A non-uniform birthday problem with applications to discrete logarithms

We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends...

Full description

Saved in:
Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 160; no. 10-11; pp. 1547 - 1560
Main Authors: Galbraith, Steven D., Holmes, Mark
Format: Journal Article
Language:English
Published: Elsevier B.V 01.07.2012
Subjects:
Algorithms
Birthday paradox
Color
Colour
Discrete logarithm problem
Logarithms
Mathematical analysis
Paradoxes
Placing
Probabilistic analysis of randomised algorithms
Sampling
Birthday paradox
Probabilistic analysis of randomised algorithms
Discrete logarithm problem
ISSN:0166-218X, 1872-6771
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends on the colour of the ball. We determine the expected number of trials until two balls of different colours are placed in the same urn. As an aside we present an amusing “paradox” about birthdays.
AbstractList We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in groups. More precisely, we consider sampling coloured balls and placing them in urns, such that the distribution of assigning balls to urns depends on the colour of the ball. We determine the expected number of trials until two balls of different colours are placed in the same urn. As an aside we present an amusing “paradox” about birthdays.
Author Galbraith, Steven D.
Holmes, Mark
Author_xml – sequence: 1
  givenname: Steven D.
  surname: Galbraith
  fullname: Galbraith, Steven D.
  email: S.Galbraith@math.auckland.ac.nz
  organization: Mathematics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
– sequence: 2
  givenname: Mark
  surname: Holmes
  fullname: Holmes, Mark
  email: mholmes@stat.auckland.ac.nz
  organization: Department of Statistics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
BookMark eNp9kE9LAzEUxINUsK1-AG979LJrXrLdP3gqRasgeFHwFrLJW5uyu1mTVOm3N7WePBQG3uU3w7yZkclgByTkGmgGFIrbbaZlnzEKLKNRUJ-RKVQlS4uyhAmZRqZIGVTvF2Tm_ZbSSEI1JetlEpPS3WBa6_qkMS5stNwno7NNh33ybcImkePYGSWDsYNPgk208cphwKSzH9JFoveX5LyVncervzsnbw_3r6vH9Pll_bRaPqeKcxrSfNFgDQ3wvAbJuS64LBlAzSnFBeZtXQONTNXQWmveIFuUWIGiGiVlWOV8Tm6OubHg5w59EH0sg10nB7Q7L6AoY3gJjEYUjqhy1nuHrRid6aXbC6DiMJrYijiaOIwmaFSsMSflP48y4ffx4KTpTjrvjk6M338ZdMIrg4NCbRyqILQ1J9w_k9GIew
CitedBy_id crossref_primary_10_1002_rsa_21166
crossref_primary_10_1016_j_ipl_2021_106225
crossref_primary_10_1214_16_AAP1176
crossref_primary_10_1016_j_ejc_2019_06_007
crossref_primary_10_4236_jmf_2016_64044
crossref_primary_10_4213_dm1257
crossref_primary_10_1002_rsa_20856
crossref_primary_10_1137_18M1184461
crossref_primary_10_1007_s10623_018_0492_3
crossref_primary_10_1002_rsa_21017
crossref_primary_10_1007_s10623_021_00966_5
Cites_doi 10.1214/ss/1177012015
10.1007/BF00053956
10.1007/BF02252867
10.1515/dma.1995.5.1.73
10.1214/aop/1176996359
10.1016/0166-218X(92)90177-C
10.1214/aop/1176991491
10.1016/j.jspi.2003.11.015
10.1007/s001450010010
ContentType Journal Article
Copyright 2012 Elsevier B.V.
Copyright_xml – notice: 2012 Elsevier B.V.
DBID 6I.
AAFTH
AAYXX
CITATION
7SC
8FD
JQ2
L7M
L~C
L~D
DOI 10.1016/j.dam.2012.02.019
DatabaseName ScienceDirect Open Access Titles
Elsevier:ScienceDirect:Open Access
CrossRef
Computer and Information Systems Abstracts
Technology Research Database
ProQuest Computer Science Collection
Advanced Technologies Database with Aerospace
Computer and Information Systems Abstracts – Academic
Computer and Information Systems Abstracts Professional
DatabaseTitle CrossRef
Computer and Information Systems Abstracts
Technology Research Database
Computer and Information Systems Abstracts – Academic
Advanced Technologies Database with Aerospace
ProQuest Computer Science Collection
Computer and Information Systems Abstracts Professional
DatabaseTitleList
Computer and Information Systems Abstracts
DeliveryMethod fulltext_linktorsrc
Discipline Mathematics
EISSN 1872-6771
EndPage 1560
ExternalDocumentID 10_1016_j_dam_2012_02_019
S0166218X12000807
GroupedDBID -~X
6I.
AAFTH
ADEZE
AFTJW
AI.
ALMA_UNASSIGNED_HOLDINGS
FA8
FDB
OAUVE
VH1
WUQ
AAYXX
CITATION
7SC
8FD
JQ2
L7M
L~C
L~D
ID FETCH-LOGICAL-c330t-45be91b13491a33d63a72119300e5e4f991045b8b09dd3be257e81c0dea02e843
ISICitedReferencesCount 13
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000304217000013&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0166-218X
IngestDate Thu Jul 10 22:11:06 EDT 2025
Tue Nov 18 22:35:43 EST 2025
Sat Nov 29 02:59:31 EST 2025
Sat Apr 29 22:45:09 EDT 2023
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 10-11
Keywords Birthday paradox
Probabilistic analysis of randomised algorithms
Discrete logarithm problem
Language English
License http://www.elsevier.com/open-access/userlicense/1.0
https://www.elsevier.com/tdm/userlicense/1.0
https://www.elsevier.com/open-access/userlicense/1.0
LinkModel OpenURL
MergedId FETCHMERGED-LOGICAL-c330t-45be91b13491a33d63a72119300e5e4f991045b8b09dd3be257e81c0dea02e843
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
OpenAccessLink https://dx.doi.org/10.1016/j.dam.2012.02.019
PQID 1671347120
PQPubID 23500
PageCount 14
ParticipantIDs proquest_miscellaneous_1671347120
crossref_primary_10_1016_j_dam_2012_02_019
crossref_citationtrail_10_1016_j_dam_2012_02_019
elsevier_sciencedirect_doi_10_1016_j_dam_2012_02_019
PublicationCentury 2000
PublicationDate July 2012
2012-07-00
20120701
PublicationDateYYYYMMDD 2012-07-01
PublicationDate_xml – month: 07
  year: 2012
  text: July 2012
PublicationDecade 2010
PublicationTitle Discrete Applied Mathematics
PublicationYear 2012
Publisher Elsevier B.V
Publisher_xml – name: Elsevier B.V
References DasGupta (br000030) 2005; 130
Nishimura, Sibuya (br000065) 1988; 40
Chen (br000025) 1975; 3
Pollard (br000075) 1978; 32
Pollard (br000080) 2000; 13
Arratia, Goldstein, Gordon (br000010) 1990; 5
Camarri, Pitman (br000015) 2000; 5
Grimmett, Stirzaker (br000060) 1992
S.D. Galbraith, J.M. Pollard, R.S. Ruprai, The discrete logarithm problem in an interval, Math. Comput. (in press).
Chatterjee, Diaconis, Meckes (br000020) 2004
Galbraith, Ruprai (br000045) 2009; vol. 5921
Arratia, Goldstein, Gordon (br000005) 1989; 17
Flajolet, Gardy, Thimonier (br000035) 1992; 39
Nishimura, Sibuya (br000070) 1990; 2
Galbraith, Ruprai (br000050) 2010; vol. 6056
Gaudry, Schost (br000055) 2004; vol. 3076
Selivanov (br000085) 1995; 5
Pollard (10.1016/j.dam.2012.02.019_br000075) 1978; 32
Arratia (10.1016/j.dam.2012.02.019_br000005) 1989; 17
Nishimura (10.1016/j.dam.2012.02.019_br000065) 1988; 40
Chatterjee (10.1016/j.dam.2012.02.019_br000020) 2004
Chen (10.1016/j.dam.2012.02.019_br000025) 1975; 3
Arratia (10.1016/j.dam.2012.02.019_br000010) 1990; 5
Camarri (10.1016/j.dam.2012.02.019_br000015) 2000; 5
Pollard (10.1016/j.dam.2012.02.019_br000080) 2000; 13
Nishimura (10.1016/j.dam.2012.02.019_br000070) 1990; 2
Galbraith (10.1016/j.dam.2012.02.019_br000045) 2009; vol. 5921
DasGupta (10.1016/j.dam.2012.02.019_br000030) 2005; 130
Gaudry (10.1016/j.dam.2012.02.019_br000055) 2004; vol. 3076
Flajolet (10.1016/j.dam.2012.02.019_br000035) 1992; 39
Galbraith (10.1016/j.dam.2012.02.019_br000050) 2010; vol. 6056
Selivanov (10.1016/j.dam.2012.02.019_br000085) 1995; 5
Grimmett (10.1016/j.dam.2012.02.019_br000060) 1992
10.1016/j.dam.2012.02.019_br000040
References_xml – year: 1992
  ident: br000060
  article-title: Probability and Random Processes
– volume: 39
  start-page: 207
  year: 1992
  end-page: 229
  ident: br000035
  article-title: Birthday paradox, coupon collectors, caching algorithms and self-organizing search
  publication-title: Discrete Appl. Math.
– year: 2004
  ident: br000020
  article-title: Exchangeable pairs and Poisson approximation
  publication-title: Electron. Encyclopedia Probab.
– volume: 130
  start-page: 377
  year: 2005
  end-page: 389
  ident: br000030
  article-title: The matching, birthday and the strong birthday problem: a contemporary review
  publication-title: J. Statist. Plann. Inference
– volume: vol. 6056
  start-page: 368
  year: 2010
  end-page: 383
  ident: br000050
  article-title: Using equivalence classes to accelerate solving the discrete logarithm problem in a short interval
  publication-title: PKC 2010
– volume: 32
  start-page: 918
  year: 1978
  end-page: 924
  ident: br000075
  article-title: Monte Carlo methods for index computation (mod
  publication-title: Math. Comput.
– volume: 5
  start-page: 1
  year: 2000
  end-page: 18
  ident: br000015
  article-title: Limit distributions and random trees derived from the birthday problem with unequal probabilities
  publication-title: Electron. J. Probab.
– volume: 5
  start-page: 403
  year: 1990
  end-page: 424
  ident: br000010
  article-title: Poisson approximation and the Chen–Stein method
  publication-title: Statist. Sci.
– volume: 13
  start-page: 437
  year: 2000
  end-page: 447
  ident: br000080
  article-title: Kangaroos, monopoly and discrete logarithms
  publication-title: J. Cryptology
– reference: S.D. Galbraith, J.M. Pollard, R.S. Ruprai, The discrete logarithm problem in an interval, Math. Comput. (in press).
– volume: vol. 5921
  start-page: 368
  year: 2009
  end-page: 382
  ident: br000045
  article-title: An improvement to the Gaudry–Schost algorithm for multidimensional discrete logarithm problems
  publication-title: Twelfth IMA International Conference on Cryptography and Coding
– volume: 5
  start-page: 73
  year: 1995
  end-page: 82
  ident: br000085
  article-title: On waiting time in the scheme of random allocation of coloured particles
  publication-title: Discrete Math. Appl.
– volume: 40
  start-page: 77
  year: 1988
  end-page: 91
  ident: br000065
  article-title: Occupancy with two types of balls
  publication-title: Ann. Inst. Statist. Math.
– volume: vol. 3076
  start-page: 208
  year: 2004
  end-page: 222
  ident: br000055
  article-title: A low-memory parallel version of Matsuo, Chao and Tsujii’s algorithm
  publication-title: ANTS VI
– volume: 3
  start-page: 534
  year: 1975
  end-page: 545
  ident: br000025
  article-title: Poisson approximation for dependent trials
  publication-title: Ann. Probab.
– volume: 2
  start-page: 13
  year: 1990
  end-page: 22
  ident: br000070
  article-title: Probability to meet in the middle
  publication-title: J. Cryptology
– volume: 17
  start-page: 9
  year: 1989
  end-page: 25
  ident: br000005
  article-title: Two moments suffice for poisson approximations: the Chen–Stein method
  publication-title: Ann. Probab.
– volume: 5
  start-page: 403
  issue: 4
  year: 1990
  ident: 10.1016/j.dam.2012.02.019_br000010
  article-title: Poisson approximation and the Chen–Stein method
  publication-title: Statist. Sci.
  doi: 10.1214/ss/1177012015
– volume: vol. 3076
  start-page: 208
  year: 2004
  ident: 10.1016/j.dam.2012.02.019_br000055
  article-title: A low-memory parallel version of Matsuo, Chao and Tsujii’s algorithm
– volume: vol. 6056
  start-page: 368
  year: 2010
  ident: 10.1016/j.dam.2012.02.019_br000050
  article-title: Using equivalence classes to accelerate solving the discrete logarithm problem in a short interval
– volume: 40
  start-page: 77
  issue: 1
  year: 1988
  ident: 10.1016/j.dam.2012.02.019_br000065
  article-title: Occupancy with two types of balls
  publication-title: Ann. Inst. Statist. Math.
  doi: 10.1007/BF00053956
– volume: 2
  start-page: 13
  issue: 1
  year: 1990
  ident: 10.1016/j.dam.2012.02.019_br000070
  article-title: Probability to meet in the middle
  publication-title: J. Cryptology
  doi: 10.1007/BF02252867
– volume: 5
  start-page: 73
  issue: 1
  year: 1995
  ident: 10.1016/j.dam.2012.02.019_br000085
  article-title: On waiting time in the scheme of random allocation of coloured particles
  publication-title: Discrete Math. Appl.
  doi: 10.1515/dma.1995.5.1.73
– volume: 3
  start-page: 534
  issue: 3
  year: 1975
  ident: 10.1016/j.dam.2012.02.019_br000025
  article-title: Poisson approximation for dependent trials
  publication-title: Ann. Probab.
  doi: 10.1214/aop/1176996359
– ident: 10.1016/j.dam.2012.02.019_br000040
– volume: 39
  start-page: 207
  issue: 3
  year: 1992
  ident: 10.1016/j.dam.2012.02.019_br000035
  article-title: Birthday paradox, coupon collectors, caching algorithms and self-organizing search
  publication-title: Discrete Appl. Math.
  doi: 10.1016/0166-218X(92)90177-C
– year: 2004
  ident: 10.1016/j.dam.2012.02.019_br000020
  article-title: Exchangeable pairs and Poisson approximation
  publication-title: Electron. Encyclopedia Probab.
– year: 1992
  ident: 10.1016/j.dam.2012.02.019_br000060
– volume: 17
  start-page: 9
  issue: 1
  year: 1989
  ident: 10.1016/j.dam.2012.02.019_br000005
  article-title: Two moments suffice for poisson approximations: the Chen–Stein method
  publication-title: Ann. Probab.
  doi: 10.1214/aop/1176991491
– volume: 130
  start-page: 377
  year: 2005
  ident: 10.1016/j.dam.2012.02.019_br000030
  article-title: The matching, birthday and the strong birthday problem: a contemporary review
  publication-title: J. Statist. Plann. Inference
  doi: 10.1016/j.jspi.2003.11.015
– volume: 32
  start-page: 918
  issue: 143
  year: 1978
  ident: 10.1016/j.dam.2012.02.019_br000075
  article-title: Monte Carlo methods for index computation (mod p)
  publication-title: Math. Comput.
– volume: 5
  start-page: 1
  issue: 2
  year: 2000
  ident: 10.1016/j.dam.2012.02.019_br000015
  article-title: Limit distributions and random trees derived from the birthday problem with unequal probabilities
  publication-title: Electron. J. Probab.
– volume: vol. 5921
  start-page: 368
  year: 2009
  ident: 10.1016/j.dam.2012.02.019_br000045
  article-title: An improvement to the Gaudry–Schost algorithm for multidimensional discrete logarithm problems
– volume: 13
  start-page: 437
  issue: 4
  year: 2000
  ident: 10.1016/j.dam.2012.02.019_br000080
  article-title: Kangaroos, monopoly and discrete logarithms
  publication-title: J. Cryptology
  doi: 10.1007/s001450010010
SSID ssj0001218
ssj0000186
ssj0006644
Score 2.066263
Snippet We consider a generalisation of the birthday problem that arises in the analysis of algorithms for certain variants of the discrete logarithm problem in...
SourceID proquest
crossref
elsevier
SourceType Aggregation Database
Enrichment Source
Index Database
Publisher
StartPage 1547
SubjectTerms Algorithms
Birthday paradox
Color
Colour
Discrete logarithm problem
Logarithms
Mathematical analysis
Paradoxes
Placing
Probabilistic analysis of randomised algorithms
Sampling
Title A non-uniform birthday problem with applications to discrete logarithms
URI https://dx.doi.org/10.1016/j.dam.2012.02.019
https://www.proquest.com/docview/1671347120
Volume 160
WOSCitedRecordID wos000304217000013&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVESC
  databaseName: Elsevier SD Freedom Collection Journals 2021
  customDbUrl:
  eissn: 1872-6771
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0001218
  issn: 0166-218X
  databaseCode: AIEXJ
  dateStart: 19950101
  isFulltext: true
  titleUrlDefault: https://www.sciencedirect.com
  providerName: Elsevier
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3PT9swFLZQuwM7IBhMY2OTkTitCnJqJ46P1cr4IYF2AKk3y04cAeoCalPEn7_nxHZDBQgOk6KoShM3zfvy_PnZ73sIHRDocnkJ4C3TMolYUopIZ8xEKoHeL1VCm6Joik3wi4tsMhF_XArBvCknwKsqe3wU9__V1HAMjG1TZ99h7tAoHIDPYHTYg9lh_ybDjwYwoo8WlU25-jvQN7P62uZ6uMoxLpmtM21t2afNzZ0BfR6AJ4TBc33tNMwdax37rz1nPQ9ir4GSH6spDLxdmKatlTYYHwbUhMyHkBzkQg3xcllqiD6maQScYPLEfbb1ADxOSOScZ-sQgaHxTudq87afddxtDOEWftvKA9gALWzOmT4RyV7pvMKSQr9a7VZCE9I2IQlsVhG2P-SJyHqoPzo9mpx1xMWsct66D8ctZ5-AhTGnCd_-XT8b3qwLXLnDl_jMSs_e0JXLTbThxhl41OJjC62Z6hP62LHbNjoe4Q5SsEcKdkjBFim4ixRc32GPFLxEyg66-n10-eskcnU1opxSUsMbqY2ItRWmjBWlRUqVjQMISohJDCthyABEX2eaiKKg2oBXN1mck8IoMjQZo59RD-7PfEE44TxnVBWpSDkThCuuch2rgg3hXCPELiL-6cjcic7b2idT-aK9dtHPcMl9q7jy2snMP3LpKGNLBSUA67XL9r15JLhTO0emKnO3mMs4bZKr4yH5-p77-IbWly_MHurVs4X5jj7kD_XNfPbDIe8fn9uUBA
linkProvider Elsevier
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+non-uniform+birthday+problem+with+applications+to+discrete+logarithms&rft.jtitle=Discrete+Applied+Mathematics&rft.au=Galbraith%2C+Steven+D.&rft.au=Holmes%2C+Mark&rft.date=2012-07-01&rft.issn=0166-218X&rft.volume=160&rft.issue=10-11&rft.spage=1547&rft.epage=1560&rft_id=info:doi/10.1016%2Fj.dam.2012.02.019&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_dam_2012_02_019
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0166-218X&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0166-218X&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0166-218X&client=summon