Topics in Algorithms and Complexity Theory: Elliptic Curve and Pairing-based Cryptography

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Nov 21, 2013 (3 years and 9 months ago)

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Syllabus
--

CSE 209
B

--

UCSD
--

Winter quarter 2005

Kristin Lauter, Microsoft Research, Cryptography and Anti
-
Piracy


Topics in Algorithms and Complexity Theory:

Elliptic Curve and Pairing
-
based Cryptography


1.

Introduction to Elliptic Curves (1
-
2 weeks)


2.

E
lliptic
C
urve
C
ryptography: ECDH, ECDSA, EC El
Gamal

(1 week)



3.

Security questions: security
proofs of protocols

(1 week)


4.

Algorithmic number theory questions related to ECC
:

(3
-
4

weeks)


(*)
a. Discrete
log attacks (Pollard Rho, BSGS
,

MOV,

Weil desc
ent, index calculus
)


(*)
b. CDH



c.

DDH (pairing solution)


(*)
d. point
-
counting algorithms


(Schoof including d
ivision polynomials and modular
polynomials, AGM, …)


(*)

e. complex multiplication methods for generating ellip
tic curves


(*)


f. efficient group law implementations

(special “Koblitz” curves, general techniques
, fast exponentiation, …)



5.

Pairing
-
based cryptosystems

(protocols covered in CSE 208)

(2
-
4 weeks)

a.

Weil and Tate pairings: definitions and


b.

(*) Efficient
implementations

c.

DDH solution in EC
-
groups
/MOV attack above

d.

(*)
BDH assumption

e.

(*) Generation of suitable/special curves (MNT curves, …)

f.

(*)
Additional pairing systems




6. (*) Generalization of IV to Jacobians of higher genus c
urves (if time permits)


(*) indicates active area of resea
r
ch with interesting research problems


Text: Blake, Seroussi, Smart.
Elliptic Curves in Cryptography
, London Mathematical
Society Lecture Note Series, volume 265, Cambridge University Press, 2000.