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Courses
Algebraic Number Theory
(Fabien Pazuki and Valerio Talamanca)
This course offers a concrete introduction to algebraic number
theory, delving more specifically into the arithmetic of number fields. We wish the
course to have a strong emphasis on practical, computational examples. By the
end of this course, students will have gained a solid grasp of the basics of number
theory, which should enable them to follow the more advanced courses.
Detailed contents:
Basic definitions: Number fields, algebraic
integers, Dedekind domains, integral closure, ideals, invertibility for ideals. Some
review of Galois theory; Prime ideal factorisation, Kummer-Dedekind theorem,
ramification, inertia, residue fields.
Norm, trace, discriminant. Quadratic and
cyclotomic fields. Prime ideal decomposition in Galois extensions; Class group
and unit group of a number field. Minkowski and Dirichlet theorems; Quadratic
and number field sieve for integer factorisation (if time permits).
References :
D.A. Marcus, Number Fields.
J. S. Milne, Algebraic number theory, available at:
https://www.jmilne.org/math/CourseNotes
P. Stevenhagen, Number rings, available at: https://websites.math.leidenuniv.nl/algebra
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Elliptic Curves
(Francesco Campagna and Richard Griffon)
Elliptic curves are fundamental objects both in number theory and in
modern cryptography. The goal of this course is to give an introduction to the
theory of elliptic curves, focusing on those parts of the theory which are more
relevant for their concrete applications. Particular attention will be given to
effective results, and explicit and computational examples.
Detailed contents:
Elliptic curves as algebraic curves: Weierstrass equation, group law, isogenies,
Weil pairing, endomorphism rings;
Elliptic curves over number fields: Mordell-Weil theorem, canonical height, torsion structure, Mazur's theorem, rank,
regulator; Elliptic curves over finite fields: Frobenius endomorphism, group of
points, Hasse-Weil bound, ordinary and supersingular elliptic curves; Schoof's
algorithm for counting points and isogeny graphs.
References :
J. H. Silverman
and J. T. Tate, Rational points on elliptic curves,
J. H. Silverman, The arithmetic
of elliptic curves |
Lattices
(Michel Waldschmidt and Adeline Roux-Langlois)
The theory of lattices is deeply intertwined with all the courses
present in this summer school: lattices naturally appear in algebraic number
theory when studying the unit group of a number field, in the theory of elliptic
curves when studying the Mordell-Weil group of an elliptic curve over a number
field, in the theory of modular forms when studying theta functions, in
cryptography when dealing with lattice-based cryptosystems, and in coding
theory, in the theory of weight enumerators of codes. This course provides an
introduction to lattices, including a generous sample of applications.
Detailed contents:
Lattices: quadratic forms, determinants, duality,
sublattices, index module, Hadamard inequality. Even and unimodular lattices,
famous examples. Geometry of numbers: Convex bodies, successive minima,
Minkowski two theorems and applications; Shortest vector problem, the Hermite
constant, lattice basis reduction algorithms, LLL algorithm; Theta function of a
lattice, sphere packings, densities. Applications to coding theory and
cryptography.
References :
H. W. Lenstra, Lattices, available at:
https://www.math.leidenuniv.nl/~psh/ANTproc/filesheet.shtml
N. Elkies, lecture
notes for the 2019 Harvard university course Rational Lattices and their Theta
Functions . Available at:
https://people.math.harvard.edu/~elkies/M272.19/index.html
J. H. Conway and N.
J. A. Sloane, Sphere Packings, Lattices and Groups. |
Modular Forms
(Samuele Anni and Pee Chon Toh)
This course introduces the student to the basic theory of modular
forms, which are one of the unifying themes in modern number theory. Modular
forms are indeed used in studying various topics going from Diophantine
equations to analytic number theory questions.
Detailed contents
Introduction to the course and motivation. The action of the modular group SL2(Z)
on the upper half plane in C: its fundamental domain, cusps. Lattices, modular
functions; Modular forms and cusp forms. The Eisenstein series as examples of
modular forms: transformation law and convergence. The k/12 -formula for
modular functions. The graded algebra of modular forms: dimension and
generators of the spaces of forms of weight k. The Fourier expansion of the
modular forms Ek, especially E4 and E6. Arithmetic identities derived from the
Fourier coefficients. The discriminant function Δ and the Ramanujan τ function.
Hecke operators, application to the Ramanujan conjecture. Congruences for
Fourier coefficients and applications (if time permits).
References :
J.-P. Serre, A
course in arithmetic
F. Diamond and J. Shurman, A first course in modular
forms
J.H. Bruinier, G. van der Geer, G. Harder, D. Zagier, The 1-2-3 of Modular
Forms, Lectures at a Summer School in Nordfjordeid, Norway.
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Cryptography
(Any Muanalifah and Indah Emilia Wijayanti)
In the first part of the course we will describe the basic protocols
and algorithms used in modern cryptography, such as RSA and Diffie-Hellman key
exchange protocol. Along the way we will also discuss the importance of primality
testing and integer factorisation in cryptography, providing some examples of
algorithms that perform these tasks. In the second part of the course we will
discuss more specialised topics such as elliptic and lattice-based cryptography.
The precise choice of the topics will also depend on the interest of the audience.
Detailed contents
Finite fields and primitive roots. Primality and
pseudo primality tests. Agrawal-Kayal-Saxena's test. RSA method: first
description, attacks. Rabin's method and its connection with integer factorization.
Discrete logarithm methods. Symmetric methods (historical ones, DES, AES),
asymmetric methods and attacks. Key exchange, Key exchange in three steps,
secret splitting, secret sharing, secret broadcasting, timestamping. Signatures
with RSA and discrete log; Lattice based and elliptic curve based cryptography:
ideas and attacks.
References :
N. Koblitz A Course in Number Theory and
Cryptography
J.H. Silverman , J. Pipher , J. Hoffstein An Introduction to
Mathematical Cryptography
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Coding Theory
(Anna Maria Iezzi and Elisa Lorenzo Garcia)
The goal of coding theory is to develop systems and methods that
allow to detect and correct errors that occur when a piece of information is
transmitted through a noisy channel. This course will be an introduction to this
theory.
Detailed contents
Error-correcting Codes: repetition code,
parity check code, Hamming code. Hamming distance. Linear codes: generator
matrix, parity check matrix, dual code; Decoding and error probability: Basic
decoding, symmetric channel. Equivalent codes. Shannon Theorem. Weight
enumerator; Codes constructions and bounds: Punturing, restriction, extension,
short, augmentation, direct sum, juxtaposition, product and concatenation of
codes. Singleton, Griesmer, Plotkin, Hamming, Gilbert Varshamov and asymptotic
bounds; Cyclic codes: cyclic codes as ideals. Encoding cyclic codes. Parity check
polynomial. BCH bound. Examples; Algebraic geometry codes and applications:
Codes on curves, Goppa codes, Reid-Solomon codes. The McEliece cryptosystem;
Codes on graphs: Some graph theory. Cycle code and graph code of a graph.
References :
R. Pellikaan, X.-W. Wu, S. Bulygin and R. Jurrius, Codes, Cryptology
and Curves with Computer Algebra
W. Trappe and L. C. Washington, ntroduction to Cryptography with Coding Theory
W. Ebeling, Lattices and Codes
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Programming Session
Introduction to Sage
(Samuele Anni, Francesco Campagna, Elisa Lorenzo Garcia and Richard Griffon)
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