scispace - formally typeset
Search or ask a question
Proceedings ArticleDOI

Secure communication over Trellis: Graph theoretic approach

TL;DR: This paper has used a class of group codes called Kernel codes and its trellis, to show that private key cryptosystem can be used over Trellis and fundamental cut-set acts as a key to encrypt and decrypt message at sender and receiver respectively.
Abstract: Forney's representation of Trellis code received wide attention by researchers and engineers with its simplicity in representing codes and elegant nature to analyze codes with sys- tem theoretic properties, graph theoretic properties with efficient encoding and decoding procedures. In this paper, we consider the connected graph nature of trellis and propose security feature over Trellis using fundamental cut-set and fundamental circuits principle. We use the graph theoretic approach, by generating limited spanning trees of trellis, fundamental cut-sets and fundamental circuits, private key cryptosystem is defined in which fundamental cut-set acts as a key to encrypt and decrypt. We have used a class of group codes called Kernel codes and its trellis, to show that private key cryptosystem can be used over Trellis and fundamental cut-set acts as a key to encrypt and decrypt message at sender and receiver respectively.
References
More filters
Journal ArticleDOI
N.R. Malik1
01 Oct 1975
TL;DR: Graph Theory and Its Applications to Problems of Society and its Applications to Algorithms and Computer Science.
Abstract: Introductory Graph Theory with ApplicationsGraph Theory with ApplicationsResearch Topics in Graph Theory and Its ApplicationsChemical Graph TheoryMathematical Foundations and Applications of Graph EntropyGraph Theory with Applications to Engineering and Computer ScienceGraphs Theory and ApplicationsQuantitative Graph TheoryApplied Graph TheoryChemical Graph TheoryA First Course in Graph TheoryGraph TheoryGraph Theory with ApplicationsGraph Theory with ApplicationsSpectra of GraphsFuzzy Graph Theory with Applications to Human TraffickingApplications of Graph TheoryChemical Applications of Graph TheoryRecent Advancements in Graph TheoryA Textbook of Graph TheoryGraph Theory and Its Engineering ApplicationsGraph Theory, Combinatorics, and ApplicationsAdvanced Graph Theory and CombinatoricsTopics in Intersection Graph TheoryGraph Theory with Applications to Engineering and Computer ScienceGraph Theory and Its Applications, Second EditionHandbook of Research on Advanced Applications of Graph Theory in Modern SocietyGraph Theory with Applications to Algorithms and Computer ScienceGraph TheoryGraph Theory with Algorithms and its ApplicationsGraph TheoryGraph Theory with ApplicationsGraph Theory ApplicationsHandbook of Graph TheoryGraph Theory and Its Applications to Problems of SocietyBasic Graph Theory with ApplicationsTen

809 citations


"Secure communication over Trellis: ..." refers background or methods in this paper

  • ...Such a cut-set S containing exactly one branch of a tree T and remaining branches from G is called a fundamental cutset with respect to T [14]....

    [...]

  • ...Spanning Trees A tree T is said to be a spanning tree of a connected graph G if T is sub graph of G and T contains all vertices of G [14]....

    [...]

  • ..., bk } is a fundamental circuit with respect to T [14]....

    [...]

  • ...Following theorems 1 and 2 [14] are useful in achieving security over fundamental cut-set and fundamental circuit, which are extended to trellis in the proposed method to provide security feature over trellis....

    [...]

Journal ArticleDOI
25 Jun 2000
TL;DR: Any state realization of a code can be put into normal form without essential change in the corresponding graph or in its decoding complexity; this fundamental result has many applications, including to dual state spaces, dual minimal trellises, duals to Tanner (1981) graphs, dual input/output (I/O) systems, and dual kernel and image representations.
Abstract: A generalized state realization of the Wiberg (1996) type is called normal if symbol variables have degree 1 and state variables have degree 2. A natural graphical model of such a realization has leaf edges representing symbols, ordinary edges representing states, and vertices representing local constraints. Such a graph can be decoded by any version of the sum-product algorithm. Any state realization of a code can be put into normal form without essential change in the corresponding graph or in its decoding complexity. Group or linear codes are generated by group or linear state realizations. On a cycle-free graph, there exists a well-defined minimal canonical realization, and the sum-product algorithm is exact. However, the cut-set bound shows that graphs with cycles may have a superior performance-complexity tradeoff, although the sum-product algorithm is then inexact and iterative, and minimal realizations are not well-defined. Efficient cyclic and cycle-free realizations of Reed-Muller (RM) codes are given as examples. The dual of a normal group realization, appropriately defined, generates the dual group code. The dual realization has the same graph topology as the primal realization, replaces symbol and state variables by their character groups, and replaces primal local constraints by their duals. This fundamental result has many applications, including to dual state spaces, dual minimal trellises, duals to Tanner (1981) graphs, dual input/output (I/O) systems, and dual kernel and image representations. Finally a group code may be decoded using the dual graph, with appropriate Fourier transforms of the inputs and outputs; this can simplify decoding of high-rate codes.

643 citations

Journal ArticleDOI
TL;DR: It is shown that soft decision maximum likelihood decoding of any (n,k) linear block code over GF(q) can be accomplished using the Viterbi algorithm applied to a trellis with no more than q^{(n-k)} states.
Abstract: It is shown that soft decision maximum likelihood decoding of any (n,k) linear block code over GF(q) can be accomplished using the Viterbi algorithm applied to a trellis with no more than q^{(n-k)} states. For cyclic codes, the trellis is periodic. When this technique is applied to the decoding of product codes, the number of states in the trellis can be much fewer than q^{n-k} . For a binary (n,n - 1) single parity check code, the Viterbi algorithm is equivalent to the Wagner decoding algorithm.

612 citations


"Secure communication over Trellis: ..." refers background in this paper

  • ...Further, Wolf [8] showed the possibility of decoding of linear binary block codes constructed by...

    [...]

  • ...Further, Wolf [8] showed the possibility of decoding of linear binary block codes constructed by Bahl et. al. using Viterbi algorithm....

    [...]

Journal ArticleDOI
D.J. Muder1
TL;DR: It is shown that minimal proper trellises exists for all block codes and bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes.
Abstract: Basic concepts in the study of trellises of block codes are defined. It is shown that minimal proper trellises exists for all block codes. Bounds on the size of such trellises are established. These bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes. >

238 citations


"Secure communication over Trellis: ..." refers methods in this paper

  • ...The properties such as Observability, Controllability and minimal trellises for codes have been studied with emphasis on Algebra and Graph theory as well [9] [6] [10]....

    [...]

Journal ArticleDOI
27 Jun 1994
TL;DR: The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete and the number of distinct minimal linear block code trellises is a Stirling number of the second kind.
Abstract: The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete. Examples are provided that show that (1) the minimal trellis for a nonlinear code may not be observable, i.e. some codewords may be represented by more than one path through the trellis and (2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis produce is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of nonattacking rooks on a triangular chess board is established and used to show that the number of distinct minimal linear block code trellises is a Stirling number of the second kind. Various bounds on trellis size are reinterpreted in this context.

200 citations


"Secure communication over Trellis: ..." refers background or methods in this paper

  • ...Kschischang and Sorokine defined [6], Trellis for a block code C of length n is an edge labeled directed graph with a distinguished ”root” vertex having in-degree zero and a distinguished ”goal” vertex having out-degree zero, and with the following properties: 1) all vertices can be reached from the root; 2) the goal can be reached from all vertices; 3) the number of edges traversed in passing from the root to the goal along any path is n; and 4) the set of n-tuples obtained by ’reading off’ the edge labels encountered in traversing all paths from the root to the goal is C....

    [...]

  • ...Forney introduced ’Trellis’ to represent linear block codes [6]....

    [...]

  • ...The properties such as Observability, Controllability and minimal trellises for codes have been studied with emphasis on Algebra and Graph theory as well [9] [6] [10]....

    [...]