Designs, Codes and Cryptography最新文献

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can be used to obtain error-correcting codes with prescribed parameters and an efficient decoding algorithm.

The hardness of discrete logarithm problem (DLP) over finite fields forms the security foundation of many cryptographic schemes. When the characteristic is not small, the state-of-the-art algorithms for solving the DLP are the number field sieve (NFS) and its variants. NFS first computes the logarithms of the factor base, which consists of elements of small norms. Then, for a target element, its logarithm is calculated by establishing a relation with the factor base. Although computing the factor-base elements is the most time-consuming part of NFS, it can be performed only once and treated as pre-computation for a fixed finite field when multiple logarithms need to be computed. In this paper, we present a method for accelerating individual logarithm computation by utilizing two subfields. We focus on the case where the extension degree of the finite field is a multiple of 6 within the extended tower number field sieve framework. Our method allows for the construction of an element with a lower degree, while maintaining the same coefficient bound compared to Guillevic’s method, which uses only one subfield. Consequently, the element derived from our approach enjoys a smaller norm, which will improve the efficiency in individual logarithm computation.

We introduce the concept of a sum–rank saturating system and outline its correspondence to covering properties of a sum–rank metric code. We consider the problem of determining the shortest length of a sum–rank-(rho )-saturating system of a fixed dimension, which is equivalent to the covering problem in the sum–rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.

We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the (2^r 3^s)th cyclotomic field for (r ge 3) and (s ge 1). Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with ({mathcal {O}}(n log n)) arithmetic operations, where n is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the pth cyclotomic field versus the maximal totally real subextension of the 4pth cyclotomic field for a reasonable set of parameters of cryptographic size.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Constructing binary cyclic codes with parameters ([n, frac{n+1}{2}, d ge sqrt{n}]) is an interesting topic in coding theory, as their minimum distances have a square-root bound. Let (n=2^lambda -1), where (lambda ) has three forms: (p^2, p_1p_2, 2p_2) for odd primes (p, p_1, p_2). In this paper, we mainly construct several classes of binary cyclic codes with parameters ([2^lambda -1, k ge 2^{lambda -1}, d ge sqrt{n}]). Specifically, the binary cyclic codes ({mathcal {C}}_{(1, p^2)}), ({mathcal {C}}_{(1, 2p_2)}), ({mathcal {C}}_{(2, 2p_2)}), and ({mathcal {C}}_{(1, p_1p_2)}) have minimum distance (d ge sqrt{n}) though their dimensions satisfy (k > frac{n+1}{2}). Moreover, two classes of binary cyclic codes ({mathcal {C}}_{(2, p^2)}) and ({mathcal {C}}_{(2, p_1p_2)}) with dimension (k= frac{n+1}{2}) and minimum distance d much exceeding the square-root bound are presented, which extends the results given by Sun, Li, and Ding [30]. In fact, the rate of these two classes of binary cyclic codes are around (frac{1}{2}) and the lower bounds on their minimum distances are close to (frac{n}{log _2 n}). In addition, their extended codes are also investigated.

In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates. Transversal gates are of significant interest to the quantum information community for their central role in fault tolerant quantum computing. Most unitary (text {t})-designs have never been realized as the transversal gate group of a quantum code. We, for the first time, find nontrivial quantum codes realizing nearly every finite group which is a unitary 2-design or better as the transversal gate group of some error-detecting quantum code.

Let p be a prime number, m be a positive integer, and (q=p^m). For any fixed locality r such that (pnot mid r(r+1)), we construct infinite families of locally recoverable codes with availabilty of nodes lower bounded by (q/r!+O(sqrt{q})) and number of locality sets equal to (q^2/(r+1)!+O(q^{3/2})).

In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.

The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by (x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + cdots + y) over the finite field of cardinality (q^s), where u is a positive divisor of (frac{q^s - 1}{q - 1}). As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters.

Nonlinear feedback shift registers (NFSRs) are widely used in the design of stream ciphers and the cycle structure of an NFSR is a fundamental problem still open. In this paper, a new configuration of Galois NFSRs, called F-Ring NFSRs, is proposed. It is shown that an n-bit F-Ring NFSR generates n sequences with the same period simultaneously, that is, sequences from all bit registers have the same period. Recall that the ring-like cascade connection proposed by Zhao et al. (Des Codes Cryptogr 86:2775–2790, 2018) also has such period property. But it is abnormal that if every component shift register is nonsingular, then the ring-like cascade connection is singular. F-Ring NFSRs proposed in this paper could fix this weakness. Moreover, it is proved that when an n-stage m-sequence is input to the internal state of an F-Ring NFSR by xor, the periods of its internal state are multiples of (2^n-1). At last, two toy examples are given to illustrate the new configuration.