By Vasyl Ustimenko
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Extra resources for Algebraic graphs and security of digital communications
We say the point (p) is incident with the line [l], and we write (p)I[l], if the relations (2. 3) between their co-ordinates hold: For each positive integer k ≥ 2 we obtain an incidence structure (Pk , Lk , Ik ) as follows. 7. The jump to commutative rings, dynamical systems and fast implementations simply projecting each vector onto its k initial coordinates with respect to the above order. The incidence Ik is then deﬁned by imposing the ﬁrst k−1 incidence equations and ignoring all others. The incidence graph corresponding to the structure (Pk , Lk , Ik ) is denoted by D(k, K).
To guess the encoding arc. In fact the quality is good for graphs which are close to the Erd¨os bound, deﬁned by the Even Cycle Theorem. In the case of parallelotopic graphs there is a uniform way to match arcs with strings in the certain alphabet. Among parallelotopic graphs we distinguish linguistic graphs of aﬃne type whose vertices (messages) and arcs (encoding tools) both could be tuples over certain ﬁnite commutative ring. This unit presents the description of graph theoretical approach to symmetric encryption as well as to the construction of public key algorithm.
Let Γ be a bipartite (a, b)-biregular graph of girth g, and (Γ, n) be a scheme of high girth, with inputs of valency b. Then pkey (Γ, n) = 1/(b(a − 1)[n/2] (b − 1)[n/2] ). Corollary 3. We are counting probabilities pmes and pkey in a equiprobable space. Thus the condition pmes = pkey corresponds to one time pad encryption scheme. Let (Γi , ni ) be a family of regular or bipartite biregular schemes of high girth with non decreasing ki and non decreasing bidegrees ai and bi such that ai + bi + ni is unbounded.
Algebraic graphs and security of digital communications by Vasyl Ustimenko