On estimations of distribution of the length of aperiodicity segment in the graph of k-fold iteration of uniform random mapping | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2018. № 42. DOI: 10.17223/20710410/42/1

Given k,n £ N, x₀ £ S = {1,... ,n}, and f : S ^ S, define x_i+1 = f^k(xj) for every i £ {0,1,...} and Туk(x₀) as the least integer i such that f^k(xj) = Xj for some j, j < i. For the local probability Р {Туk (x₀) = z} and for the distribution function F_Tfk (_x0) (z), the following estimates are obtained. If kz < n, then P {r_fk Ы=4 > - £ e^-(l+m) m² + y: Г² 1-1-^r + ^k - _m> 1, _m>1, Г + k \ V - (m,k) (m,k) { } 1 (m-1)² 1 (r-1)² / / Г \ k ^p {Tfk (xo) = z} <- E e^-^ + E -e^-^ ¹ - ¹ - - ⁿ m> 1, m> 1, ^r \ ^{v n/} m,. - 1 ) k. If k²z ^ n, then (m, k) 1 E e-(1+m)m2 + Л - E e-(1+n) ⁿ m^1, V J ^- m>1, _m_<_z (m,k) <^Z (m,k) (r-1)² { } 1 ( m- 1)² k < P{ Tf k (xo) = z}<- E e ^ + - E e" k ⁿ m> 1, ⁿ m> 1, 256 and more). Also, if kz ^ л/-, then r / r (m + r)\ _(₁₊m)m² r + 1 (m-¹ E - 1 - ^-- e ⁽¹+nbn f(xo)(z) < E -e ²n г>1, - V 2- J f^k ( 0) m>1, - (m,k) ^ (m,k) where r = m + ( z - , ^m,. ) k. In some cases, the obtained results allow to estimate (m, k) the allowable period of usage of the encryption keys generated by iterative algorithms and to build criteria for quality assessment of random sequences.