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1、TechnischeUniversit?atDarmstadtDepartmentofComputerScienceCryptographyandComputeralgebraDiplomaThesisJuly2011FullyHomomorphicEncryption:OverviewandCryptanalysisPatrickSchmidtTechnischeUniversit?atDarmstadtDepartmentofMathematicsSupervisedbyProf.Dr.JohannesBuchmannDr.RichardLindnerMichaelSchneide
2、rAcknowledgementsFirstofall,IwouldliketothankProf.JohannesBuchmannforgivingmetheopportunitytowritemythesisunderthistopicandgainknowledgeinthe eldoffullyhomomorphicschemesandmoreoverintheareaoflattices.Furthermore,manythanksgotomytwosupervisorsDr.RichardLindnerandMichaelSchneiderwhoo eredandspent
3、muchtimeansweringmeallmyinvestigatingquestionsandguidingmeintherightdirection.InadditionIwouldliketothankallpeoplethatsupportedmyworkonthisthesisbyproofreadingit, ndingspellingmistakesandlogicalerrorsaswellassupportingmeinimprovingthestructureofmythesisandits
uentreading;asthereare:RichardFrankl
4、and,MainaOlembo,DeniseDemirel,andDr.MelanieVolkamer.Lastly,IthankallwhomIhaveforgottentomentionbefore!Withoutyourcontributionsthisworkwouldnotbewhatitactuallyis.WarrantyIherebydeclarethatthecontentofthisthesisisthedirectresultofmyownworkandthatanyuseofpublishedorunpublishedmaterialinthisthesisis
5、fullyandcorrectlyreferenced.Date:July14th,2011Signature:P.SchmidtiiiContentsIntroduction1I.ProgressonHomomorphicEncryptionandSignatureSchemes1.Preliminaries31.1.Lattices.....................................41.2.ComputationalProblems............................61.3.SecurityModels.................
6、................82.HomomorphicEncryptionSchemes132.1.Lattice-BasedEncryptionSchemes......................132.2.Integer-ArithmeticEncryptionSchemes...................243.HomomorphicSignatureSchemes293.1.AdditionalWorktoHomomorphicSignatureSchemes...........32II.CryptanalysisoftheGentry-HaleviFHES
7、cheme4.TheGentry-HaleviScheme334.1.BriefRecalloftheConstruction........................334.2.ParameterSelection..............................345.AttackingtheGentry-HaleviScheme355.1.AttackMethodologyandStrategies............
