Random mappings with a given number of cyclical points Jennie C. Hansen∗ and Jerzy Jaworski† ‡
Abstract In this paper we consider a random mapping, Tˆn , of the finite ˆ n is set {1, 2, ..., n} into itself for which the digraph representation G ˆ constructed by: (1) selecting a random number, Ln , of cyclic vertices, (2) constructing a uniform random forest of size n with the selected cyclic vertices as roots, and (3) forming ‘cycles’ of trees by applying a random permutation to the selected cyclic vertices. We investigate ˆ n , and, under the assumption kˆn , the size of a ‘typical’ component of G that the random permutation on the cyclical vertices is uniform, we ˆ n = m(n). obtain the asymptotic distribution of kˆn conditioned on L As an application of our results, we show in Section 3 that provided ˆ n is of order much larger than √n, then the joint distribution of the L ˆ n converges to normalized order statistics of the component sizes of G the Poisson-Dirichlet(1) distribution as n → ∞. Other applications and generalizations are also discussed in Section 3.
1
Introduction
In this paper we consider the distribution of the size of a ‘typical’ component in a random mapping from the set Vn = {1, 2, ..., n} into Vn conditioned ∗
Actuarial Mathematics and Statistics, Heriot–Watt University,Edinburgh EH14 4AS, UK. E-mail address:
[email protected] † Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Pozna´ n, Poland. E-mail address:
[email protected] ‡ J.Jaworski acknowledges the generous support by the Marie Curie Intra-European Fellowship No. 501863 (RANDIGRAPH) within the 6th European Community Framework Programme.
1
on the total number of cyclical vertices in the mapping. Before discussing the motivation for this investigation, we introduce some notation and review some well-known results for the component structure of the uniform random mapping. Throughout this paper, for n ≥ 1, let Mn denote the set of mappings f : Vn → Vn . For any f ∈ Mn , we can represent f by a directed graph G(f ) on n labelled vertices such that a directed edge from vertex i to vertex j exists in G(f ) if and only if f (i) = j. We say that vertex i in G(f ) is a cyclic vertex if the m-fold composition f (m) (i) = i for some m ≥ 1 and we let L(f ) denote the number of cyclic vertices in G(f ). We also note that since each vertex in G(f ) has out-degree 1, the components of G(f ) consist of directed cycles with directed trees attached. Finally, let Tn denote the uniform random mapping of Vn into Vn with distribution given by n o 1 Pr Tn = f = n n for each f ∈ Mn and let Gn ≡ G(Tn ) denote the corresponding random directed graph which represents the random mapping Tn . Much is known about the component structure of the random digraph Gn which represents Tn (see for example the monograph by Kolchin [13], survey by Mutafchiev [14], Aldous [1], Arratia et.al. [2], [3], Hansen [8]). In particular, it is known that, given Ln , the set of cyclic vertices of Tn , the random mapping Tn restricted to Ln is a uniformly distributed random permutation on the vertices in Ln . This observation motivates the following, alternative construction of the random digraph Gn as a special case of the general random mapping model described below. A random mapping model ˆ n is a discrete random variable such that 1 ≤ L ˆ n ≤ n. Also, for Suppose that L 1 ≤ m, let σ ˆm be a random, but not necessarily uniform, permutation of the set {1, 2, ..., m} such that for such that for any subsets B, C ⊆ {1, 2, ..., m} with |B| = |C|, we have Pr{ˆ σm B is a cyclic permutation } = Pr{ˆ σm C is a cyclic permutation }. (1.1) For any subset A ⊆ Vn such that |A| = m, we use the natural ordering of the set A to identify the elements of A with the elements of {1, 2, ..., m}, and we define σ ˆA to be the random permutation of A which is induced by 2
the action σ ˆm via the identification of A with {1, 2, .., m}. Now the random ˆ ˆ n and the random permutations mapping Tn : Vn → Vn is constructed using L ˆ σ ˆm as follows: Given Ln = m, let Am denote a uniform random subset of size m from the vertices Vn (i.e. all subsets of size m are equally likely). Given Am = A ⊆ Vn , let Fn (A) denote the uniform random rooted forest on the vertices Vn , where A is the set of roots, and the edges in the trees of Fn (A) are directed such that any path from a vertex to a root is directed towards the root. Finally, suppose that σ ˆA is a random permutation of A (as described above) which is also independent of the random forest Fn (A). ˆ n from the rooted forest Fn (A) by adding a We form the directed graph G directed edge from i ∈ A to j ∈ A if σ ˆA (i) = j, and we let Tˆn denote the ˆ n. random mapping which is represented by G The construction described above gives us a class of random mapping models ˆ n , the number of cyclic which are determined by the distribution chosen for L vertices, and by the distributions of the random permutations σ ˆm . We note d ˆ n has the same distribution as Ln (denoted L ˆ n ∼ Ln ), where Ln ≡ that if L L(Tn ) is the number of cyclic vertices in the uniform random mapping Tn , and if, for every A ⊆ Vn , σ ˆA = σA , the uniform random permutation of A, d d ˆ ˆ ˆ n ≡ n (i.e. L ˆ n is then Gn ∼ Gn and Tn ∼ Tn . On the other hand, if L degenerate), then Tˆn is just the random permutation σ ˆn . ˆ n for various In general, it is of interest to determine the structure of G ˆ n and of the random permutations σ choices for the distributions of L ˆm . In ˆ n arise in the analysis particular, questions concerning the structure of G of cryptographic systems (e.g. DES see [16]), in applications of Pollard’s algorithm (see [15], [18]), in simulations of shift register sequences, and in computational number theory and random number generation. We note that since both uniform random mappings and uniform random permutations are special cases of our general model, we can view these models as part of a ‘continuum’ of random mapping models. Specifically, we note√that for uniform random mappings, it is known (see [11]) that Ln = O( n), and Ln more precisely, the normalized variable √ converges in distribution to L n as n → ∞, where L is a continuous random variable with density √ given by 2 ˆ n is of order much greater than n (but fL (x) = xe−x /2 , x ≥ 0. So, if L not necessarily of order n) and if, for m ≥ 1, σ ˆm , is a uniform permutation, ˆ then we obtain a model Gn which is ‘sandwiched’ (in some sense) between the uniform random mapping model and the uniform random permutation
3
on Vn . One goal of this paper is to investigate the component structure of such models. We begin our investigation by noting that if Tˆn is a random mapping as described above, and if φ : Mn → Z is a discrete functional, then the ˆ n and by the distribution of φ(Tˆn ) is determined by the distribution of L conditional probabilities ˆ n = m} . Pr{φ(Tˆn ) = k L (1.2) Hence the conditional probabilities in (1.2) are fundamental to any investiˆ n . We note that gation of Tˆn and of the corresponding random digraph G sometimes it can be very easy to compute the conditional probabilities in (1.2). For example, suppose that, for f ∈ Mn , φ(f ) equals the number of components in G(f ), then the conditional distribution of φ(Tˆn ) given that ˆ n = m is the same as the distribution of the number of cycles in the ranL dom permutation σ ˆm . In the case where σ ˆm = σm , the uniform random permutation of {1, 2, .., m}, the distribution of the number of cycles in σm is well-known (see Feller [7], p.258). However, for many functionals it can be much more complicated to compute the conditional probabilities in (1.2). In this paper we investigate the conditional distribution of kˆn ≡ kn (Tˆn ) ˆ n , where, for f ∈ Mn , kn (f ) is defined to be the size of the component given L in G(f ) which contains the vertex 1. Since vertex 1 is ‘arbitrary’, we can say ˆ n . In Section 2, we determine that kˆn is the size of a ‘typical’ component in G a general formula (see Fact 1) which can be used to compute the conditional probabilities ˆ n = m}. Pr{kˆn = k L (1.3) In the special case where, for m ≥ 1, σ ˆm = σm , the uniform random permuˆ tation on {1, 2, ..., m}, but Ln is arbitrary we obtain exact formulae for the conditional probabilities (1.3). A key observation in this case is that, as a consequence of the construction of Tˆn , we have ˆ n = m} = Pr{kn = k Ln = m} . Pr{kˆn = k L (1.4) where kn ≡ k(Tn ). So, in this case, it is enough to consider the conditional distribution of kn given Ln . In Section 2, we also determine the asymptotic distribution of kn conditioned on L√n = m(n) under four distinct regimes: (i) m(n) √ = O(1), (ii) √m(n) = o( n) but m(n) → ∞ as n → ∞, (iii) m(n) = α n, and (iv) n = o(m(n)). We remark here that it is well-known 4
for the uniform random mapping Tn that Z b 1 kn √ du lim Pr a < 0 (by (2.3)) Pr n − kn = ` Ln = m = Pr kn = n − ` Ln = m min{n−`,m}
=
X j=max{1,m−`}
`−m+j n−`−j m−j n−m ` ` 1− . (2.4) `m ` − m + j n n
Therefore for ` = 1, 2, . . . , m − 1 we obtain Pr n − kn = ` Ln = m m m X X ` − (` − m + j) ``−m+j m − j ``−m+j −` e = e−` ∼ m` (` − m + j)! m` (` − m + j)! j=m−` j=m−` m m 1 X 1 X 1 `` −` ``−m+j ``−m+j−1 −` = e − e−` = e . m j=m−` (` − m + j)! m j=m−`+1 (` − m + j − 1)! m `!
9
Similarly for any bounded ` = m, m + 1, . . . we have Pr n − kn = ` Ln = m m m X X m − j ``−m+j ` − (` − m + j) ``−m+j ∼ e−` = e−` m` (` − m + j)! m` (` − m + j)! j=1 j=1 m
m
1 X ``−m+j 1 X ``−m+j−1 −` = e − e−` m j=1 (` − m + j)! m j=1 (` − m + j − 1)! =
1 `` −` 1 ``−m e − e−` . m `! m (` − m)!
To show that this is a proper probability distribution one can use another discrete distribution (see [12]), namely pk = a
(a + k)k−1 −a−k e , k = 0, 1, . . . , where a is a positive integer . k!
√ Theorem 2. Suppose that m = o( n) but m → ∞ as n → ∞, and suppose y is fixed, 0 < y < ∞. Then n − kn 1 1 1 Pr = y Ln = m ∼ 2 √ 1 − exp − . m2 m 2πy 2y √ Proof. Let us assume that m = o( n) but m → ∞ as n → ∞ , j = mx, where x is fixed, 0 < x < 1 and n − kn = ` = ym2 , where 0 < y < ∞ is fixed. Then −m + xm + m` ` − m + j − (n − m) n` 1 x n q q = ∼ −√ + √ , √ y y (n − m) n` 1 − n` ` 1− m 1 − n` n √ since ` = o(n) and j = o( n) . Therefore the approximation given by the local de Moivre-Laplace theorem is applicable for the binomial probability
n−m `−m+j
`−m+j n−`−j ` ` 1− n n
and together with (2.4) it implies that 10
Pr
Z 1 2 n − kn 1 − x 1 1 (1 − x) dx = y Ln = m ∼ 2 √ √ exp − m2 m 0 y 2y y 2π (1 − x)2 1−x , we have du = − dx and 2y y Z 1 2y n − kn 1 1 √ exp {−u} du , Pr = y L = m ∼ n m2 m2 2πy 0
Taking u =
which immediately leads to the assertion. Theorem 3. Suppose that α > 0, 0 < δ < 1/2, 0 < δ < x < 1 − δ < 1 and √ 5 n is such that δ n > α3 ∨ 1. Then Ln kn Pr = x √ = α n n Z 1 α 1 (x − y)2 α2 = p (1 − y) exp − dy(1 + ε(n, α, x)) (2.5) n 2πx(1 − x)3 0 2x(1 − x) 3 2 /δ 2 ) α ∨1 √ where |ε(n, α, x)| ≤ C min(δ5 ,α) exp(α and C is a constant which does n not depend on n, α, or δ. √ Proof. Let m = α n and k + 1 = xn, then it follows from Corollary 2 and a careful application of the usual deMoivre-Laplace local limit theorem for the binomial distribution (see Feller) , that Pr{ kn = k + 1 Ln = m} n−m−k+j−2 min{m} X m − j n − m k + 1 k−j+1 k+1 = 1− (2.6) nm k − j + 1 n n j=1 ( ) m X (x − mj )2 α2 1 j 1 α p = (1 − ) exp − (1 + ∆(n, α, x)) , n 2πx(1 − x)3 j=1 m m 2x(1 − x) 3 α ∨1 ˜ is a constant which does not √1 and C where |∆(n, α, x)| ≤ C˜ min(δ 5 ,α) n depend on n, α, or δ. Next, since m ( ) Z X 1 1 (x − mj )2 α2 j (x − y)2 α2 (1 − ) exp − − (1 − y) exp − dy m m 2x(1 − x) 2x(1 − x) 0 j=1 11
≤
4 m
(2.7)
and, for 0 < δ < x < 1 − δ < 1, Z 1 (x − y)2 α2 1 (1 − y) exp − dy ≥ exp(−α2 /δ 2 ), 2x(1 − x) 2 0
(2.8)
we can obtain (2.5) from (2.6)-(2.8). We note that one can show that Z 1 Z 1 α (x − y)2 α2 p (1 − y) exp − dydx = 1 2x(1 − x) 2πx(1 − x)3 0 0 (i.e. that the local limit is a the proper density function) by using the substitution (x − y)α z=p , x(1 − x) changing the order of integration and some “routine” calculations. Ln We note that in Theorem 3 above, the value √ = α is a parameter in the n kn L Ln n limiting conditional distribution of n given √n . Recall that as n → ∞, √ n 2
converges in distribution to a variable L with density fL (α) = αe−α /2 , α > 0. It can be checked that the integral over (0, ∞) of the local limit in Theorem 3 with respect to the density fL (α) yields, in light of (1.5), the expected result, i.e. Z Z 1 2 (x − y)2 α2 α2 eα /2 1 1 1 ∞ p √ (1 − y) exp − . dydα = n 0 2x(1 − x) n2 1−x 2πx(1 − x)3 0 √ Finally, in Theorem 4 below, we consider the case where n = o(m). In order to apply this result to prove our main result in the next section, we have given the error bounds for the local limit in the statement of the theorem. √ Theorem 4. Suppose that n = o(m), a is fixed , 0 < a < 12 , and let fix x , 0 < a < x < 1 − a < 1. Then, as n → ∞, 1 kn Pr = x Ln = m = 1 + ε(n, m, x) , n n 1/4 √ . where ε(n, m, x) ≤ (n−m) am 12
Proof. First, we suppose that n − m → ∞ as n → ∞. We also fix 0 < a < 12 and suppose that k + 1 = n x, where√x is fixed, 0 < a < x < 1 − a < 1. Let ∆(j) ≡ j − mx and let δ(n, m, a) ≡ a m(n − m)1/4 . Then by Corollary 2, we have Pr{kn = k + 1 Ln = m} = S1 + S2 + S3 = (2.9) min{k+1,m} X 1{|∆(j)|≤δ(n,m,a)} n − m xn−j x (1 − x)n−m−xn+j n xn − j j=max{1,m−n+k+1} min{k+1,m}
−
X j=max{1,m−n+k+1} min{k+1,m}
+
X j=max{1,m−n+k+1}
1{|∆(j)|≤δ(n,m,a)} ∆(j) n − m xn−j x (1 − x)n−m−xn+j n(1 − x) m xn − j 1{|∆(j)|>δ(n,m,a)} m − j n − m xn−j x (1 − x)n−m−xn+j n(1 − x) m xn − j
We consider the three sums inp (2.9). Let X be a Binomial(n − m, x) variable and let Y = (X − (n − m)x)/ (n − m)x(1 − x) . Then we have min{k+1,m}
S3
1{|∆(j)|>δ(n,m,a)} n − m xn−j ≤ x (1 − x)n−m−xn+j n(1 − x) xn − j j=max{1,m−n+k+1} ( ) √ δ(n, m, a) 1 1 n−m Pr |Y | > p ≤ ≤ n(1 − x) n am (n − m)x(1 − x) X
where the last inequality follows from Chebyshev’s inequality. Next, observe that if |∆(j)| ≤ δ(n, m, a), then √ 1/4 ∆(j) ≤ a(n√− m) , m m and it follows that 1 |S2 | ≤ n(1 − x)
√
a(n − m)1/4 1 √ ≤ n m
13
(n − m)1/4 √ am
.
Finally, it is clear that S1 ≤ n1 . A lower bound for the sum S1 is given by min{n−m,(n−m)x+δ(n,m,a)} X 1 n−m S1 ≥ (x)` (1 − x)n−` n ` `=max{0,(n−m)x−δ(n,m,a)} ( ) √ δ(n, m, a) 1 1 n−m Pr |Y | ≤ p 1− , = ≥ n n am (n − m)x(1 − x) where the random variable Y is as defined above. So it follows from (2.9) and the calculations above that for k + 1 = xn, where 0 < a < x < 1 − a < 1, n − m → ∞, we have 1 Pr{kn = k + 1 Ln = m} = (1 + ε(n, m, x)) (2.10) n 1/4
√ where |ε(n, m, x)| ≤ max{ (n−m) , am
√
n−m }. am
Finally, we consider the case when n − m = i = O(1) and, as previously, fix 0 < a < 21 and k + 1 = xn, where 0 < a < x < 1 − a < 1. Then Pr{kn = k + 1 Ln = m} k−j+1 n−m−k+j−2 k+1 X m−j n−m k+1 k+1 = 1− nm k − j + 1 n n j=k+1−i i X m − mx + mx − xn + t i (x)t (1 − x)i−t−1 = nm t t=0 i X 1 1 −ix + t i t 1 = + x (1 − x)i−t = (1 + ε(n, m, x)) n n(1 − x) t=0 m t n where, in this case, |ε(n, m, x)| ≤ desired result.
3
i . am
This together with (2.10) gives the
Applications and discussion
In this section we prove our main result, Theorem 5. We also discuss some related applications of our results and suggest some directions for future work. 14
ˆ n where, for m ≥ 1, We begin by considering a random mapping digraph G σ ˆm = σm , uniform random permutation on {1, 2, ..., m}. If we consider the ˆ n as a ‘function’ of the number of cyclic vertices, component structure of G √ ˆ n , then, in some sense, n is a threshold for the number of cyclic vertices. L √ ˆ n < m(n)} → 1 as Specifically, if there exists m(n) = o( n) such that Pr{L n → ∞, then it follows from Theorems 1 and 2 that for any constant C > 0 we have ) ( C(m(n))2 kˆn ≥1− → 1 as n → ∞. Pr n n ˆ n consists of one large component In other words, with high probability, G of order n and √ the remaining components are of order at most (m(n))2 . If ˆ n is of order n (as in the case of the uniform random mapping) then L ˆ ˆn L and the conditional distribution of nk is parameterised by the value of √ n ˆ ˆ n in a the asymptotic distribution of knn may depend on the distribution of L √ ˆ n is of order greater than n then the condicomplicated way. Finally, if L ˆ ˆ n is asymptotically uniform on [0, 1] and is tional distribution of knn given L ˆ n . We exploit this ‘independence’ ‘independent’ of the exact distribution of L ˆ ˆ n to prove the our main result which completely characterises of knn and L of the asymptotic joint distribution of the order statistics of the normalised ˆ n in this case: component sizes of G ˆ 1, L ˆ 2 , ... is a sequence of discrete random variTheorem 5. Suppose that L ˆ ables such that for √ each n ≥ 1, 1 ≤ Ln ≤ n. Also, supposeˆ that there exists m(n) such that n = o(m(n)) and such that α(n) ≡ Pr{Ln < m(n)} → 0 as n → ∞. Finally, suppose that, for m ≥ 1, σ ˆm = σm , a uniform permutation on {1, 2, ..., m}. Then the joint distribution of the order statistics of ˆ n converges to the Poisson-Dirichlet(1) the normalised component sizes of G distribution on the simplex X ∇ = {xi } : xi ≤ 1, xi ≥ xi+1 ≥ 0 for every i ≥ 1 as n → ∞. To describe the main steps in the proof, we need to introduce some notation and state a sufficient condition for convergence to the Poisson-Dirichlet(1) distribution on ∇. 15
ˆ n , let K ˆ n(1) denote the component in G ˆ n which contains First, given G (1) (2) ˆ n 6= G ˆ n , then let K ˆ n denote the component in vertex labelled 1. If K (1) ˆ ˆ ˆ n(2) = ∅. For Gn \ Kn which contains the smallest vertex; otherwise, set K ˆ n(i) iteratively: If G ˆ n \ (K ˆ n(1) ∪ ... ∪ K ˆ n(i−1) ) 6= ∅, then let i > 2, we define K ˆ n(i) denote the component in G ˆ n \ (K ˆ n(1) ∪ ... ∪ K ˆ n(i−1) ) which contains the K ˆ n(i) = ∅. For i ≥ 1, let kˆn(i) = |K ˆ n(i) | and define smallest vertex; otherwise, set K (1) (2) the sequence (ˆ zn , zˆn , . . .) by zˆn(1)
(i) (1) (2) kˆn kˆn kˆn (i) (2) , . . . , zˆn = , ... = , zˆn = (1) (1) (2) (i−1) n n − kˆn n − kˆn − kˆn − . . . − kˆn
(i) (1) (2) (i−1) where zˆn = 0 if n − kˆn − kˆn − . . . − kˆn = 0. For i ≥ 1, we also define (i) ˆ n . Finally, for the uniform dˆn to be the size of the ith largest component in G (i) (i) ( (i) random mapping digraph Gn , the definitions of Kn , kn , zn i), and dn are analogous to the definitions given above. Now it is well-known (see, for example, Hansen [9] and references therein) that (1) to(2)show that the joint distribution of the normalized order statistics, ˆ dˆn , dnn , . . . , converges to the PD(1) distribution on ∇, it is sufficient to n show that for each t ≥ 1 and 0 < ai < bi < 1 , i = 1, 2, . . . , t, we have
lim Pr ai
0 is a fixed parameter, and in this case we show that the limiting distribution cannot be PD(θ). In light of Theorem 5 and our results for ‘trimmed’ random mappings, ˆ it √ would be of interest to determine whether there is some Ln (of order n) such that the joint distribution of the normalized order statistics of the ˆ n (constructed using uniform permutations) converges component sizes of G to the PD(θ) for some 1/2 < θ < 1. In another direction, we mention that 18
functional central limit theorems have been obtained for the number of cycles in a uniform random permutation (see [5]) and for the number of components in Gn (see [8]). It is likely that a functional central limit theorem also holds ˆ n , and we expect that the normalization in for the number of components G ˆ n. the functional central limit theorem will depend on the distribution of L Finally, we note that the results in this paper have been obtained under ˆ n are uniformly the assumption that the permutations used to construct G distributed. If we consider Fact 1 above, we see that when σ ˆm is a uniform random permutation we can compute the values of pm,j exactly to obtain Corollary 2. One may ask how far we can perturb the product mj jpm,j from 1 and still obtain the same asymptotic results. In another direction, it θ θ ˆ n , where σ would be interesting to use permutations σ ˆm to construct G ˆm is a random permutation on {1, 2, ..., m} with cycle structure given by the Ewens sampling formula (see [4], p.60) with parameter θ > 0. We note that the case θ = 1 corresponds to the uniform random permutation on {1, 2, ..., m}. ˆ n varies as So it would be interesting to investigate how the structure of G the parameter θ varies and to determine the corresponding threshold for the number of cyclic vertices in this case as well.
References [1] Aldous, D., Exchangeability and related topics, Lecture Notes in Math, 1117, Springer-Verlag, (1985), New York. [2] Arratia, R. and Tavar´e, S., Limit theorems for combinatorial structures via discrete process approximations, Random Structures and Algorithms, 3, (1992), 321–345. [3] Arratia, R., Stark, D. and Tavar´e, S., Total variation asymptotics for Poisson process approximations of logarithmic combinatorial assemblies, The Annals of Probability, 23, (1995), 1347–1388 . [4] Arratia, R., Barbour, A. D. and Tavar´e, S., Logarithmic Combinatorial Structures: a Probabilistic Approach, European Mathematical Society, (2003), Zurich.
19
[5] DeLaurentis, J. M. and Pittel, B., Random permutations and Brownian motion, Pacific J. Math., 119, (1985), 287–301. [6] DeLaurentis, J. M., Components and cycles of a random function, Lecture Notes in Computer Science, 293, Advances in Cryptology CRYPTO’87 Proceedings, (1988), 231–242. [7] Feller, W., An Introduction to Probability Theory and its Applications, Vol I, 3rd edition, John Wiley and Sons, New York (1970). [8] Hansen, J. C., A functional central limit theorem for random mappings, Ann. of Probab., 17, (1989), 317–332. [9] Hansen, J. C., Order statistics for decomposable combinatorial structures, Random Structures and Algorithms, 5, (1994), 517–533. [10] Hansen, J. C. and Jaworski, J., A cutting process for random mappings, submitted (http://www.staff.amu.edu.pl/ jaworski/), (2005). [11] Harris, B., Probability distribution related to random mappings, Ann. Math. Statist. 31, (1960), 1045–1062. [12] Jaworski, J., Epidemic processes on digraphs of random mappings, Appl. Prob. 36 (1999), 1–19.
J.
[13] Kolchin, V. F., Random Mappings, Optimization Software Inc., New York (1986). [14] Mutafchiev, L., On some stochastic problems of discrete mathematics, in: Mathematics and Education in Mathematics, Bulg.Akad.Nauk, Sofia, (1984), 57–80. [15] Pollard, J. M., A Monte Carlo method for factorization, 15(3), (1975), 331-334.
BIT,
[16] Quisquater, J.-J. and Delescaille, J.-P., How easy is collision search? Application to DES, Lecture Notes in Computer Science, 434, Advances in Cryptology - Eurocrypt Proceedings, 429–434. [17] Vershik, A. M. and Schmidt, A. A., Limit measures arising in the asymptotic theory of symmetric groups. I, Theor. Probab. Appl., 22, (1977), 70–85. 20
[18] van Oorschot, P. C., and Wiener, M. J., Parallel collision search with cryptanalytic applications, Journal of Cryptology, 12(1), (1999), 1-28.
21
