A joint CIMPA-ICTP research school on
Lattices and applications to cryptography and coding theory
Saigon University, Ho Chi Minh City, Vietnam.
August 1st - 12th, 2016



Introductory Classes

Introduction to lattices,
René Schoof
  • Scalar product, Euclidean vector space, Gram-Schmidt orthonormalization;
  • Lattices, Gram matrix, isometries, sphere packing, Kissing number, Hermite constant.
Introduction to complex analysis,
Francesco Pappalardi

  • Complex numbers,series, exponential, sinus and cosinus of comple numbers;
  • Complex differentiation, holomorphic functions and the Cauchy Riemann Equations;
  • Complex integration, Cauchy theorem and Cauchy integral formula;
  • Taylor series, Laurent series and residue theorem.
Introduction to algebraic number theory,
Peter Stevenhagen

  • Unique factorization in $\mathbb{Z}$ and $\mathbb{Z}[i]$;
  • Uniquue ideal factorization in rings of integers: units, class groups;
  • Finiteness results from embeddings in Euclidean spaces and Minkowski's theorem;
  • Explicit computations.


Lattices and geometry of numbers,
Michel Waldschmidt

Lecture 1 :
  • Subgroups of $\mathbb{R}^n$: discrete, closed, dense;
  • Topological groups;
  • Lattices;
  • Fundamental parallelepiped, volume;
  • Packing, covering, tiling;
  • Subgroup of $\operatorname{Hom}_{\mathbb{R}}(\mathbb{R}^n, \mathbb{R})$ associated with a subgroup of $\mathbb{R}^n$.
Lecture 2:
  • Convex Sets, Star Bodies and Distance Functions;
  • Minkowski's convex body theorem;
  • Minkowski's theorems on linear forms;
  • Gauge functions;
  • Minkowski's theorem on successive minima.
Lecture 3
  • Minima of quadratic forms;
  • Sum of two squares;
  • Sum of four squares;
  • Primes of the form $x^2 + ny^2$;
  • Discriminant of a number field;
  • Units of a number field: Dirchlet's s theorem;
  • Geometry of numbers and transcendence.
Coding theory,
Dung Duong
  • Hamming distance;

  • Linear codes;

  • Cyclic codes;

  • Quadratic residue codes;

  • dual codes;

  • MacWilliams identities;

  • Perfect codes.
Lattices and cryptography,
Phong Q. Nguyen

1. Basic Algorithms:
  • Gram-Schmidt Orthogonalization and Computing a Basis. Notes.
2. Finding Short Lattice Vectors:
  • Hermite's inequality and the LLL algorithm;
  • Mordell's inequality and Blockwise algorithms;
  • Minkowski's inequality, worst-case to average-case reductions and sieve algorithms;
  • Lattice enumeration.
3. Cryptography from lattices:
  • SIS and LWE;
  • One-way Functions from Lattices;
  • Lattice-based Key Exchange and Public-Key Encryption;
Lattices in number theory,
Ha Tran

1. The Arakelov class group: Notes
  • Arakelov divisors (degree, covolume,...);
  • Ideal lattices;
  • The Arakelov class group and its structure;
  • Some examples.
2. Reduced Arakelov divisors: Notes
  • Metric on the Arakelov class group;
  • Definition and examples (in quadratic fields) of Reduced Arakelov divisors;
  • Properties of Reduced Arakelov divisors.
3. Computations with reduced Arakelov divisors:Notes
  • Reduction algorithm;
  • Composition algorithm and inversion algorithm;
  • Scan algorithm.
Lattices and Mordell-Weil groups,
Francesco Pappalardi and Peter Stevenhagen

1. Elliptic curves over number fields:
  • Projective coordinates and projective space, homogeneous equations, singular points;
  • Weierstrass equation, sum of points, duplication of points, points at infinity, explicit formulas for operations;
  • The group of rational points over a field, points of order two and points of order three;
  • Points of finite order and endomorphisms of elliptic curves.
2. Three proofs of the associativity of the group law:
  • Computer assisted proof;
  • Proof via the Pappus theorem
  • Proof via Picard group
3. Mordell-Weil lattices
  • The weak Mordell-Weil theorem $+$ descent $\Rightarrow$ the Mordell-Weil theorem
  • heights on projective spaces
  • Neron-Tate height
  • Mordell-Weil lattices
Lattices and Lie algebras,
Laura Geatti

  • Abstract root systems;
  • Existence of a base;
  • The lattice associated to an abstract root system (root lattice);
  • The classification of root systems;
  • Examples of root lattices.
Lecture 2.
  • Complex semisimple Lie algebras;
  • Cartan subalgebras and root decomposition;
  • The Lie algebraa $\mathfrak{sl}_2$ and its representations.
Lecture 3.
  • The root system of a semisimple Lie algebra
  • Cartan matrix and Dynkin diagram.
  • The weight lattice.


  • J. E. Humphreys, Introduction to Lie algebras and Representation theory, Graduate Texts in Mathematics, Springer Verlag, New York, 1972
  • J.-P. Serre, Complex semisimple Lie algebras, Springer Verlag, New York, 1966
  • Course notes.
Lattices and modular forms,
Valerio Talamanca

Lecture 1
  • Lattices in the complex plane and elliptic curves;
  • Action of $\operatorname{SL}_2(\mathbb{Z})$ on the set of lattices in the complex plane;
  • Moduli space for complex elliptic curves.
Lecture 2
  • Classsical theta functions and their properties;
  • Modular forms for $\operatorname{SL}_2(\mathbb{Z})$, cusp forms. The space $M_k$ of weight 2k modular forms;
  • Invariants of elliptic curves as modular functions;
  • Eisenstein series and the explicit determination of a basis of $M_k$.
Lecture 3
  • Theta series of lattices;
  • Even Unimodular lattices;
  • The theta series of even unimodular lattices is a modular form;
  • Explicit examples.

  • References
    J.-P. Serre, A Course in Arithmetic, Springer Verlag, New York, 1973 (Chapter VII)


Poisson summation,
René Schoof

  • Dual lattices;

  • Fourier analysis;

  • Poisson summation formula.
Elliptic curves over ${\mathbb C}$,
Peter Stevenhagen

  • calculus for algebraic integrals: Riemann surfaces;

  • elliptic functions, Weierstrass parametrization of elliptic curves;

  • Eisenstein series, modular functions;

  • Endomorphism rings, CM elliptic curves.
Notes on complex elliptic curves
The Leech lattice,
René Schoof

  • Construction and properties of the Leech lattice.

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