Designs, Codes and Cryptography最新文献

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.

A De Bruijn cycle is a cyclic sequence in which every word of length n over an alphabet (mathcal {A}) appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words, which shorten De Bruijn cycles using a wildcard character, we introduce universal partial tori and matrices. We find them computationally and construct infinitely many of them using one-dimensional variants of universal cycles, including a new variant called a universal partial family.

Nonlinear feedback shift registers (NFSRs) are used in many recent stream ciphers as their main building blocks. Two NFSRs are said to be isomorphic if their state diagrams are isomorphic, and to be equivalent if their sets of output sequences are equal. So far, numerous work has been done on the equivalence of NFSRs with same bit number, but much less has been done on their isomorphism. Actually, the equivalence problem of NFSRs with same bit number can be transformed to their isomorphism problem. The latter can be solved if the bijection between their states and its inverse can be explicitly expressed, which are quite hard to get in general. This paper studies the isomorphism of NFSRs by building a general framework for bijections. It first gives basic bijections. It then presents a unified formula for bijections, and discloses that any bijection can be expressed as a composite of finite basic bijections, setting up a general framework for bijections. Based on the general framework, the paper discloses in theory how to obtain all Galois NFSRs that are isomorphic to a given NFSR, and then reveals the bijections between the states of the previous types of Galois NFSRs and their own equivalent Fibonacci NFSRs. Finally, it proposes a new type of Galois NFSRs that are isomorphic and further equivalent to Fibonacci NFSRs, covering and improving most previous types of Galois NFSRs known to be equivalent to Fibonacci NFSRs.

In this paper, for an odd prime power q and an integer (mge 2), let (mathcal {C}(q,m)) be a one-weight irreducible cyclic code with parameters ([q^m-1,m,(q-1)q^{m-1}]), we consider the complete weight enumerator and the weight distribution of the square (big (mathcal {C}(q,m)big )^2), whose dual has (lfloor frac{m}{2}rfloor +1) zeros. Using the character sums method and the known result of counting (mtimes m) symmetric matrices over (mathbb {F}_q) with given rank, we explicitly determine the complete weight enumerator of (left( mathcal {C}(q,m)right) ^2) and show that (left( mathcal {C}(q,m)right) ^2) is a ((2lfloor frac{m}{2}rfloor +1))-weight cyclic code with parameters ([q^{m}-1,frac{m(m+1)}{2},(q-1)(q^{m-1}-q^{m-2})]). Moreover, we get the weight distribution of the square of the simplex code by puncturing the last (frac{(q-2)(q^m-1)}{q-1}) coordinates of (left( mathcal {C}(q,m)right) ^2).

The learning parity with noise (LPN) problem underlines several classic cryptographic primitives. Researchers have attempted to show the algorithmic difficulty of this problem by finding a reduction from the decoding problem of linear codes, for which several hardness results exist. Earlier studies used code smoothing as a technical tool to achieve such reductions for codes with vanishing rate. This has left open the question of attaining a reduction with positive-rate codes. Addressing this case, we characterize the efficiency of the reduction in terms of the parameters of the decoding and LPN problems. As a conclusion, we isolate the parameter regimes for which a meaningful reduction is possible and the regimes for which its existence is unlikely.

In this paper, we introduce and study the problem of binary stretch embedding of edge-weighted graphs in both integer and fractional settings. Roughly speaking, the binary stretch embedding problem for a weighted graph G is to find a mapping from the vertex set of G, to the vertices of a hypercube graph such that the distance between every pair of the vertices is not reduced under the mapping, hence the name binary stretch embedding. The minimum dimension of a hypercube for which such a stretch embedding exists is called the binary addressing number of G. We show that the binary addressing number of weighted graphs is the optimum value of an integer program. The optimum value for the corresponding linear relaxation problem is called the fractional binary addressing number of G. This embedding type problem is closely related to the well-known addressing problem of Graham and Pollak and isometric hypercube embedding problem of Firsov. Using tools and techniques such as Hadamard codes and the linear programming theory help us to find upper and lower bounds, approximations, or exact values of the binary addressing number and the fractional variant of graphs. As an application of our results, we derive improved upper bounds or exact values of the maximum size of Lee metric codes of certain parameters.

A (2-(v, k, lambda )) design is additive if, up to isomorphism, the point set is a subset of an abelian group G and every block is zero-sum. This definition was introduced in Caggegi et al. (J Algebr Comb 45:271-294, 2017) and was the starting point of an interesting new theory. Although many additive designs have been constructed and known designs have been shown to be additive, these structures seem quite hard to construct in general, particularly when we look for additive Steiner 2-designs. One might generalize additive Steiner 2-designs in a natural way to graph decompositions as follows: given a simple graph (Gamma ), an additive ((K_v,Gamma ))-design is a decomposition of the graph (K_v) into subgraphs (blocks) (B_1,dots ,B_t) all isomorphic to (Gamma ), such that the vertex set (V(K_v)) is a subset of an abelian group G, and the sets (V(B_1), dots , V(B_t)) are zero-sum in G. In this work we begin the study of additive ((K_v,Gamma ))-designs: we develop different tools instrumental in constructing these structures, and apply them to obtain some infinite classes of designs and many sporadic examples. We will consider decompositions into various graphs (Gamma ), for instance cycles, paths, and k-matchings. Similar ideas will also allow us to present here a sporadic additive 2-(124, 4, 1) design.