P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354. Home/Search Document Details and Download
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The Asymptotic Distribution of the Diameter of a Random Mapping - Aldous, Pitman (2002) (Correct)
....loi limite, g en eralisant la formule pour sa moyenne due a Flajolet et Odlyzko (1990) Research supported in part by N.S.F. Grants DMS 9970901 and DMS 0071448 1 Introduction Let F n be a uniformly distributed random mapping from the set [n] f1; 2; ng to itself, as studied in [10, 8, 1] and papers cited there. The diameter of F n is the random variable Delta n : max i2[n] T n (i) where T n (i) is the number of iterations of F n starting from i until some value is repeated: T n (i) minfj 1 : F n (i) F n (i) for some 0 k jg where F n (i) i and F n (i) F ....
....1 jL j v, while I(v) is the expected number of j with jB 1 jL j v and jB 1 jL j 2jB 1 jM j v. The probability of the event jB 1 j Delta v, that there is no j with jB 1 jL j 2jB 1 jM j v, is therefore e . The conclusion of Theorem 1 is now evident. 4 Related Results As indicated in [8] there are companions to (1) for other functionals of F n besides the diameter, in particular the total length of cycles and the maximum tree height. One advantage of the present approach is that all these results can be understood in terms of the Poisson representation of Lemma 3. For instance, ....
P. Flajolet and A. Odlyzko. Random mapping statistics. In J.-J. Quisquater and J. Vandewalle, editors, Advances in Cryptology -- EUROCRYPT '89, pages 329--354. Springer-Verlag, 1990. Lecture Notes in C.S. 434. 8
A Poisson * geometric convolution law for the number of components .. - Hwang (1995) (Correct)
....1. Random mapping patterns. By random mapping [26] we mean a random singled valued mapping of the set 2, 3, n into itself. These structures have been extensively studied in the literature due both to their intrinsic interest and their wide applications to many di#erent fields, cf. [26, 9, 7, 1]. Two mappings 1 and 2 are said to be equivalent if there exists a permutation # of 2, 3, n such that 1 (i) j i# 2 (#(i) #(j) for all pairs (i, j) Random mapping patterns are equivalence classes of mapping functions. Structurally, they are multisets of cycles of rooted ....
P. Flajolet and A. M. Odlyzko. Random mapping statistics. In EUROCRYPT '89, Lecture Notes in Computer Science, 434, pages 329--354. Springer, Berlin, 1990.
Asymptotics of Poisson approximation to random discrete.. - Hwang (1998) (Correct)
....and Arratia et al. 7] and on record values, Nevzorov [68] and Borovkov and Pfeifer [18] Random mappings. Structurally, random mappings are set of cycles of rooted labeled trees. In terms of the notation of 3. 1, the generating function for the number of cycles (T structures) is given by (cf. [34, 65]) 1 K(z) where K(z) ze K(z) z e 1 ) By the implicit function theorem (or, in this case, one considers n#1 n e n nu n by Mellin transform) and the analytic continuation of K, we have 1 ez log 2 O ez uniformly for z in any compact region contained in ....
....2 O ez uniformly for z in any compact region contained in a certain # 0 domain; and consequently 1 2n . Theorem 3 applies. An explicit expression for t n is easily obtained by the Lagrange inversion formula t n = n 0#j n . For further information on random mappings, see [2, 29, 34, 47, 60]. Profiles of increasing trees. A plane recursive tree (cf. 16] is a (plane) increasing tree, namely, a labeled rooted tree in which labels along any path from the root form an increasing sequence, with degree set 0, 1, 2, Let L n,m denote the expected number of nodes at depth m in ....
[Article contains additional citation context not shown here]
Flajolet, P. and Odlyzko, A. M. (1990) Random mapping statistics.Lecture Notes in Computer Science, 434, EUROCRYPT '89 Springer, Berlin, pp. 329--354.
Large deviations of combinatorial distributions II: Local limit.. - Hwang (1997) (11 citations) (Correct)
....viewed as a set (partitional complex) of connected components each of which is a cycle of rooted labeled (or Cayley) trees. The bivariate generating function for random mappings is given by w log 1 (jzj e ) where C(z) z e C(z) enumerates Cayley trees. From the singular expansion (cf. [9]) C(z) 1 Gamma 2 (1 Gamma ez) Gamma (1 Gamma ez) c k (1 Gamma ez) z e ; 1) 23) we get log 1 2 log 1 1 Gamma ez H( 1 Gamma ez) z e ; 1) where the logarithm takes its principal value and H(t) is analytic at t = 0 with H(0) 0. Theorem 3 applies to n , ....
P. Flajolet and A. M. Odlyzko, Random mapping statistics, In Lecture Notes in Computer Science, 434, EUROCRYPT '89, pages 329--354, Springer, Berlin, 1990.
A Poisson * geometric convolution law for the number of components .. - Hwang (1995) (Correct)
....1. Random mapping patterns. By random mapping [26] we mean a random singled valued mapping of the set f1; 2; 3; ng into itself. These structures have been extensively studied in the literature due both to their intrinsic interest and their wide applications to many di erent elds, cf. [26, 9, 7, 1]. Two mappings 1 and 2 are said to be equivalent if there exists a permutation of f1; 2; 3; ng such that 1 (i) j i 2 ( i) j) for all pairs (i; j) Random mapping patterns are equivalence classes of mapping functions. Structurally, they are multisets of cycles of rooted ....
P. Flajolet and A. M. Odlyzko. Random mapping statistics. In EUROCRYPT '89, Lecture Notes in Computer Science, 434, pages 329-354. Springer, Berlin, 1990.
Uniform asymptotics of some Abel sums arising in coding theory - Hwang (1997) (Correct)
.... by a change of variables, S n;k = n k) jzj= z) dz (0 e = n k) e Z c i c i n )B k (e ) d (c 0) j j ( 62 (1; 0] Analytic continuation of B k (z) follows from that of B(z) which in turn is obtained from that of T (z) see Flajolet and Odlyzko [12] or Corless et al. 5] We now make explicit the local behavior of ) as 0. From the Mellin inversion formula (cf. 9] 1 (s)w ds ( w 0; a 0) it follows, by absolute convergence, that 1 (s) Y k (s) ds (a k ) 14) Y k (s) s k 2 ) The singularities ....
P. Flajolet and A. M. Odlyzko, Random mapping statistics, in Lecture Notes in Computer Science, 434, EUROCRYPT '89, pp. 329-354, Springer, Berlin, 1990.
Uniform asymptotics of some Abel sums arising in coding theory - Hwang (Correct)
.... (6) we have, by a change of variables, S n,k = # z =# z n k 1 B(z)B (z) dz (0 # e 1 ) k) e c i# c i# n# B(e 1 # )B k (e 1 # ) d# (c 0) Analytic continuation of B k (z) follows from that of B(z) which in turn is obtained from that of T (z) see Flajolet and Odlyzko [12] or Corless et al. 5] We now make explicit the local behavior of B k (e 1 # ) as # 0. From the Mellin inversion formula (cf. 9] e w = 1 #(s)w s ds (#w 0, a 0) it follows, by absolute convergence, that B k (e 1 # ) 1 #(s)# s Y k (s) ds (a k ) where Y k (s) ....
P. Flajolet and A. M. Odlyzko, Random mapping statistics, in Lecture Notes in Computer Science, 434, EUROCRYPT '89, pp. 329--354, Springer, Berlin, 1990.
Large deviations of combinatorial distributions II: Local limit.. - Hwang (1997) (11 citations) (Correct)
....be viewed as a set (partitional complex) of connected components each of which is a cycle of rooted labeled (or Cayley) trees. The bivariate generating function for random mappings is given by w log 1 ( z e 1 ) where C(z) z e enumerates Cayley trees. From the singular expansion (cf. [9]) C(z) 1 2 (1 ez) c k (1 (23) we get log 1 1 ez H( # 1 ez) z where the logarithm takes its principal value and H(t) is analytic at t = 0 with H(0) 0. Theorem 3 applies to # n , the number of connected components in a random mapping of size n, and we ....
P. Flajolet and A. M. Odlyzko, Random mapping statistics, In Lecture Notes in Computer Science, 434, EUROCRYPT '89, pages 329--354, Springer, Berlin, 1990.
Asymptotics of Poisson approximation to random discrete.. - Hwang (1998) (Correct)
....and Arratia et al. 7] and on record values, Nevzorov [68] and Borovkov and Pfeifer [18] Random mappings. Structurally, random mappings are set of cycles of rooted labeled trees. In terms of the notation of x 3. 1, the generating function for the number of cycles (T structures) is given by (cf. [34, 65]) 1 1 Gamma K(z) where K(z) ze K(z) jzj e ) By the implicit function theorem (or, in this case, one considers Gamman Gammanu =n by Mellin transform) and the analytic continuation of K, we have 1 1 Gamma ez log 2 O j1 Gamma ezj uniformly for z in ....
....uniformly for z in any compact region contained in a certain Delta 0 domain; and consequently 1 2n Theorem 3 applies. An explicit expression for t n is easily obtained by the Lagrange inversion formula t n = n Gamma 1) 0j n : For further information on random mappings, see [2, 29, 34, 47, 60]. Profiles of increasing trees. A plane recursive tree (cf. 16] is a (plane) increasing tree, namely, a labeled rooted tree in which labels along any path from the root form an increasing sequence, with degree set f0; 1; 2; g. Let L n;m denote the expected number of nodes at depth m in a ....
[Article contains additional citation context not shown here]
Flajolet, P. and Odlyzko, A. M. (1990) Random mapping statistics.Lecture Notes in Computer Science, 434, EUROCRYPT '89 Springer, Berlin, pp. 329--354.
Analysis of a Randomized Rendezvous Algorithm - Métivier, Saheb, Zemmari (Correct)
....(z) c(t(z) where t(z) ze t(z) is the EGF (exponential generating function) of the number of labeled trees and c(z) 1 z z z is the EGF of the number of cycles of length at least 3, see [8] The unique singularity of F (z) c(t(z) is z 0 = 1=e, since t(z 0 ) 1 i z 0 = 1=e. In [7,8], the authors show that t(z) 1 2 1=2 p 1 ez: F (z) 1 ez) which yields the number of failure calls N(K n ) on the complete graph K n : N(K n ) n [z ]F (z) n : In order to get the failure probability over K n , we have to divide N(K n ) by which is the total ....
P. Flajolet, and A. Odlyzko, Random Mapping Statistics, in Advances in Cryptology, Lecture Notes in Computer Science vol 434, (1990), 329-354. 23
Moderately Hard, Memory-bound Functions - Abadi, Burrows, Manasse, Wobber (2003) (4 citations) (Correct)
....experiments further. A posteriori, we have sketched a proof of the quadratic size, there assuming an independent random function at each tree level. A more sophisticated analysis might be possible using tools from research on random functions, a rich field with many theorems (see for instance [9]) In light of the quadratic size of trees, it is tempting to use very deep trees, so as to increase the work ratio between S and R. There are, however, important limitations on tree depth. At each level in a tree, S may try to invert all the leaves simultaneously, somehow. When there are enough ....
P. Flajolet and A. Odlyzko. Random mapping statistics. In J.-J. Quisquater and J. Vandewalle, editors, Advances in Cryptology -- EUROCRYPT ' 89, volume 434 of Lecture Notes in Computer Science, pages 329--354. Springer, 1990.
Boltzmann Samplers For The Random Generation Of.. - Duchon, Flajolet, .. Self-citation (Flajolet) (Correct)
....here a function from [1; n] into [1; n] Obviously, there are n of these. We x a nite set and restrict attention to degree constrained mappings f such that for each x in the domain, the cardinality of f ( 1) x) lies in (In the combinatorics literature, such mappings are surveyed in [2, 25]. For instance, in a nite eld, a non zero element has either 0 or 2 predecessors under the mapping f ; x 7 x , so that (neglecting one exceptional value) a quadratic function may be regarded as an element of the set of mappings constrained by = f0; 2g. Mappings are of interest in ....
Flajolet, P., and Odlyzko, A. M. Random mapping statistics. In Advances in Cryptology (
On the Security of Double and 2-key Triple Modes of Operation - Handschuh, Preneel (1999) (Correct)
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P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354.
On the Security of Double and 2-key Triple Modes of Operation - Handschuh, Preneel (1999) (Correct)
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P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354.
TMD-Tradeoff and State Entropy Loss Considerations of.. - Hong, Kim (2005) (Correct)
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P. Flajolet and A. M. Odlyzko, Random mapping statistics. Eurocrypt '89, LNCS 434, pp. 329-354, Springer-Verlag, 1990.
Daniel Adkins Karthik Lakshminarayanan Adrian Perrig Ion Stoica - Uc Berkeley Uc (Correct)
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P. Flajolet and A. Odlyzko. Random Mapping Statistics. In Advances in Cryptology --- EUROCRYPT '89, volume 434 of Lecture Notes in Computer Science. International Association for Cryptologic Research, Springer-Verlag, 1990.
Improving Implementable Meet-in-the-Middle Attacks by.. - van Oorschot, Wiener (1996) (Correct)
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P. Flajolet and A.M. Odlyzko, "Random Mapping Statistics", Lecture Notes in Computer Science 434: Advances in Cryptology - Eurocrypt '89 Proceedings, Springer-Verlag, pp. 329-354.
On the Security of Double and 2-key Triple - Modes Of Operation (Correct)
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P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354.
Modes of Operation of a Block Cipher - Preneel (Correct)
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P. Flajolet, A.M. Odlyzko, "Random mapping statistics," Advances in Cryptology, J. Stern, Ed., Springer-Verlag, 1999, pp. 329--354.
On the Security of Double and 2-key Triple Modes of Operation - Handschuh, Preneel (1999) (Correct)
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P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354.
The Width of Galton-Watson Trees Conditioned by the Size - Drmota, Gittenberger (2004) (1 citation) (Correct)
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P. Flajolet and A. M. Odlyzko, Random mapping statistics, In Advances in Cryptology (
On the Security of Double and 2-key Triple Modes of Operation - Handschuh, Preneel (1999) (Correct)
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P. Flajolet, A. M. Odlyzko, "Random mapping statistics," EUROCRYPT'89, LNCS 434, Springer-Verlag, 1990, pp. 329--354.
On the Implementation of Huge Random Objects - Oded Goldreich Shafi (2003) (1 citation) (Correct)
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P. Flajolet and A.M. Odlyzko. Random mapping statistics. In EuroCrypt'89, SpringerVerlag, LNCS 434, pages 329--354.
On the Implementation of Huge Random Objects - Oded Goldreich Shafi (2003) (1 citation) (Correct)
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P. Flajolet and A.M. Odlyzko. Random mapping statistics. In EuroCrypt'89, Springer-Verlag, LNCS 434, pages 329--354.
On the Implementation of Huge Random Objects - Oded Goldreich Sha (2003) (1 citation) (Correct)
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P. Flajolet and A.M. Odlyzko. Random mapping statistics. In EuroCrypt'89, SpringerVerlag, LNCS 434, pages 329-354.
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