Designs, Codes and Cryptography最新文献

This paper focuses on non-existence results for Cameron–Liebler k-sets. A Cameron–Liebler k-set is a collection of k-spaces in ({{,mathrm{textrm{PG}},}}(n,q)) or ({{,mathrm{textrm{AG}},}}(n,q)) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron–Liebler k-sets with parameter x. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron–Liebler k-set in ({{,mathrm{textrm{PG}},}}(n,q)) should be larger than (q^{n-frac{5k}{2}-1}), which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter x of Cameron–Liebler k-sets in ({{,mathrm{textrm{PG}},}}(n,q)) with (x<frac{q^{n-k}-1}{q^{k+1}-1}), (nge 2k+1), (n-k+1ge 7) and (n-k) even. In the affine case we show a similar result for (n-k+1ge 3) and (n-k) even. This is a generalization of earlier known modular equalities in the projective and affine case.

The HQC post-quantum cryptosystem enables two parties to share noisy versions of a common secret binary string, and an error-correcting code is required to deal with the mismatch between both versions. This code is required to deal with binary symmetric channels with as large a transition parameter as possible, while guaranteeing, for cryptographic reasons, a decoding error probability of provably not more than 2^-128. This requirement is non-standard for digital communications, and modern coding techniques are not amenable to this setting. This paper explains how this issue is addressed in the last version of HQC: precisely, we introduce a coding scheme that consists of concatenating a Reed–Solomon code with the tensor product of a Reed–Muller code and a repetition code. We analyze its behavior in detail and show that it significantly improves upon the previous proposition for HQC, which consisted of tensoring a BCH and a repetition code. As additional results, we also provide a better approximation of the weight distribution for HQC error vectors, and we remark that the size of the exchanged secret in HQC can be reduced to match the protocol security which also significantly improves performance.

Tang and Ding (IEEE Trans Inf Theory 67(1):244–254, 2021) opened a new direction of searching for t-designs from elementary symmetric polynomials, which are used to construct the first infinite family of linear codes supporting 4-designs. In this paper, we first study the properties of elementary symmetric polynomials with 6 or 7 variables over (textrm{GF}(3^{m})). Based on them, we present more infinite families of 3-designs that contain some 3-designs with new parameters as checked by Magma for small numbers m. We also construct an infinite family of cyclic codes over (textrm{GF}(q^2)) and prove that the codewords of any nonzero weight support a 3-design. In particular, we present an infinite family of 6-dimensional AMDS codes over (textrm{GF}(3^{2m})) holding an infinite family of 3-designs and an infinite family of 7-dimensional NMDS codes over (textrm{GF}(3^{2m})) holding an infinite family of 3-designs.

In this paper, we propose a formal security model and a construction methodology of interactive aggregate message authentication codes with detecting functionality (IAMDs). The IAMD is an interactive aggregate MAC protocol which can identify invalid messages with a small amount of tag-size. Several aggregate MAC schemes that can detect invalid messages have been proposed so far by using non-adaptive group testing in the prior work. In this paper, we utilize adaptive group testing to construct IAMD scheme, and we show that the resulting IAMD scheme can identify invalid messages with a small amount of tag-size compared to the previous schemes. To this end, we give the formalization of adaptive group testing and IAMD, and propose a generic construction starting from any aggregate MAC and any adaptive group testing method. In addition, we compare instantiations of our generic constructions, in terms of total tag-size and several properties. Furthermore, we show advantages of IAMD by implementing constructions of (non-)adaptive aggregate message authentication with detecting functionality and comparing these ones in terms of the data-size and running time of verification algorithms.

Given a polynomial f over the finite field (mathbb {F}_q), its intersection distribution provides fruitful information on how lines in the affine plane intersect the graph of f over (mathbb {F}_q). The intersection distribution in its simplest cases gives rise to oval polynomials in finite geometry and Steiner triple systems in design theory. Previously, the intersection distribution of degree two and degree three polynomials has been computed. In this paper, we determine the intersection distribution of all degree four polynomials over finite fields. As an application, we present a direct construction of Steiner systems using polynomials with prescribed intersection distribution.

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.