-invertible cycles for multivariate quadratic (mq) public key cryptographynew

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1、-InvertibleCyclesforMultivariateQuadratic(MQ)PublicKeyCryptographyJintaiDing1,,ChristopherWolf2,andBo-YinYang3,1UniversityofCincinnatiandTechnischeUniversitätDarmstadtding@math.uc.edu2EcoleNormaleSuperieurchris@christopher-wolf.de3InstituteofInformationSci

2、ence,AcademiaSinicaandTWISCby@moscito.orgAbstract.WeproposeanewbasictrapdoorIC(-InvertibleCycles)ofthemixedfieldtypeforMultivariateQuadraticpublickeycryptosys-tems.ThisisthefirstnewbasictrapdoorsincetheinventionofUn-balancedOilandVinegarin1997.ICcanbeconsider

3、edanextended∗formofthewell-knownMatsumoto-ImaiSchemeA(alsoMIAorC),andsharesomefeaturesofstagewisetriangularsystems.HoweverIChasverydistinctivepropertiesofitsown.Inpractice,ICismuchfasterthanMIA,andcanevenmatchthespeedofsingle-fieldMQschemes.Keywords:PublicKey

4、,MQ,Trapdoor,Encryption,Signing.1IntroducingMQPublicKeyCryptosystemsWeworkoverafinitefieldFofqelements(thebasenmfield).P∈MQ(F,F)isasystemofmquadraticpolynomialsinnvariablesinF,calledthecentralmapx=(x1,...,xn)anditscomponentscentralpolynomials.Compositionwithth

5、eaffinemapsS,TmasksthestructureofPprivate:Sandgivesthepublicmap:xpublic:P=(p1,...,pm):=T◦P◦S(1)private:P(p1,...,pm)Weusuallywrite,for1≤i≤m,1≤j≤k≤n,PPynpi(x1,...,xn):=γi,j,kxjxk+βi,jxj+αi1≤j≤k≤nj=1private:Twhereαiisusuallynormalizedtozero.ThePublickeyoutpu

6、ty∈Fmcomprisethemn(n+3)/2coefficientsγijk,βij∈F.Fig.1.IllustrationofTerminologyandNotationforamodernMQ-trapdoorMultivariateQuadratic(MQ)public-keycryptographyfirstappearedintheEnglishliteratureinthemid’80s[FD85,IM85]asalternativestotraditionalAlsopartiallyspons

7、oredbygrantsfromtheCharlesPhelpsTaftResearchCenterandtheAlexandervonHumboldtFoundation.AlsosponsoredbyTaiwan’sNationalScienceCouncilproject95-2115-M-001-021andindirectlyviaTWISC(TaiwanInformationSecurityCenter)@NTUST.T.OkamotoandX.Wang(Eds.):PKC2007,LNCS4450

8、,pp.266–281,2007.cInternationalAssociationforCryptologicResearch2007-InvertibleCyclesforMQPublicKeyCryptography267PKCs.Acommonexcusegiventostudythemis“forecologicaldiversity”,in-