Theta functions and applications in cryptography  Fonctions thêta et  applications en cryptographie
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Theta functions and applications in cryptography Fonctions thêta et applications en cryptographie

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60 pages
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ThetafunctionsandapplicationsincryptographyFonctionsthêtaetapplicationsencryptographieThèsed’informatique1DamienRobert1Caramelteam,NancyUniversités,CNRS,INRIANancyGrandEst21/07/2010(Nancy)Outline1 Public-keycryptography2 Abelianvarieties3 Thetafunctions4 Pairings5 Isogenies6 PerspectivesDamienRobert (Caramel,LORIA) Thetafunctionsandcryptography 21/07/2010(Nancy) 2/40Public-keycryptographyOutline1 Public-keycryptography2 Abelianvarieties3 Thetafunctions4 Pairings5 Isogenies6 PerspectivesDamienRobert (Caramel,LORIA) Thetafunctionsandcryptography 21/07/2010(Nancy) 3/40Public-keycryptography Public-keysystemsA briefhistoryofpublic-keycryptographySecret-keycryptography:Vigenère(1553),Onetimepad(1917),AES(NIST,2001).Public-keycryptography:Diffie–Hellmankeyexchange(1976).RSA(1978):multiplication/factorisation.∗ElGamal:exponentiation/discretelogarithminG =F .qECC/HECC(1985):discretelogarithminG = A(F ).qLattices,NTRU(1996),IdealLattices(2006):perturbatealatticepoint/ClosestVectorProblem,BoundedDistanceDecoding.Polynomialsystems,HFE(1996):evaluatingpolynomials/findingroots.Coding-basedcryptography,McEliece(1978):Matrix.vector/decodingalinearcode.⇒ Encryption,Signature(+PseudoRandomNumberGenerator,ZeroKnowledge).Pairing-basedcryptography(2000–2001).Homomorphiccryptography(2009).DamienRobert (Caramel,LORIA) Thetafunctionsandcryptography 21/07/2010(Nancy) 4/40Public-keycryptography Public-keysystemsRSA versus(H)ECCSecurityRSA ...

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eta functions and applications in cryptography Fonctions thêta et applications en cryptographie èse d’informatique
Damien Robert
Caramel team, Nancy Universités, CNRS, INRIA Nancy Grand Est
// (Nancy)
maraC(trAIROL,leDabeRoenmipyotdnrcyhrgpatafu)Theonsancti
Pairings
eta functions
Perspectives
Isogenies
/
Abelian varieties
Public-key cryptography
naN()yc//
Outline
l,meRILOThA)afettcnusnoicdnatpyrDmaeiRnbore(taCar
Outline
Perspectives
Isogenies
Pairings
eta functions
Abelian varieties
//hpygoar
Public-key cryptography
/ncy)(NablPuhyrgpapyotyerccik-
-cilcyekbuPyPphliubptryraogysyskec-msteptrydcanyphraogfatehT)Asnoitcnu
Abrief history of public-key cryptography
Public-key cryptography: Diffie–Hellman key exchange (). RSA ():multiplication/factorisation. ElGamal:exponentiation/discrete logarithminG=Fq. ECC/HECC():discrete logarithminG=A(Fq). Lattices, NTRU (), Ideal Lattices ():perturbate a lattice point/Closest Vector Problem, Bounded Distance Decoding. Polynomial systems, HFE ():evaluating polynomials/finding roots. Coding-based cryptography, McEliece ():Matrix.vector/decoding a linear code. Encryption, Signature (+Pseudo Random Number Generator, Zero Knowledge).
Secret-key cryptography: Vigenère (), One time pad (), AES (NIST, ).
/
Pairing-based cryptography(–). Homomorphic cryptography ().
//cn)y(aNert(Caramel,LORIDmaeiRnbo
/
RSAversus (H)ECC
Factorisation of a -bit RSA modulus [Kle+]. Currently: attempt to attack a -bit Koblitz elliptic curve.
Key length comparison between RSA and ECC
     
Security (bits level)
RSA
ECC
     
     
Pybuil-ceksysyetmsilbuPryyckec-phraogpt//(Nancy)nacdyrtpgoarhpyThA)afetctunnsio(trearaC,lemIROLDnRobamie
ographyDkeycryptPbuil-capgrhycrintoypragomhtircsileteyrcdnasnhpargotpetThA)RIioctunaf(taCbore,lOLaremienRDam)y/
Discrete logarithm
Exponentiation:O(logp). DLP:O(√p)(in a generic group). ̃ G=Fp: sub-exponential attacks. Findsecuregroups withefficient law,compact representation.
Definition (DLP) LetG=be a cyclic group of prime order. LetxNandh=x.ediscrete logarithmlog(h)isx.
ab=(b)a=(a)b.
Protocol [Diffie–Hellman Key Exchange] Alice sendsaBob sendsb, the common key is ,
y//(Nanc
hyapgrtoypcrnd)ycnaN(//
e(,)ab c=e(b,c)a=e(c,a)b=e(a,b)cG.
Identity-based cryptography [BF]. Short signature [BLS]. One way tripartite Diffie–Hellman [Jou]. Self-blindable credential certificates [Ver]. Attribute based cryptography [SW]. Broadcast encryption [Goy+].
Definition Apairingis a bilinear applicationeG×GG.
Tripartite Diffie–Helman
Alice sendsa, Bob sendsb, Charlie sendsc.e common key is
Pairing-based cryptography
/neimeboRaDel,LORIArt(CaramcnitnoasT)ehatufrgotpyrcnimhtirahyapublic-keycryptogarhpDysirctelegoP
ieamobnRt(erraCaDesleaibAeiitvnra
Perspectives
Outline
eta functions
Abelian varieties
Isogenies
Pairings
//phyogra
Public-key cryptography
/cn)y(aNafetThA)RILOl,metpyrcdnasnoitcnu
UseG=A(k)withk=Fqfor the DLP. Pairing-based cryptography with theWeilorTatepairing. (Only available on abelian varieties.)
(aNcn)y/
Abelian varieties
Abelian variety =pointson a projective space (locus of homogeneous polynomials) + an abelian group law given byrational functions.
Definition AnAbelian varietyis a complete connected group variety over a base fieldk.
raogyph//tcnusnoicdnatpyrraCat(erobnRieamfatehT)AIROL,lemDsruevebAiravnailacsJieetfcnoiaob
ibnafoucitseaJocianvarieAbelesrvrt(CRobemienDaT)ehROAIleL,ramacrndsaontincfuta//yhpargotpycy)/(Nan
Elliptic curves
P+Q=R=(xR,yR) λ=yQyP xQxP xR=λxPxQ yR=yP+λ(xRxP)
Definition (cark, ) Ey=x+ax+b. a+b. An elliptic curve is a plane curve of genus . Elliptic curves = Abelian varieties of dimension .
Dam
Mumford coordinates:D=(u,v)u=(xxi),v(xi)=yi. Cantor algorithm:addition law.
k D=(PiP)k, symmetricPiPj i=
Divisor: formal sumD=niPi,PiC(k). degD=ni.
y)ncNa
Jacobian ofC= Divisors of degree  modulo principal divisors = Abelian variety of dimension. Divisor classDuniquerepresentative (Riemann–Roch):
Principal divisor:PC(k)vP(f).P;fk(C).
Cy=f(x), hyperelliptic curve of genus. (degf=)
Jacobian of hyperelliptic curves
/arhpy//(ctionsandcryptogOL,l)AIRtehTnufanRieerobCat(meravnaieiralebAvrsebianofcutiesJaco
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