. Based on both, the paper constructs some long-period Galois NFSRs including maximum-period Galois NFSRs with stage numbers up to 41. Finally, it analyzes their hardware implementation via the technology mapping obtained by synthesizing the NFSRs with Synopsys Design Compiler L(-)2016.03-Sp1 using the TSMC 90nm CMOS library, and the results show that they have good hardware performance.

It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one (-1) entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. The proofs rely on solving polynomial systems of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension 6.

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.

In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size (M=2^k) so that codewords can conveniently be labeled with binary vectors of length k. For every integer (ell ge 3), we construct a ((n=2^ell , M=2^{k_{ell }}, d=2)) constant weight code ({{{mathcal {C}}}}[ell ]) of weight (ell ) by encoding information in the gaps between successive 1’s of a vector, and call them as cyclic-gap constant weight codes. The code is associated with a finite integer sequence of length (ell ) satisfying a constraint defined as anchor-decodability that is pivotal to ensure low complexity for encoding and decoding. The time complexity of the encoding algorithm is linear in the input size k, and that of the decoding algorithm is poly-logarithmic in the input size n, discounting the linear time spent on parsing the input. Both the algorithms do not require expensive computation of binomial coefficients, unlike the case in many existing schemes. Among codes generated by all anchor-decodable sequences, we show that ({{{mathcal {C}}}}[ell ]) has the maximum size with (k_{ell } ge ell ^2-ell log _2ell + log _2ell - 0.279ell - 0.721). As k is upper bounded by (ell ^2-ell log _2ell +O(ell )) information-theoretically, the code ({{{mathcal {C}}}}[ell ]) is optimal in its size with respect to two higher order terms of (ell ). In particular, (k_ell ) meets the upper bound for (ell =3) and one-bit away for (ell =4). On the other hand, we show that ({{{mathcal {C}}}}[ell ]) is not unique in attaining (k_{ell }) by constructing an alternate code (mathcal{{hat{C}}}[ell ]) again parameterized by an integer (ell ge 3) with a different low-complexity decoder, yet having the same size (2^{k_{ell }}) when (3 le ell le 7). Finally, we also derive new codes by modifying ({{{mathcal {C}}}}[ell ]) that offer a wider range on blocklength and weight while retaining low complexity for encoding and decoding. For certain selected values of parameters, these modified codes too have an optimal k.

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.