Fully Homomorphic Encryption Overview and Cryptanalysis

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1、TechnischeUniversit•atDarmstadtDepartmentofComputerScienceCryptographyandComputeralgebraDiplomaThesisJuly2011FullyHomomorphicEncryption:OverviewandCryptanalysisPatrickSchmidtTechnischeUniversit•atDarmstadtDepartmentofMathematicsSupervisedbyProf.Dr.JohannesBuchmannDr.RichardLindnerMichaelSchneide

2、rAcknowledgementsFirstofall,IwouldliketothankProf.JohannesBuchmannforgivingmetheopportunitytowritemythesisunderthistopicandgainknowledgeintheeldoffullyhomomorphicschemesandmoreoverintheareaoflattices.Furthermore,manythanksgotomytwosupervisorsDr.RichardLindnerandMichaelSchneiderwhooeredandspent

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............