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Algebra, Codes, Crypto, Quantum (ACCQ)

The ACCQ study track gives an introduction to several domains of computer science and telecommunications:  symbolic computation, coding theory, cryptography, and quantum information theory, which all rely to a certain extent on a common algebra based mathematical background.
These domains will be studied mostly from a theoretical point of view which, together with the MITRO track, might constitute a first step toward a research career

Students can also choose courses from related but more practical and applied  study tracks, such as SOCOM, RES, and SR2I as a complement. Such a choice would help them prepare for careers in communication systems, networks, or security engineering.
Students are expected to have previously acquired a high level in mathematics, even if no official prerequisite is needed.

2nd year courses

ACCQ 2nd year program (192 hours)  Fall semester Spring semester
Period 1 Period 2 Period 3 Period 4
Time slot C1 ACCQ201 Finite Algebraic Structures ACCQ203 Computational Algebra ACCQ203 Computational Algebra ACCQ206 Introduction to Quantum Information and Computation
Time slot C2 ACCQ202 Information Theory ACCQ204 Error Correcting Codes ACCQ205 Introduction to Algebraic Curves ACCQ207 Selected Topics in Mathematical Cryptography

Details :

Fall semester, period 1

  • ACCQ 201 Finite Algebraic Structures (24 hrs) (Hugues Randriam)
    This course is an introduction to basic algebraic structures, with emphasis on finite aspects and a view toward applications in coding theory and cryptography. Topics covered will include: quotient structures, groups and group actions, cyclic groups, Euler's totient function, structure theorem for finite Abelian groups, modular arithmetic, Chinese remainder theorem, Euler's criterion, quadratic reciprocity, inite fields, primitive polynomials and primitive elements, norm and trace.
  • ACCQ 202 Information Theory (24 hrs) (Aslan Tchamkerten)
    In this course we present the main tools and results of Information Theory. More precisely, we first introduce entropy, divergence, and mutual information. Then we state and discuss the two Shannon theorems on source compression and transmission over a noisy channel. Last, more recent results and applications to other domains will be addressed depending on the time available.

Fall semester, period 2

  • ACCQ 204 Error Correcting Codes (48 hrs) (Bertrand Meyer)
    Coding theory is aesthetically pleasing from a mathematical point of view but it raises several practical issues. In this regard, Elwyn R. Berlekamp, one of the leading experts in the domain, asks the fundamental questions: How good are the best codes? How can we design good codes? How can we decode them? This course presents the basics and the state of the art on these three problems, which are still of current interest.. We will insist on algebraic constructions and hard decoding algorithms, in contrast with soft decoding algorithms that will be presented in more advanced courses.

Fall & spring semesters, periods 2 & 3

  • ACCQ 203 Computational Algebra (24 heures) (Ghaya Rekaya)
    This class reviews, from an algorithmic perspective,  some of the basic notions of algebra and number theory that are useful for their applications in telecommunication and computer science :Modules over rings (their stucture, Hermite and Smith reduction of matrices, invariant factors, linear recurrence sequence - LFSR), Euclidean lattices (connections with classical arithmetic and coding theoretic problems, LLL algorithm), Factorisation of polynomials, Primality testing, factorisation of integers, and discrete log, Manipulation of polynomial equations (Groebner bases), Introduction to various algebraic complexity questions (products of polynomials and of matrices).

Spring semester, period 3

  • ACCQ 205 Introduction to Algebraic Curves (24 hrs) (David Madore)
    This course is an introduction to algebraic geometry in dimension 1 (algebraic curves): notions covered will include that of the function field of a curve, valuations, morphisms between curves, ramification, and divisors and differentials on curves. We will state and discuss the Riemann-Roch theorem and the notion of genus of a curve. The case of elliptic and hyperelliptic curves will be highlighted. Emphasis will be put on computational aspects, and applications to cryptography and to the construction of algebraic codes.

Spring semester, period 4

  • ACCQ 206 Introduction to Quantum Information and Computation (24 hrs) (Romain Alléaume)
    This course aims at introducing students to radically new information technologies based on the principles of quantum mechanics. This includes quantum communication for channel security, storage and processing of quantum information, quantum error correcting codes, and quantum computation, leading to algorithms which are much more powerful than existing classical algorithms. At the end of the course, the student will master the tools and basic principles of quantum communication and computation, which will then be covered more extensively in the Quantum-Safe Cryptography program during the 3rd year (1st semester) .
  • ACCQ 207 Selected Topics in Mathematical Cryptography (24 hrs) (Hugues Randriam)
    This course gives an introduction to some of the most mathematical aspects of cryptography: elliptic curves in cryptography, lattice based cryptography, links between coding theory and cryptography. Other topics such as pseudo-random generators, pairings, information theoretic security, etc. may  be addressed if time permits.

3rd year options

The students can choose

-       To stay  on the school premises (courses and a 6-month internship) :

  • Quantum Safe cryptography program (advisor: Romain Alleaume)
  • Digital Communications program (advisor: Philippe Ciblat)

-       Or to apply for one of the following two Master of Science programs (M2) :

  • M2  AFP (ex MPRI),  Algorithmics and Foundations of Programming (University of Paris Saclay), see AFP  program
  • M2  STN , Systèmes de Télécommunications numériques (UPMC), see STN  program