Final report on CIMPA-ICTP school on Lattices and applications to cryptography and coding theory at Saigon University, Ho Chi Minh, Vietnam August 01–12, 2016 Summary The CIMPA-ICTP research school “Lattices and applications to cryptography and coding theory" was organized by Saigon University (SGU) and University of Science (HCMUS, VNU-HCMC) at Ho Chi Minh city, Vietnam. It was held at Saigon University during August 01–12, 2016. The website of the school with updated information is located at http://ricerca.mat.uniroma3.it/users/valerio/hochiminh16.html The classes were given by nine lecturers including two Vietnamese lecturers (one from Finland and one from Japan), two from France, three from Italy and two from The Netherlands; there were two female lecturers which is 22.2% of total lecturers. There were 104 participants from 22 countries including 54 from Vietnam (3 from Dong Nai, 3 from Hue, 10 from Hanoi, 1 from Thai Nguyen and 37 from Ho Chi Minh) and 50 from other countries: 8 from Indonesia, 9 from Philippines, 1 from Russia, 1 from Turkey, 4 from China, 2 from Pakistan, 1 from Cambodia, 2 from Malaysia, 3 from India, 2 from Mongolia, 1 from Srilanka, 6 from Thailand, 2 from Italy, 1 from Spain, 1 from Canada, 1 from The United States, 1 from Brazil, 1 from South Korea, 1 from Israel, 1 from Egypt and 1 from Finland. Among participants, there were 66 males (32 from Vietnam and 34 from other countries) and 38 females (22 from Vietnam and 16 from other countries). Female participants were 36.2% of total participants. The list of participants is included. The school obtained generous support from CIMPA, ICTP, HCMUS, SGU, International Mathematical Union (IMU), Number Theory Foundation (NTF), the Italian Department of foreign affairs, G.N.S.A.G.A research group of Istituto Nazionale di Alta Matematica, Leiden Universiteit, Consulate General of France, Italy and The Netherlands in Ho Chi Minh city. The school started on August 01. The opening ceremony took place on the first morning with the participation and speeches of the Italian Ambassador in Vietnam and of representatives of the General Consul of France, Italy and The Netherlands in Ho Chi Minh city. The school ended on August 12 with a short closing ceremony and participants were given certificates of attendance.

Goal of the school The aim of this school was to introduce participants to the ubiquity of lattices in number theory, algebra, arithmetic algebraic geometry, cryptography and coding theory. To this goal, the theory of lattices was developed from its very beginning and the basic notions required for the applications in number theory, algebra, arithmetic algebraic geometry were provided. This was achieved by a very cohesive and well planned series of courses as well as a few introductory lectures on specific basic topics (see the detailed plan of lectures and courses for more details). The more advanced courses covered appearances of lattices tin various settings: the natural lattice structure of Mordell-Weil groups and unit groups, root lattices, the lattice basis reduction algorithm "LLL", modular functions associated to even unimodular lattices.

Organization Scientific Committee • Laura Geatti (Università di Roma Tor Vergata) • Phong Nguyen (INRIA and JFLI) • René Schoof (Università di Roma Tor Vergata) • Anton Mellit (SISSA-ICTP) • Peter Stevenhagen (Universiteit Leiden) • Valerio Talamanca (Università di Roma Tre)

Local Organizing Committee • Nguyen Dinh Thuc (University of Science, Vietnam National University) • Pham Hoang Quan (Saigon University) • Duong Hoang Dung (Kyushu University) • Tran Nguyen Thanh Ha (Aalto University)

ICTP local organizer • Fernando Rodriguez Villegas (ICTP)

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Lecturers • Duong Hoang Dung (Kyushu University) • Laura Geatti (Università di Roma Tor Vergata) • Phong Nguyen (INRIA and JFLI ) • Francesco Pappalardi (Università di Roma Tre) • Peter Stevenhagen (Universiteit van Leiden) • René Schoof (Università di Roma Tor Vergata) • Valerio Talamanca (Università di Roma Tre) • Tran Nguyen Thanh Ha (Aalto University) • Michel Waldschmidt(Université Pierre et Marie Curie)

School Program Introductory classes Introduction to lattices René Schoof Content: Scalar products, Euclidean vector spaces, Gram-Schmidt orthonormalization. Lattices, Gram matrix, isometries, sphere packings, Kissing number, Hermite constant. Dual lattices, Fourier analysis, Poisson summation formula. Construction and properties of the Leech lattice.

Introduction to Complex analysis Francesco Pappalardi Content: Complex numbers, series, exponential, sine and cosine of complex 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 Stevenheagen Content: Unique factorization in Z and Z[i ], Unique ideal factorization in rings of integers: units, class groups, Finiteness results from embeddings in Euclidean spaces and Minkowski’s theorem, Explicit computations.

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Courses Coding theory Duong Hoang Dung Content: Hamming distance, Linear codes; Cyclic codes; Quadratic residue codes; Dual codes; MacWilliams identities; Golay codes; Perfect codes.

Lattices and cryptography Phong Nguyen Content: Basic Algorithms: Gram-Schmidt Orthogonalization and Computing a Basis; 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; Cryptography from lattices: SIS and LWE, One-way Functions from Lattices, Lattice-based Key Exchange and Public-Key Encryption.

Lattices and Lie algebras Laura Geatti Content: 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; Complex semisimple Lie algebras, Cartan subalgebras and root decomposition, The Lie algebra sl2 and its representations; The root system of a semisimple Lie algebra, Cartan matrix and Dynkin diagram.

Lattices and modular forms Valerio Talamanca Content: Lattices in the complex plane and elliptic curves, action of SL2 (Z) on the set of lattices in the complex plane, Moduli space for complex elliptic curves; Classsical theta functions and their properties; Modular forms for SL2 (Z), cusp forms, The space Mk of weight 2k modular forms, Invariants of elliptic curves as modular functions, Eisenstein series and the explicit determination of a basis of Mk ; Theta series of lattices, Even Unimodular lattices, The theta series of even unimodular lattices is a modular form, Explicit examples.

Lattices in Number theory Ha Tran Content: The Arakelov class group: Arakelov divisors (degree, covolume,...), Ideal lattices, The Arakelov class group and its structure, Some examples; Reduced Arakelov divisors: Metric on the Arakelov class group, Definition and examples

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(in quadratic fields) of Reduced Arakelov divisors, Properties of Reduced Arakelov divisors; The reduction algorithm.

Lattices and Mordell-Weil groups Francesco Pappalardi and Peter Stevenhagen Content: 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; Three proofs of the associativity of the group law: Computer assisted proof, Proof via the Pappus theorem, Proof via Picard group; Mordell-Weil lattices: the Mordell-Weil theorem, heights on projective spaces, Neron-Tate height, Mordell-Weil lattices.

Lattices and geometry of numbers Michel Waldschmidt Content: Subgroups of Rn : discrete, closed, dense; Topological groups; Lattices; Fundamental parallelepiped, volume of a lattice; Packing, covering, tiling; Subgroup of HomR (Rn , R) associated with a subgroup of Rn . 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. Minima of quadratic forms; Sum of two squares; Sum of four squares; Primes of the form x2 + ny2 ; Discriminant of a number field; Units of a number field: Dirchlet’s s theorem; Geometry of numbers and transcendence.

Lectures Poisson summation René Schoof Content: Dual lattices, Fourier Analysis, Poisson summation formula. Examples and applications: a proof of Minkowski theorem using the Poisson summation formula.

Elliptic curves over C Peter Stevenhagen Content: Elliptic curves over C: calculus for algebraic integrals: Riemann surfaces, Elliptic functions, Weierstrass parametrization of elliptic curves, Eisenstein series, modular functions, Endomorphism rings, CM elliptic curves.

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The Leech Lattice by René Schoof Content: Construction of even unimodular lattices of low rank using binary Golay codes. Constuction of the Leech lattice.

Schedule The school started on Monday August 1st and ended on Friday August 12th at Saigon University. The opening ceremony took place on the first Monday and lasted one hour. Every class consisted in two periods of 50 minutes each. The two wednesday afternoons were left free of classes. Below you find the full schedule of classes.

Financial Report Sponsors We received generous support from the following sponsors:

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Organizations

Amount (in EUR)

Italian minister of foreign affairs GNSAGA Leiden University INRIA JSPS CIMPA (Centre International de ) Mathématiques Pures et Appliquées) ICTP (International Centre for Theoretical Physics) IMU (International Mathematical Union) NTF (Number Theory Foundation) 3000 USD HCMUS (University of Science) SGU (Saigon University) TOTAL

Expenses details

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2435 900 1234 1700 410 10,255 4849.64 2000 2689.5 2000 2000 30473.52

Sponsor Italian minister of foreign affairs of GNSAGA Leiden University INRIA JSPS ICTP

900 1234 1700 410 4849.64

IMU

2000

NTF

2689.5

CIMPA

Total 2435

10,255

SGU and HCMUS

4000

TOTAL

30473.52

Item 80% ticket for Geatti, Pappalardi and Valerio 90% ticket for Schoof Ticket for Stevenhagen Ticket and lodging for Phong Nguyen Ticket for Duong Hoang Dung Ticket of 26 participants Tickets for 13 Vietnamese Tickets for 5 international Hotel and lunch for 2 Vietnamese Lodging and lunch for 4 international Lodging and lunch for 1 Vietnamese Fruits, cakes for coffee breaks Opening ceremony Notebooks, pens, printing 10% ticket for Schoof Tickets for Ha and Waldschmidt 20% tickets for Geatti, Pappalardi and Talamanca Lodging for 28 CIMPA participants Lodging for 6 lecturers Lodging 13 Vietnamese Participant Lunches 28 CIMPA participants Lunches 9 lecturers Lunches 13 Vietnamese Participants Pocket money for 28 international Pocket money for 18 Vietnamese T-shirts for participants Banners Hotel and lunch for Andrea Local service (cleaning, pickup, hire assistant students)

Amount 2435 900 1234 1700 410 4849.64 1135 550 315 653 157.5 1200 350 329 100 1532.13 609 3003 1117 1394.25 1400 450 650 1400 900 900 100 140 560

Conclusion This CIMPA-ICTP school had several positive side effects: • This was the first CIMPA-ICTP academic school in the area of Mathematics, Cryptography and Coding Theory happening in Ho Chi Minh city. It was preceded by the preparatory SEAMS School last year, also in Ho Chi Minh city. It introduced students both in Mathematics and Computer Science to

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these fascinating research areas. It also gave Vietnamese and international participants a chance to make friends and work together in group discussions from which some future collaboration may occur. • The School has been an opportunity for Vietnamese mathematicians and computer scientists from Hanoi and Ho Chi Minh to have useful exchanges involving future programs of cooperation. • The School has attracted several young mathematicians working in the area of Algebra to Ho Chi Minh city. They all obtained their Ph.D’s from abroad, mostly from european countries. Since last year, together with the two vietnamese organizers (Dung H. Duong and Ha Tran) they started a study group on Cryptography. A weekly seminar is being held in HCMUS and there are several master students going to defend their thesis in this area by the end of this year. There are several students of this year that decided to do their master/bachelor thesis in this area. These are the first steps for building up a research group working in Cryptography in the near future. • A future IACR-SEAMS School "Cryptography: Foundations and New Directions" will be held at Vietnam Institute for Advanced Study in Mathematics (VIASM) in Hanoi in the period November 27–December 02, 2016. It is aimed as a good preparation for AsiaCrypt 2016 happening the week after in Hanoi. These events are very important for the the development of Mathematics and Cryptology research in Vietnam in the future. However, the school also highlighted some problems: • There are no experts working in the area of Mathematical Cryptography or Lattices residing in Vietnam at the moment. It is then very difficult for students to take appropriate courses, study or do research in those fields. • The background of the students is not balanced and it is not easy, especially for undergraduate students, to follow advanced courses and discuss with researchers. Hence it is necessary to have some preliminary courses/training for students before they take such a great research school. In conclusion, the School has been very successful and we hope this will enable many such events in the area of Mathematical Cryptography and Coding Theory in the region. We would like to thank all the supporters for their generous support and we hope they will continue to support us in the future for the development of mathematics in Vietnam and Southeast Asia.

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Some photos

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