Designs, Codes and Cryptography最新文献

In distributed storage systems, an r-Locally Repairable Code (r-LRC) ensures that a failed symbol can be recovered by accessing at most r other symbols. Prakash et al. in (Proceedings of IEEE International Symposium on Information Theory, pp. 2776–2780, 2012) further introduced the concept of ((r, delta ))-LRC, where (delta ge 2), which can deal with the symbol failure in the presence of extra (delta -2) symbol failures still by accessing at most r other symbols. In particular, an r-LRC is just an (r, 2)-LRC. Luo and Ling in (Des Codes Cryptogr 90:1271–1287, 2022) obtained some alphabet-optimal r-LRCs concerning the Cadambe–Mazumdar bound from optimal linear codes constructed by special projective spaces. In this paper, we generalize the results of Luo and Ling in (Des Codes Cryptogr 90:1271–1287, 2022). Firstly, we generalize the result of constructing optimal linear codes to larger code length. In particular, we present the conditions for the constructed linear codes to qualify as Griesmer codes or distance-optimal codes. Secondly, we explore the locality of the constructed codes. The novelty of our work lies in establishing the locality as ((r,delta ))-locality and ((r,delta ))-locality with availability, in contrast to the previous literature that only considered r-locality. In addition, through the analysis combining the code parameters and the Cadambe–Mazumdar-like bound for ((r,delta ))-LRCs, we obtained some alphabet-optimal ((r, delta ))-LRCs and alphabet-optimal ((r, delta ))-LRCs with availability.

We investigate CSS and CSS-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a CSS code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that CSS-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of CSS-T codes. We conclude with a simple construction of CSS-T codes from Hermitian curves. The paper also offers a concise introduction to CSS and CSS-T codes from the point of view of classical coding theory.

In situations where every item in a data set must be compared with every other item in the set, it may be desirable to store the data across a number of machines in such a way that any two data items are stored together on at least one machine. One way to evaluate the efficiency of such a distribution is by the largest fraction of the data it requires to be allocated to any one machine. The all-to-all comparison (ATAC) data limit for m machines is a measure of the minimum of this value across all possible such distributions. In this paper we further the study of ATAC data limits. We begin by investigating the data limits achievable using various classes of combinatorial designs. In particular, we examine the cases of transversal designs and projective Hjelmslev planes. We then observe relationships between data limits and the previously studied combinatorial parameters of fractional matching numbers and covering numbers. Finally, we prove a lower bound on the ATAC data limit that improves on one of Hall, Kelly and Tian, and examine the special cases where equality in this bound is possible.

Motivated by applications in DNA storage, we study a setting in which strings are affected by tandem-duplication errors. In particular, we look at two settings: disjoint tandem-duplication errors, and equal-length tandem-duplication errors. We construct codes, with positive asymptotic rate, for the two settings, as well as for their combination. Our constructions are duplication-free codes, comprising codewords that do not contain tandem duplications of specific lengths. Additionally, our codes generalize previous constructions, containing them as special cases.

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup (Grtimes {{,textrm{End},}}(G)) for a finite semigroup G. Given (gin G, sigma in {{,textrm{End},}}(G)), and (h=prod _{i=0}^{t-1}sigma ^i(g)) for some integer t, the SDLP((G,sigma )), for g and h, asks to determine t. As Shor’s algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. For generic semigroups, the best known algorithm for the SDLP is based on Kuperberg’s subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of semidirect product key exchange. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group G, we describe quantum algorithms for the SDLP in (Grtimes {textrm{Aut}}(G)) for the following two classes of instances: the first one is when G is solvable and the second is when G is a matrix group and a power of (sigma ) with a polynomially small exponent is an inner automorphism of G. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor’s factoring and discrete logarithm algorithms and well-known generalizations.

We introduce quaternary modified four (mu )-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four (mu )-circulant Hermitian self-dual codes. We also construct quaternary modified four (mu )-circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.

In this work, we propose two new types of codes with locality, namely, locally maximal recoverable (LMR) codes and (lambda )-maximally recoverable ((lambda )-MR) codes. The LMR codes are a subclass of codes with ((r, delta ))-locality such that they can correct h additional erasures in any one local set, in addition to having ((r, delta ))-locality. These codes are a restricted case of maximally recoverable (MR) codes, which enable recovery from all information-theoretically correctable erasure patterns in a local set. The (lambda )-MR codes are a subclass of LMR codes which can also handle (lambda ) erasures from any coordinate positions. We give constructions for both of these families of codes. We also study the LMR codes that satisfy the complementary dual property. It is well known that codes with this property are capable of safeguarding communication systems against fault injection attacks. We give a construction of distance-optimal cyclic LMR codes that satisfy the complementary dual property.

LCD codes and (almost) optimally extendable codes can be used to safeguard against fault injection attacks (FIA) and side-channel attacks (SCA) in the implementations of block ciphers. The first objective of this paper is to use a family of binary self-orthogonal codes given by Ding and Tang (Cryptogr Commun 12:1011–1033, 2020) to construct a family of binary LCD codes with new parameters. The parameters of the binary LCD codes and their duals are explicitly determined. It turns out that the codes by Ding and Tang are almost optimally extendable codes. The second objective is to prove that two families of known q-ary linear codes given by Heng et al. (IEEE Trans Inf Theory 66(11):6872–6883, 2020) are self-orthogonal. Using these two families of self-orthogonal codes, we construct another two families of q-ary LCD codes. The parameters of the LCD codes are determined and many optimal codes are produced. Besides, the two known families of q-ary linear codes are also proved to be almost optimally extendable codes.

Balanced generalized weight matrices are used to construct optimal constant weight codes that are monomially inequivalent to codes derived from the classical simplex codes. What’s more, these codes can be assumed to be generated entirely by (omega )-shifts of a single codeword where (omega ) is a primitive element of a Galois field. Additional constant weight codes are derived by projecting onto subgroups of the alphabet sets. These too are shown to be optimal.

In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field (mathbb {F}_{q}) with q elements. Given a finite separable extension (mathcal {M}/mathcal {F}) of function fields and an iso-dual AG-code (mathcal {C}) defined over (mathcal {F}), we provide a general method to lift the code (mathcal {C}) to another iso-dual AG-code (tilde{mathcal {C}}) defined over (mathcal {M}) under some assumptions on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian p-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the GGS function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.