By Vasyl Ustimenko
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Additional info for Algebraic graphs and security of digital communications
Fk (a2 (u), . . , at+1 (u)) is given by the list of coeﬃcients (public key). He can encrypt but could not decrypt in case of properly chosen parameters and polynomials Fi , i = 1, . . , k. You know graph D(k, q) and its invariants ai (k). If you apply to e consecutively transformations Nds−1 , Nds−2 , . . , Nd1 , Nd0 , where di = Fi (a2 (e), . . , at+1 (e)), i = 1, . . , k − 1 and d0 is the colour of plaintext, you obtain the plaintext u. Here we use the fact that u and c are vertices from the same component of D(k, q).
For instance, if we have a text in English, we can consider an injective function f from the alphabet, which contains letters, comma, dot, empty space, to GF (29). We could apply the function f to each character in our text to identify of it with an element of the ﬁnite ﬁeld. After that we can use a coding by a folder of linguistic graph of increasing girth of 39 40 2. Simple graphs with special arcs and Cryptography aﬃne type over GF (29), which will guarantee that the encoded text will be diﬀerent from the initial text.
The only way to get into the account is to ”guess” the message. Let pmes (Γ, n) be the probability of a success in this ”dark” situation. The purpose of this paper is to consider special cases of graphs, for which there is some similarity with linear coding, it satisﬁes some of the properties A, B, C and completely avoids situation D. For some of them the probability pkey (Γ, n) can be computed. We may assume without loss of generality that the edges of k-regular graph are marked by symbols from some alphabet A (set of colors) in such way that neighbors of each vertex are of diﬀerent color.
Algebraic graphs and security of digital communications by Vasyl Ustimenko