Designs, Codes and Cryptography最新文献

Let ({mathcal {K}}) be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space ({mathbb {R}}^n), there is a well-studied notion of “ultrametric orthogonality” in ({mathcal {K}}^n). In this paper, motivated by a question of Erdős in the real case, given integers (k ge ell ge 2), we investigate the maximum size of a subset (S subseteq {mathcal {K}}^n {setminus }{textbf{0}}) satisfying the following property: for any (E subseteq S) of size k, there exists (F subseteq E) of size (ell ) such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.

We consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal ((2, delta ))-LRCs over (mathbb {F}_q) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a ((2, delta ))-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe–Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.

A functional commitment (FC) scheme enables committing to a vector ({textbf{x}}) and later producing an opening proof (pi ) for a function value (y=f({textbf{x}})) with function f in some function set ({mathcal {F}}). Everyone can verify the validity of the opening proof (pi ) w.r.t. the function f and the function value y. Up to now, the largest function set is the bounded-depth circuits and achieved by FC schemes in [Peikeit et al. TCC 2021, De Castro et al. TCC 2023, Wee et al. Eurocrypt 2023, Wee et al. Asiacrypt 2023] with the help of the homomorphic encoding and evaluation techniques from lattices. In fact, these FC schemes can hardly support circuits of large depth, due to the fast accumulation of noises in the homomorphic evaluations. For example, if the depth of the circuit is linear to the security parameter (lambda ), then the underlying (textsf {GapSVP}_{gamma }) problem will be accompanied with a super-exponentially large parameter (gamma >(lambda log lambda )^{Theta (lambda )}) and can be easily solved by the LLL algorithm. In this work, we propose a new FC scheme supporting arbitrary circuits of bounded sizes. We make use of homomorphic encoding and evaluation as well, but we disassemble the circuit gate by gate, process the gates, and reassemble the processed gates to a flattened circuit of logarithm depth (O(log lambda )). This makes possible for our FC scheme to support arbitrary polynomial-size circuits. Our FC scheme has the common reference string (CRS) growing linear to the size of the circuit. So CRSs of different sizes allow our FC scheme to support circuits of different (bounded) sizes. Just like the recent work on FC schemes [Wee et al. Eurocrypt 2023, Asiacrypt 2023], our FC scheme achieves private opening and target binding based on a falsifiable family of “basis-augmented” SIS assumptions. Our FC scheme has succinct commitment but not succinct opening proof which of course does not support fast verification. To improve the running time of verification, we resort to the non-interactive GKR protocol to outsource the main computation in verification to the proof generation algorithm. As a result, we obtain an improved FC scheme which decreases the computational complexity of verification with a factor (O(lambda )).

In this paper, we study properties and constructions of a general family of involutions of finite abelian groups, especially those of finite fields. The involutions we are interested in have the form (lambda +gcirc tau ), where (lambda ) and (tau ) are endomorphisms of a finite abelian group and g is an arbitrary map on this group. We present some involutions explicitly written as polynomials for the special cases of multiplicative and additive groups of finite fields.

We describe a heuristic polynomial-time algorithm for breaking the NTRU problem with multiple keys when given a sufficient number of ring samples. Following the linearization approach of the Arora-Ge algorithm (ICALP ’11), our algorithm constructs a system of linear equations using the public keys. Our main contribution is a kernel reduction technique that extracts the secret vector from a linear space of rank n, where n is the degree of the ring in which NTRU is defined. Compared to the algorithm of Kim-Lee (Designs, Codes and Cryptography, ’23), our algorithm does not require prior knowledge of the Hamming weight of the secret keys. Our algorithm is based on some plausible heuristics. We demonstrate experiments and show that the algorithm works quite well in practice, with close to cryptographic parameters.

Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of permutation binomials is still unknown. Let (q=2^n) for a positive integer n. In this paper, we start classifying permutation binomials of the form (x^i+ax) over ({mathbb {F}}_{q}) in terms of their indices. After carrying out an exhaustive search of these permutation binomials over ({mathbb {F}}_{2^n}) for n up to 12, we gave three new infinite classes of permutation binomials over ({mathbb {F}}_{q^2}), ({mathbb {F}}_{q^3}), and ({mathbb {F}}_{q^4}) respectively, for (q=2^n) with arbitrary positive integer n. In particular, these binomials over ({mathbb {F}}_{q^3}) have relatively large index (frac{q^2+q+1}{3}). As an application, we can completely explain all the permutation binomials of the form (x^i+ax) over ({mathbb {F}}_{2^n}) for (nle 8). Moreover, we prove that there does not exist permutation binomials of the form (x^{2q^3+2q^2+2q+3}+ax) over ({mathbb {F}}_{q^4}) such that (ain {mathbb {F}}_{q^4}^*) and (n=2,m) with (mge 2).

In this paper, we study block-transitive automorphism groups of t-((k^2,k,lambda )) designs. We prove that a block-transitive automorphism group G of a t-((k^2,k,lambda )) design must be point-primitive, and G is either an affine group or an almost simple group. Moreover, the nontrivial t-((k^2,k,lambda )) designs admitting block-transitive automorphism groups of almost simple type with sporadic socle and alternating socle are classified.

A finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over (mathbb {F}_q) equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-((n,k,lambda )) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly (lambda ) members of Y. Nontrivial examples are currently only known for (tle 2). We show that t-((n,k,lambda )) designs in polar spaces exist for all t and q provided that (k>frac{21}{2}t) and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

This article analyzes a key exchange protocol based on a modified tropical structure proposed by Ahmed et al. in 2023. It is shown that the modified tropical semiring is isomorphic to the (2times 2) tropical circular matrix semiring. Therefore, matrices in this modified tropical semiring can be represented as tropical matrices, and the key exchange protocol is actually based on the tropical matrix semiring. Tropical irreducible matrices exhibit almost linear periodic property. Efficient algorithms for calculating the linear period and defect of irreducible matrices are designed. Based on the public information of the protocol, the equivalent private key can be computed and then the shared key is easily obtained. The analysis shows that the key exchange protocol based on this modified tropical structure is not secure.