Arithmetic on abelian varieties and related topics

67 Pages·2014·0.58 MB·English

by  Damien Robert

Arithmetic on abelian varieties and related topics
2014/03/03—Neuchâtel
DamienR󰀻󰀮󰀱󰀾󰁀
ÉquipeLFANT,InriaBordeauxSud-Ouest
February27,2014
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Discrete logarithm
Definition(DLP)
LetG =〈g〉beacyclicgroupofprimeorder.Letx(cid:0)(cid:78)andh=gx.The
discretelogarithmlog (h)isx.
g
(cid:112)
Exponentiation:O(logp).DLP:O(cid:101)( p)(inagenericgroup).Sowecan
usetheDLPforpublickeycryptography.
⇒ Wewanttofindsecuregroupswithefficientadditionlawandcompact
representation.
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Elliptic curves
Definition(chark(cid:54)=2,3)
Anellipticcurveisaplanecurvewithequation
y2=x3+ax+b 4a3+27b2(cid:54)=0.
2
R
1 Exponentiation:
Q
((cid:96),P)(cid:55)→(cid:96)P
P 0
-1.5 -1 -0.5 0 0.5 1 1.5 2
Discretelogarithm:
-1 (P,(cid:96)P)(cid:55)→(cid:96)
-R
-2
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Scalar multiplication on an elliptic curve
3
2
1
P
0 2P
-2 -1 0 1 2 3
-2P
-1
-2
-3
-4
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Scalar multiplication on an elliptic curve
3
2
1
-3P P
0 2P
-2 -1 0 1 2 3
-2P
3P -1
-2
-3
-4
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Scalar multiplication on an elliptic curve
3
2
1
-3P P
0 2P
-2 -1 0 -5P 1 2 3
5P
-2P
3P -1
-2
-3
-4
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
ECC (Elliptic curve cryptography)
Example(NIST-p-256)
E ellipticcurvey2=x3−3x+
41058363725152142129326129780047268409114441015993725554835256314039467401291over
(cid:70)
115792089210356248762697446949407573530086143415290314195533631308867097853951
Publickey:
P=(48439561293906451759052585252797914202762949526041747995844080717082404635286,
36134250956749795798585127919587881956611106672985015071877198253568414405109),
Q=(76028141830806192577282777898750452406210805147329580134802140726480409897389,
85583728422624684878257214555223946135008937421540868848199576276874939903729)
Privatekey:(cid:96)suchthatQ=(cid:96)P.
UsedbytheNSA;
UsedinEuropeansbiometricpassports.
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Pairing-based cryptography
Definition
Apairingisabilinearapplicatione :G ×G →G .
1 1 2
Example
Ifthepairinge canbecomputedeasily,thedifficultyoftheDLPinG
1
reducestothedifficultyoftheDLPinG .
2
⇒ MOVattacksonsupersingularellipticcurves.
OnewaytripartiteDiffie–Hellman[Jou00].
Identity-basedcryptography[BF03].
Shortsignature[BLS04].
Self-blindablecredentialcertificates[Ver01].
Attributebasedcryptography[SW05].
Broadcastencryption[GPS+06].
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Jacobian of curves
C asmoothirreducibleprojectivecurveofgenusg.
Divisor:formalsumD=(cid:80)n P, P (cid:0)C(k).
i i i
degD=(cid:80)n .
i
Principaldivisor:(cid:80) v (f).P; f (cid:0)k(C).
P(cid:0)C(k) P
JacobianofC =Divisorsofdegree0moduloprincipaldivisors
+Galoisaction
=Abelianvarietyofdimensiong.
DivisorclassofadivisorD(cid:0)Jac(C)isgenericallyrepresentedbyasum
ofg points.
Cryptography CurvesandJacobians Abelianvarieties Arithmetic Pairings Isogenies Isogenygraphs
Example of Jacobians
Dimension2:AdditionlawontheJacobianofanhyperellipticcurveof
genus2:
y2=f(x),degf =5.
D=P1+P2−2∞
D′=Q1+Q2−2∞
b Q2
Q1
b
b b
b P2
b
b
P1b