In this paper, we look at self-dual codes over the ring Z 16 of integers modulo 16. From any doubly even self-dual binary code, we construct codes over Z 16 and give a necessary and sufficient condition for the self-duality of induced codes. We then give an inductive algorithm for constructing all self-dual codes over Z 16 , and establish the mass formula, which counts the number of such codes.
Several constructions in binary linear block codes are also related to matroid theory topics. These constructions rely on a given order in the ground set of the matroid. In this paper we define the Gröbner representation of a binary matroid and we show how it can be used for studying different sets bases, cycles, activity intervals, etc.
RC4 Key Scheduling Algorithm (KSA) uses a secret pseudo-random index j which is dependent on the secret key. Let S N be the permutation after the complete KSA of RC4. It is known that the value of j in round y + 1 can be predicted with high probability from S N [y ] for the initial values of y and from $S^{-1}_N[y]$ for the final values of y . This fact has been exploited in several recent works on secret key recovery from S N . In this paper, we perform extensive analysis of some special sequences of indices corresponding to the j values that leak useful information for key recovery. We present new theoretical results on the probability and the number of such sequences. As an application, we explain a new secret key recovery algorithm that can recover a 16 bytes secret key with a success probability of 0.1409. Our strategy has high time complexity at this point and requires further improvement to be feasible in practice.
A full categorization of irreducible classical Goppa codes of degree 4 and length 9 is given: it is an interesting example in the context of finding an upper bound for the number of Goppa codes which are permutation non-equivalent and irreducible and maximal with fixed parameters q , n and r (${\mathbb F}_q$ is the field of the Goppa code, the Goppa polynomial has coefficients in ${\mathbb F}_{q^n}$ and its degree is r ) using group theory techniques.
We propose a new efficient certificateless aggregate signature scheme which has the advantages of both aggregate signatures and certificateless cryptography. The scheme is proven existentially unforgeable against adaptive chosen-message attacks under the standard computational Diffie-Hellman assumption. Our scheme is also efficient in both communication and computation. The proposal is practical for message authentication in many-to-one communications.
A new reduction on the size of the search space for cocyclic Hadamard matrices over dihedral groups D 4t is described, in terms of the so called central distribution. This new search space adopt the form of a forest consisting of two rooted trees (the vertices representing subsets of coboundaries) which contains all cocyclic Hadamard matrices satisfying the constraining condition. Experimental calculations indicate that the ratio between the number of constrained cocyclic Hadamard matrices and the size of the constrained search space is greater than the usual ratio.
In this notice we describe the ideal structure of all cases of cyclic and negacyclic codes of length p s over a Galois ring alphabet that have not yet been discussed in the literature. Unlike in the cases reported earlier in the literature by various authors, the ambient spaces here are never chain rings. These ambient rings do nonetheless share the properties of being local and having a simple socle.
The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present new discrepancy bounds for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers of higher orders modulo a composite integer M.
In this paper we develop a generalization of the zig-zag graph product created by Reingold, Vadhan, and Widgerson[8]. We do this by using a broader definition of directed and undirected graphs in which incidence is determined by functions from the edge set to the vertex set. We introduce the sandwich product of graphs and show how our general zig-zag product is a sandwich product.
The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. This article presents a new version of the key equation and a way to use the Euclidean algorithm to solve it. A straightforward reorganization of the algorithm yields the Berlekamp-Massey algorithm.
We investigate the class of numerical semigroups verifying the property ρ i + 1 − ρ i ≥ 2 for every two consecutive elements smaller than the conductor. These semigroups generalize Arf semigroups.
The algebraic geometric tools used by Goppa to construct block codes with good properties have been also used successfully in the setting of convolutional codes. We present here this construction carried out over elliptic curves, yielding a variety of codes which are optimal with respect to different bounds. We provide a number of examples for different values of their parameters, including some explicit strongly MDS convolutional codes. We also introduce some conditions for certain codes of this class to be MDS.
An illegal re-distribution problem is a problem that a regular user who received content re-distributes without legal permutation. This becomes a social problem all over the world. As it is indicated in [2], this problem has been known since a few or more handred years ago. Boneh and Shaw formalize this problem as collusion-secure fingerprinting (c-secure code) for digital data [2]. After their work, study of c-secure code has become one of popular research stream for information security [1]. One of remarkable results is called Tardos's code [5]. Tardos's code achieved to construct approximately optimal length c-secure codes with random coding method.
We prove a necessary and sufficient condition for the existence of spreads in the projective Hjelmslev geometries $PHG(R_R^{n+1})$ . Further, we give a construction of projective Hjelmslev planes from spreads that generalizes the familiar construction of projective planes from spreads in PG(n,q).
Commutative semifields in odd characteristic can be equivalently described by planar functions (also known as PN functions). We describe a method to construct a semifield which is canonically associated to a planar function and use it to derive information on the nuclei directly from the planar function. This is used to determine the nuclei of families of new commutative semifields of dimensions 9 and 12 in arbitrary odd characteristic.
The well known Plotkin construction is, in the current paper, generalized and used to yield new families of Z2 Z4 -additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of Z2 Z4 -additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
In this paper we construct (n,k,δ) time-variant convolutional codes of period τ. We use the systems theory to represent our codes by the input-state-output representation instead of using the generator matrix. The obtained code is controllable and observable. This construction generalizes the one proposed by Ogasahara, Kobayashi, and Hirasawa (2007). We also develop and study the properties of the time-invariant equivalent convolutional code and we show a lower bound for the free distance in the particular case of MDS block codes.
Using stratified sampling a desired confidence level and a specified margin of error can be achieved with smaller sample size than under standard sampling. We apply stratified sampling to the simulation of the sum-product algorithm on a binary low-density parity-check code.
The most successful method to obtain lower bounds for the minimum distance of an algebraic geometric code is the order bound, which generalizes the Feng-Rao bound. By using a finer partition of the set of all codewords of a code we improve the order bounds by Beelen and by Duursma and Park. We show that the new bound can be efficiently optimized and we include a numerical comparison of different bounds for all two-point codes with Goppa distance between 0 and 2g − 1 for the Suzuki curve of genus g = 124 over the field of 32 elements.
We study cyclic codes of length 2p s over $\mathbb {F}_{q}$, where p is an odd prime. Using the results of [1], we compute the minimum Hamming distance of these codes.
