By Vasyl Ustimenko

ISBN-10: 8362773170

ISBN-13: 9788362773176

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**Extra resources for Algebraic graphs and security of digital communications**

**Sample text**

According to Proposition 5, uτ v implies uτ ′ v. If char GF (q) is an odd number, the converse of the last proposition is true [62]. 5. Existence of free triangular optimal folders Proposition 6. Let q be an odd number. , τ = τ ′ and EQ(q) = CD(q) (EQ(k, q) = CD(k, q)). The condition charGF (q) = 2 in the last proposition is essential. For instance, the graph EQ(k, 4)), k > 3, contains 2 isomorphic connected components. Clearly EQ(k, 2) is a union of cycles CD(k, 2). Thus neither EQ(k, 2) nor CD(k, 2) is an interesting family of graphs of high girth.

Time evaluation of the private key encryption for Alice We have implemented computer application [54], which uses family of graphs RDE(n, K) for private key cryptography. To achieve high speed property, commutative ring K = Z2k , k ∈ {8, 16, 32}, with operations +, × modulo 2k . Parameter n stands for the length of plaintext (input data) and the length of ciphertext. We mark by G1 the algorithm with k = 8, by G2 the algorithm with k = 16, and by G4 the algorithm with k = 32. So Gi, i ∈ 1, 2, 4 denotes the number of bytes used in the alphabet (and the size of 1 character in the string).

Ln , . ] and (p) = (p1 , p3 , . . , pn , . ) (line and point in ”natural coordinates”). Let ηi be the map ”deleting all coordinates with numbers > i” from D(q) to D(i, q), and ηi,j be map ”deleting all coordinates with numbers > i ” from D(j, q), j > i into D(i, q). The following statement follows directly from the deﬁnitions: Proposition 4. (see, [57], [59]) D(i, q), ηi,j is a free triangular linguistic folder over GF (q). It has a forest D(q) as a cover. Example 5. Let k ≥ 6, t = [(k + 2)/4], and let u = (ui , u11 , · · · , utt , u′tt , ut,t+1 , ut+1,t , · · · ) be a vertex of D(k, q).

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