Designs, Codes and Cryptography最新文献

In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in ({{,textrm{PG},}}(n,q)) with q odd. We define relations on the anisotropic points of such a quadric that depend on the type of line spanned by the points and whether or not they are of the same “quadratic type”. This yields an imprimitive 5-class association scheme. We calculate the matrices of eigenvalues and dual eigenvalues of this scheme. We also use this result, together with similar results from the literature concerning other classical polar spaces, to exactly calculate the spectrum of orthogonality graphs on the anisotropic points of non-degenerate quadrics in odd characteristic and of non-degenerate Hermitian varieties. As a byproduct, we obtain a 3-class association scheme on the anisotropic points of non-degenerate Hermitian varieties, where the relation containing two points depends on the type of line spanned by these points, and whether or not they are orthogonal.

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

The purpose of this paper is to establish a one-to-one correspondence between k-tuples of DNA codewords and the elements of a finite group to simulate the reverse and the complement operations of codewords by an automorphism and a translation of the group, respectively, in order to determine the structure of DNA codes. Finally, the case of vector spaces is characterized.

Let a and b be two non-zero elements of a finite field (mathbb {F}_q), where (q>2). It has been shown that if a and b have the same multiplicative order in (mathbb {F}_q), then the families of a-constacyclic and b-constacyclic codes over (mathbb {F}_q) are monomially equivalent. In this paper, we investigate the monomial equivalence of a-constacyclic and b-constacyclic codes when a and b have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than 70 new record-breaking linear codes over various finite fields, as well as new binary quantum codes.

Truncated differential cryptanalyses were introduced by Knudsen in 1994. They are a well-known family of attacks that has arguably received less attention than some other variants of differential attacks. This paper gives some new insights into the theory of truncated differential attacks, specifically the conditions of provable security of SPN ciphers with MDS diffusion matrices against this type of attack. Furthermore, our study extends to various versions within the QARMA family of block ciphers, unveiling the only valid instances of single-tweak attacks on 10-round QARMAv1-64, 10-round QARMAv1-128, and 10- and 11-round QARMAv2-64. These attacks benefit from the optimal truncated differential distinguishers as well as some evolved key-recovery techniques.

Secret sharing is a general method for distributing sensitive data among the participants of a system such that only a collection of predefined qualified coalitions can recover the secret data. One of the most widely used special cases is threshold secret sharing, where every subset of participants of size above a given number is qualified. In this short note, we propose a general construction for a generalized threshold scheme, called conjunctive hierarchical secret sharing, where the participants are divided into disjoint levels of hierarchy, and there are different thresholds for all levels, all of which must be satisfied by qualified sets. The construction is the first method for arbitrary parameters based on finite geometry arguments and yields an improvement in the size of the underlying finite field in contrast with the existing results using polynomials.

For a q-polynomial L over a finite field (mathbb {F}_{q^n}), we characterize the differential spectrum of the function (f_L:mathbb {F}_{q^n} rightarrow mathbb {F}_{q^n}, x mapsto x cdot L(x)) and show that, for (n le 5), it is completely determined by the image of the rational function (r_L :mathbb {F}_{q^n}^* rightarrow mathbb {F}_{q^n}, x mapsto L(x)/x). This result follows from the classification of the pairs (L, M) of q-polynomials in (mathbb {F}_{q^n}[X]), (n le 5), for which (r_L) and (r_M) have the same image, obtained in Csajbók et al. (Ars Math Contemp 16(2):585–608, 2019). For the case of (n>5), we pose an open question on the dimensions of the kernels of (x mapsto L(x) - ax) for (a in mathbb {F}_{q^n}). We further present a link between functions (f_L) of differential uniformity bounded above by q and scattered q-polynomials and show that, for odd values of q, we can construct CCZ-inequivalent functions (f_M) with bounded differential uniformity from a given function (f_L) fulfilling certain properties.

In this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

In the combinatorial context, one of the key problems in sequence reconstruction is to find the largest intersection of two metric balls of radius r. In this paper we study this problem for permutations of length n distorted by Hamming errors and determine the size of the largest intersection of two metric balls with radius r whose centers are at distance (d=2,3,4). Moreover, it is shown that for any (ngeqslant 3) an arbitrary permutation is uniquely reconstructible from four distinct permutations at Hamming distance at most two from the given one, and it is proved that for any (ngeqslant 4) an arbitrary permutation is uniquely reconstructible from (4n-5) distinct permutations at Hamming distance at most three from the permutation. It is also proved that for any (ngeqslant 5) an arbitrary permutation is uniquely reconstructible from (7n^2-31n+37) distinct permutations at Hamming distance at most four from the permutation. Finally, in the case of at most r Hamming errors and sufficiently large n, it is shown that at least ({varTheta }(n^{r-2})) distinct erroneous patterns are required in order to reconstruct an arbitrary permutation.

A corrector is a critical component of True Random Number Generators (TRNGs), serving as a post-processing function to reduce statistical weaknesses in raw random sequences. It is important to note that a (textit{t})-resilient Boolean function is a (textit{t})-corrector, while the converse is not necessarily true. Building upon the pioneering method introduced by Zhang in 2023 for constructing nonlinear correctors with correction order one greater than resiliency order, this paper presents for the first time two approaches for constructing nonlinear plateaued correctors with correction order at least two greater than resiliency order via Walsh spectral neutralization technique, and the resulting correctors have algebraic degree at least (text {2}). The first approach yields (textit{n})-variable plateaued correctors with correction order (textit{n}-text {2}) and resiliency order approximately (textit{n}- text {log}_text {2} textit{n}). The nonlinearity and algebraic degree of the resulting correctors are also analyzed, demonstrating that they meet both Siegenthaler’s and Sarkar-Maitra’s bounds. Another approach based on Walsh spectral neutralization technique for constructing (textit{n})-variable plateaued correctors is proposed. This approach facilitates the design of semi-bent correctors with algebraic degree (lceil frac{textit{n}}{text {2}} rceil ), correction order (lfloor frac{textit{n}}{text {2}} rfloor -text {1}) and resiliency order approximately ( frac{textit{n}}{text {4}} ).