On Graph-Based Cryptography and Symbolic Computations

We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs

Families of Small Regular Graphs of Girth 5

In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique

Bounds for graphs of given girth and generalized polygons

In this paper we present a bound for bipartite graphs with average bidegrees η and ξ satisfying the inequality η ≥ ξ α, α ≥ 1. This bound turns out to be the sharpest existing bound. Sizes of known families of finite generalized polygons are exactly on that bound. Finally, we present lower bounds for the numbers of points and lines of biregular graphs (tactical configurations) in terms of their bidegrees. We prove that finite generalized polygons have smallest possible order among tactical configuration of given bidegrees and girth. We also present an upper bound on the size of graphs of girth g ≥ 2t + 1. This bound has the same magnitude as that of Erd¨os bound, which estimates the size of graphs without cycles C₂t

On LDPC codes corresponding to affine parts of generalized polygons

In this paper we describe how to use special induced subgraphs of generalized m-gons to obtain the LDPC error correcting codes. We compare the properties of codes related to the affine parts of q-regular generalised 6-gons with the properties of known LDPC codes corresponding to the graphs D(5, q)

Semisymmetric graphs from polytopes

AbstractEvery finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope

Performance of algebraic graphs based stream-ciphers using large finite fields

Algebraic graphs D(n, q) and their analog graphs D(n, K), where K is a finite commutative ring were used successfully in Coding Theory (as Tanner graphs for the construction of LDPC codes and turbo-codes) and in Cryptography (stream-ciphers, public-keys and tools for the key-exchange protocols. Many properties of cryptography algorithms largely depend on the choice of finite field Fq or commutative ring K. For practical implementations the most convenient fields are F and rings modulo Z modulo 2m. In this paper the reader can find the first results about the comparison of D(n, 2m) based stream-ciphers for m = 8, 16, 32 implemented in C++. They show that performance (speed) of algorithms gets better when m is increased

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Networks, uncertainty reduction and strategic decision-making in social movement fields

Organisational efforts to bring about social change are riddled with choices. What is the appropriate course of action? Who best to collaborate with? How should finite resources, economic or otherwise, be spent? In this respect, the existence of Social Movement Organisations (SMO) — those entities with goals aimed at changing the state of society or protecting the status quo — is one characterised by great uncertainty. Thus a question of critical import to understanding SMOs’ capacity to bring about change is how do they go about bridging information gaps when faced with strategic decisions? In this thesis I argue that network structure affords SMOs a route to accessing information that may be used to manage uncertainty. My argument is built upon two simple observations: (a) populations of SMOs are constitutive of Social Movement Fields wherein these diverse organisations cooperate, compete and learn from one another through surveillance, comparison and mimicry; and (b) SMOs are embedded in rich webs of relations with peers, both online and offline, that enable and constrain their behaviour by governing access to informational resources that may be used for goal attainment. The core novelty of this thesis arises from my recasting of SMOs’ strategic actions as types of relationship formation in inter-organisational network scenarios that are comparatively overlooked — namely, multiplex and bipartite networks. This approach has the appealing property of making clear the effect of SMOs on each other — a key aspect of the institutional perspective on which this work is built — whilst also allowing me to more squarely address how network structure might guide action. Analytically, this leads me to focus on those micro-level network locales, i.e., the “local neighbourhoods”, within which SMOs are embedded (e.g., triads) as they relate to tie formation vis-á-vis uncertainty reduction. Methodologically, this thesis is also designed to demonstrates the sociological power of statistical models of networks in investigating the dynamics of social movement fields. The core strength of these models is their realistic handling of the constraints/benefits of social actors’ structural positions with respect to their behaviour. This is in stark contrast to the variable-centred (i.e. atomistic) statistical frameworks typical of sociological studies of SMOs (e.g., OLS or logistic regression) which fail to account for these organisations’ interdependence and thus provide poor representations of their agency as strategic actors. Empirically, this work consists of three contained case studies of strategic action: (a) a longitudinal study of tactical implementation in the Palestinian National Movement; (b) a longitudinal study of financial patronage in the US Climate Change Countermovement; and (c) a cross-sectional study of online alliance formation amongst organisational members of the Hardest Hit Coalition, a UKbased anti-austerity issue campaign. Results overwhelmingly support my assertions that information useful in managing uncertainty with respect to strategic action is encoded into oft overlooked network structure. Extant sociological work has simply missed a number of interesting, sometimes counterintuitive, dynamics of Social Movement Fields

Maximality of affine group, and hidden graph cryptosystems

We describe a new algebraic-combinatorial method of public key encryption with a certain similarity to the well known Imai-Matsumoto. We use the general idea to treat vertices of a linguistic graph (see [21] and further references) as messages and use the iterative process to walk on such graph as encryption process. To hide such encryption (graph and walk on it) we will use two affine transformation. Like in Imai - Matsumoto encryption the public rule is just a direct polynomial map from the plaintext to the ciphertext. The knowledge about graph and chosen walk on them (the key) allow to decrypt a ciphertext fast. We hope that the system is secure even in the case when the graph is Public but the walk is hidden. In case of "public" graph we can use same encryption as private key algorithm with the resistance to attacks when adversary knows several pairs:(plaintext, ciphertext). We shall discuss the general idea of combining affine transformations and chosen polynomial map of deg ≥ 2 in case of prime field Fp. As it follows from the maximality of affine group each bijection on Fp n can be obtained by such combining