Pictures and slides | Announcements | Literature | Course | Homework |
The first part will be taught by
Tanja Lange
Coding Theory and Cryptology
Eindhoven Institute for the Protection of Information
Department of Mathematics and Computer Science
Room HG 9.92
Technische Universiteit Eindhoven
P.O. Box 513
5600 MB Eindhoven
Netherlands
The easiest ways to reach me where ever I am:
e-mail:tanja@hyperelliptic.org
The second part will be taught by Alp Bassa and Ronald Cramer
This is an overview of the topics taught in the master math course
Algebraic Geometry in Cryptology. Details are filled in as time
permits. The official home page is here.
The
course takes place Tuesdays at the University of Utrecht in
Minnaertbuilding room 208, except for March 16 and April 27 when it is
in Buys Ballot Lab, room 205.
The first part consists of 7 weeks of
lectures, the dates are Feb 09, Feb 16, Feb 23, Mar 02, Mar 09, Mar
16, and Mar 23. Each class takes from 14:00 - 16:45. These 3 hours are
lectures with integrated exercises, so there are no separate exercise
sessions.
Description
Cryptology deals with mathematical techniques for design and analysis
of algorithms and protocols for digital security in the presence of
malicious adversaries. For example, encryption and digital signatures
are used to construct private and authentic communication channels,
which are instrumental to secure Internet transactions.
Some of the most fascinating advances rely on algebraic geometry:
Elliptic curves are becoming the new standard for public key
primitives and research in elliptic curve cryptography received a
recent boost with cryptographic pairings; hyperelliptic curves offer
interesting performance for hardware implementations; Goppa codes for
curves of genus 0 are used to construct secure code-based
cryptosystems; curves over finite fields also find applications in
secret sharing schemes and towers of algebraic function fields have an
important bearing on strongly multiplicative secret haring schemes
with good asymptotic properties. These schemes are highly relevant, as
they are the basis for information-theoretically secure multi-party
computation. Depending on time we will also highlight the use of
towers of function fields for constructing codes with particularly
good error correction properties.
This course provides an introduction to cryptography and to algebraic
geometry at a master course level. The main focus is on the diverse
applications that algebraic geometry has in cryptology.
The room has changed to Minnaertgebouw 208 (Zonnezaal); for exceptions see above.
There are a few books covering topics we touch in class but there is no 'textbook' that covers everything. The following list will be updated as the course proceeds.
The grade will be based on homework. During my 7 weeks you will receive 2 homework sheets.
09 Feb 2010
General introduction to cryptography; concepts public key and
symmetric key cryptography, Diffie-Hellman key exchange over the
rationals and modulo primes. The clock group and how to compute there
efficiently. Curve equation of Edwards curves.
I took pictures of all black boards before erasing them. They are here.
16 Feb 2010
Edwards curves; proof that denominators cannot vanish,
points of finite order; generalization of curve equation to
ax^{2}+y^{2}=1+dx^{2}y^{2}.
Attention, this model is called twisted Edwards curve (I
left out the 'twisted' in the definition); ElGamal encryption;
number of points;
general musings about how to do things in practice.
I took pictures of all black boards before erasing them. They are here.
23 Feb 2010
Affine and projective space, Zariski topology, connections between
affine and projective space, varieties, Jacobi criterion.
Some reading material is posted under literature (chapter 4 of HEHCC and
preprint with Frey).
I took pictures of all black boards before erasing them. They are here.
02 Mar 2010
Varieties, irreducible vs. absolutely irreducible, function field of
variety, field of constants, morphisms, Frobenius endomorphism,
isomorphisms, rational functions, birational equivalences, Montgomery
curves.
I took pictures of all black boards before erasing them. They are here.
09 Mar 2010
Model solutions of the homework; blow-ups of singularities; group of
divisors, degree of a divisor, effective divisors.
I took pictures of all black boards before erasing them. They are here.
16 Mar 2010
Principal divisors, Riemann-Roch theorem,
genus, Weierstrass form elliptic curves, divisor class group,
operations in divisor class group of elliptic curve and representation
of the classes. The slides showing the geometric interpretation of the
addition on Edwards curves are here;
the pictures start page 12. There is also a preprint giving more of the
background.
I took pictures of all black boards before erasing them. They are here.
23 Mar 2010
Hyperelliptic curves, details on imaginary hyperelliptic curves (curve
equation, involution, representation of divisor classes), pairings
(definition, protocols, Tate pairing)
The slides I used for the pairings description are
here;
some newer slides are here
I took pictures of all black boards before erasing them. They are here.