By Vasyl Ustimenko

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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). In the package CRYPTIM we use this scheme in case s = t, degFi ≤ 1, in particular, for the problem of digital signatures. Remark 1. The probability to have same invariants a2 , . . , at+1 for two random messages is about 1/q t . 41 42 2. Simple graphs with special arcs and Cryptography Remark 2. If we want to speed up the computation of c(u) we may present it to our correspondent as product of several factors.

Let Γi be a free folder over the GF (q) , where the cover Γ is a q-regular tree. We could construct the ”Theory of Γi -codes” in which the distance in the graph Γi would play the role of a Hamming metric in the classical case of linear codes. , for each k the automorphism group acts transitively on pairs of vectors at a distance k. The distance in the graph Γi may not be distance transitive, but we have an ”asymptotical” distance transitivity, because of the distance transitivity of the tree Γ and the fact that lim(Γi ) = Γ.

Nowadays RC4 is not secure enough and not recommended for use in new system. Anyway we chose it for comparison, because of its popularity and high speed. RC4 vs graph based algorithm (128 bit key) 30 1B graph 2B graph 4B graph RC4 25 Time [s] 20 15 10 5 0 0 20 40 60 80 100 120 File size [MBytes] 140 160 180 45 46 2.