ON VERIFICATION OF RESTRICTED EXTENDED AFFINE EQUIVALENCE OF VECTORIAL ...

80 Pages·2012·0.28 MB·Turkish

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ONVERIFICATIONOFRESTRICTEDEXTENDEDAFFINEEQUIVALENCEOF
VECTORIALBOOLEANFUNCTIONS
ATHESISSUBMITTEDTO
THEGRADUATESCHOOLOFAPPLIEDMATHEMATICS
OF
MIDDLEEASTTECHNICALUNIVERSITY
BY
AHMETSINAK
INPARTIALFULFILLMENTOFTHEREQUIREMENTS
FOR
THEDEGREEOFMASTEROFSCIENCE
IN
CRYPTOGRAPHY
SEPTEMBER2012
Approvalofthethesis:
ONVERIFICATIONOFRESTRICTEDEXTENDEDAFFINEEQUIVALENCEOF
VECTORIALBOOLEANFUNCTIONS
submitted by AHMET SINAK in partial fulfillment of the requirements for the degree of
Master of Science in Department of Cryptography, Middle East Technical University
by,
Prof. Dr. Bu¨lentKARASO¨ZEN
Director,GraduateSchoolofAppliedMathematics
Prof. Dr. FerruhO¨ZBUDAK
HeadofDepartment,Cryptography
Prof. Dr. FerruhO¨ZBUDAK
Supervisor,MathematicsDepartment,METU
Dr. Og˘uzYayla
Co-supervisor,CryptographyDepartment
ExaminingCommitteeMembers:
Assoc. Prof. Dr. AliDog˘anaksoy
MathematicsDepartment,METU
Prof. Dr. FerruhO¨zbudak
MathematicsDepartment,METU
Assoc. Prof. Dr. MelekDikerYu¨cel
ElectricsandElectronicsEngineeringDept.,METU
Assist. Prof. Dr. Zu¨lfu¨karSaygı
MathematicsDepartment,TOBBETU
Dr. Og˘uzYayla
CryptographyDepartment,METU
Date:
Iherebydeclarethatallinformationinthisdocumenthasbeenobtainedandpresented
in accordance with academic rules and ethical conduct. I also declare that, as required
bytheserulesandconduct,Ihavefullycitedandreferencedallmaterialandresultsthat
arenotoriginaltothiswork.
Name,LastName: AHMETSINAK
Signature :
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ABSTRACT
ONVERIFICATIONOFRESTRICTEDEXTENDEDAFFINEEQUIVALENCEOF
VECTORIALBOOLEANFUNCTIONS
Sınak,Ahmet
M.S.,DepartmentofCryptography
Supervisor : Prof. Dr. FerruhO¨ZBUDAK
Co-Supervisor : Dr. Og˘uzYayla
SEPTEMBER2012,66pages
Vectorial Boolean functions are used as S-boxes in cryptosystems. To design inequivalent
vectorial Boolean functions resistant to known attacks is one of the challenges in cryptogra-
phy. VerifyingwhethertwovectorialBooleanfunctionsareequivalentornotisthefinalstep
inthischallenge. Hence,findingafasttechniquefordeterminingwhethertwogivenvectorial
Booleanfunctionsareequivalentisanimportantproblem. Aspecialclassoftheequivalence
calledrestrictedextendedaffine(REA)equivalenceisstudiedinthisthesis. Westudythever-
ificationcomplexityofREA-equivalenceoftwovectorialBooleanfunctionsforsometypes,
namely types I to VI. We first review the verification of the REA-equivalence types I to IV
givenintherecentworkofBudaghyanandKazymyrov(2012). Furthermore,wepresentthe
complexities of the verification of REA-equivalence types I and IV in the case basic simul-
taneous Gaussian elimination method is used. Next, we present two new REA-equivalence
types V and VI with their complexities. Finally, we give the algorithms of each type I to VI
withtheirMAGMAcodes.
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Keywords: Vectorial Boolean functions, REA-equivalence, AB-functions, APN-functions,
systemoflinearequations.
v
¨
OZ
VEKTO¨RBOOLEFONKS˙IYONLARININKISITLIGEN˙IS¸LET˙ILM˙IS¸ AF˙INDENKL˙IG˘˙I
U¨ZER˙INE
Sınak,Ahmet
M.S.,KriptografiBo¨lu¨mu¨
TezYo¨neticisi : Prof. Dr. FerruhO¨zbudak
OrtakTezYo¨neticisi : Dr. Og˘uzYayla
EYLU¨L2012,66sayfa
Vekto¨r Boole fonksiyonları kriptosistemlerde S-kutularıdır. Kriptografik ataklara dayanıklı
denk olmayan vekto¨r Boole fonksiyonları tasarlamak kriptografide o¨nemli amac¸lardan biri-
sidir. ˙Iki vekto¨r Boole fonsiyonların birbiri ile denk olup olmadıg˘ını anlamak bu amacın
son basamag˘ıdır. Dolayısıyla, vekto¨r Boole fonksiyonlarının denklig˘ine bakmak ic¸in hızlı
bir yo¨ntem bulmak o¨nemli bir problemdir. Denklig˘in bir sınıfı olan kısıtlı genis¸letilmis¸ afin
(KGA)-denklig˘i bu tezde c¸alıs¸ılmıs¸tır. KGA-denklig˘inin I ile VI arasındaki c¸es¸itleri ic¸in
iki vekto¨r Boole fonksiyonların birbiri ile denk olup olmadıg˘ını anlamanın karmas¸ıklıg˘ını
c¸alıs¸tık. O¨ncelikle, yakın zamandaki Budaghyan ve Kazymyrov (2012)’un c¸alıs¸masındaki
KGA-denklig˘inin I ile IV arasındaki c¸es¸itlerini inceledik. Ayrıca, KGA-denklig˘inin I ve IV
c¸es¸itleri ic¸in temel es¸-zamanlı Gauss eleme yo¨ntemi kullanılmasıyla olus¸an karmas¸ıklıkları
sunduk. Sonra,KGA-denklig˘ininikiyenic¸es¸idiVveVI’yıkarmas¸ıklıklarıilebirlikteverdik.
Sonolarak,KGA-denklig˘ininIileVIarasındakic¸es¸itleriic¸inalgoritmalarıMAGMAkodları
ilebirliktesunduk.
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AnahtarKelimeler: Vekto¨rBoolefonksiyonlar,KGA-denklig˘i,AB-fonksiyonlar,APN-fonksiyonlar,
dog˘rusaldenklemlersistemi.
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AilemeveYeg˘enlerime
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ACKNOWLEDGMENTS
I acknowledge financial support from TUBITAK-BIDEB 2210 program during my graduate
academiclife. IwouldliketoexpressmygratitudetomysupervisorProf. Dr. FerruhO¨zbudak
forhisinspiringguidanceandvaluablesuggestions.
I am indebted to my co-supervisor Dr. Og˘uz Yayla who helped me to organize my thesis. I
am grateful to him for correcting my poor English in this paper. I also gratefully appreciate
hisconsiderableassistanceconcerningmygraduateacademiclife.
Furthermore,Ithanktoallmyfriends,whoarewithme,fortheirpatienceandunderstanding.
Specifically, mycolleaguesattheInstituteofAppliedMathematicsdeservemanythanksfor
encouragingme.
Lastly, and most importantly, I owe my thanks to my family for their continuous support
throughoutmylife.
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