Algebra, Number theory and their applications to Cryptography
Salahaddin University, Erbil, Kurdistan Region, Iraq
25/02/2024 - 7/03/2024


The school will offer comprehensive instruction on contemporary cryptographic concepts rooted in number theory, algebra, and algebraic geometry. The curriculum will cover fundamental aspects of elementary and algebraic number theory, as well as advanced topics such as elliptic cryptosystems, and computational challenges relevant to cryptography. The lectures will delve into the application of these mathematical principles in cryptographic protocols and systems.The school aims to foster and advance new research areas specifically tailored for advanced undergraduate students from Salahaddin University-Erbil and other universities in the Kurdistan Regional Government (KRG), Iraq, and neighboring countries. The program will consist of six courses, span over 10 days of classes, for a total of 40 hours of classes and 14 hours of training sessions.



Courses

Grobner basis and application, Rashid Zaare Nahandi (IASBS, Zanjan)
  1. Commutative rings, Unique factorization domains, Ring of polynomials
  2. Ideals, operations with ideals
  3. Monomial ideals and Dickson's lemma, monomial orders
  4. Buchberger criteria and Groebner basis
  5. Algorithms for computing Groebner basis
  6. First applications
  7. Solving a system of polynomial equations
  8. Systems over finite fields and cryptography


Introduction to Elliptic curves, Peter Stevenhagen (Universiteit Leiden) and Valerio Talamanca (Universitą Roma Tre)
  1. Historic introduction. Elliptic curves over the complex numbers: lattices, elliptic functions, Weierstrass p-fuction
    Generalities: Rational point on conics and their parametrization, rational points on cubic and their group structure. The projective plane. Cubic curves in the projective plane. Weierstrass equation of an elliptic curve . The group structure on rational points, formulas for addition and duplication. The invariant j, elliptic curves in characteristic 2, Endomorphisms: degree of an endomorphism, separable endomorphisms, the Frobenius endomorphism.
  2. Torsion Points: Torsion points, division polynomials. Weil pairing
  3. Elliptic curves over finite fields: Frobenius endomorphism and the Hasse bound. The problem of determining the order of the group. Curves on subfields, Symbols of Legendre, Orders of points,
  4. Nagell-Lutz theorem if time permits
Lecture notes on complex elliptic curves
Exercises 1


Elliptic curves Cryptography, Amos Turchet (Universitą Roma Tre)
Cryptosystems of Massey Omura and El Gamal. Signature scheme of El Gamal. Elliptic curve cryptosystems based on the factorization problem. Elliptic Curve Integrated Encryption Scheme of Bellare and Rogaway. Factoring whole numbers using elliptic curves. A Cryptosystem Based on the Weil Pairing. Pairing-friendly elliptic curves. Boneh-Franklin identity-based encryption. Boneh-Lynn-Shacham signatures. Boneh-Goh-Nissim homomorphic encryption.


Introduction to Cryptography, Francesco Pappalardi (Universitą Roma Tre),
Valerio Talamanca (Universitą Roma Tre), Lea Terracini, (Universitą di Torino) e Amos Turchet (Universitą Roma Tre)
  1. Basic theory of cryptography: complexity theory and security definitions, key exchange, signature and hash functions.
  2. Factorization based protocols: RSA, Rabin Cryptosystem
  3. Discrete Logarithm based Cryptosystems: Diffie Hallman Key Exchange protocol on cyclic groups. Massey Omura. ElGamal
  4. Lattice reduction cryptosystems
RSA DLP


Number Theory, Lea Terracini, (Universitą di Torino)
This course aims to introduce the basic notions concerning the theory of integer numbers, modular arithmetic and finite fields. 
  1. Natural and integer numbers. Divisibility. Prime numbers. Greatest common divisor, Euclidean algorithm and Bézout identity. Some results on the distribution of primes.
  2. Congruences. Linear congruences and Chinese Remainder Theorem. Euler function, Euler- Fermat Theorem.  Primitive roots, quadratic residues and quadratic reciprocity.  
  3. Finite fields. Algebraic extensions and splitting fields. Characteristic of a field. Existence and uniqueness of a finite field of order q for every prime power q. Algebraic closure of a finite field. Frobenius automorphism. Cyclotomic classes and factorization of a cyclotomic polynomial over a finite field.
lecture 1
lecture 2
lecture 3
lecture 4
lecture 5
lecture 5
lecture 6
Exercises 1
Exercises 2


Elementary Algorithmic Number Theory, Laura Geatti (Universitą di Roma Tor Vergata)
In this course we present some algorithms for primality testing, factoring integers and solving the discrete logarithm problem, based on elementary number theory.
For more info consult the web page of the course
  1. Complexity of an algorithm: polynomial, exponential and subexponential algorithms. Examples: the extended Euclid algorithm, the calculations of powers by successive squarings.
  2. Primality tests: Miller-Rabin test. Construction of large pseudoprimes. Factoring algorithms: trial division, Pollard ρ-1, Pollard ρ .
  3. Discrete logarithm problem: Pollard ρ, Baby Step Giant Step, Pohlig-Hellman method, index calculus


Application form



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Schedule

Number theory Number theory
Lea Terracini		 Algorithms Elementary algorithmic number theory
Laura Geatti		Cryptography Introduction to Cryptography
Francesco Pappalardi, Valerio Talamanca, Lea Terracini and Amos Turchet
Elliptic curves Introduction to Elliptic Curves
Peter Stevenhagen and Valerio Talamanca 		Elliptic Cryptography Elliptic curves Cryptography
Amos Turchet		 Grobner basis Grobner basis and applications
Rashid Zaare Nahandi


Week 1
SundayMondayTuesdayWednesdayThursday
9:00-9:50 Registration/openinig ceremony Cryptography (F) Grobner basis Elliptic curves Cryptography (A)
10:00-10:50 Number theory Cryptography (F) Grobner basis Elliptic curves Grobner basis
11:00-11:30COFFEE BREAK
11:30-12:20 Number theory Number theory Cryptography (L) Grobner basis Grobner basis
12:30-14:30LUNCH
14:30-15:20 Algorithms Elliptic curves Algorithms Cryptography (A) Elliptic curves
15:30-16:20 Algorithms Elliptic curves Elliptic curves Number Theory Elliptic curves
16:30-17:20Elliptic curves


Week 2
SundayMondayTuesdayWednesdayThursday
9:00-9:50Cryptography (V) Elliptic Cryptography Number Theory Algorithms Number Theory
10:00-10:50 Cryptography (V) Elliptic Cryptography Algorithms Algorithms Elliptic Cryptography
11:00-11:30COFFEE BREAK
11:30-12:20 Grobner basis Algorithms Cryptography Number Theory Elliptic Cryptography
12:30-14:30LUNCH
14:30-15:20 Number Theory Grobner basis Elliptic Cryptography Elliptic Cryptography
15:30-16:20 Algorithms Grobner basis Elliptic Cryptography Elliptic Cryptography


In collaboration and generously funded by



Scientific Committee

Herish Omer Abdullah (Salahaddin University-Erbil)
Laura Capuano (Universitą Roma Tre)
Laura Geatti (Universitą di Roma Tor Vergata)
Andam Ali Mustafa (Salahaddin University-Erbil)
Lea Terracini (Universitą di Torino)
Valerio Talamanca (Universitą Roma Tre)

Organizing Committee

Fuad Wahid Khdhr (Salahaddin University-Erbil)
Khwazbeen Saida Fatah (Salahaddin University-Erbil)
Rashad Rashid Haji (Salahaddin University-Erbil)
Sami Ali Hussein (Salahaddin University-Erbil)
Neshtiman N. Sulaiman (Salahaddin University-Erbil)
Gashaw A. Mohammed Saleh (Salahaddin University-Erbil)
Hardi Ali Shareef (University of Sulaimani)
Haval Mohammed Salih (Soran University)
Shadman Rahman Kareem (Koya University)
Andam Ali Mustafa (Salahaddin University-Erbil)
Hero Waisi Salih (Salahaddin University-Erbil)
Shadan Abdulkadr Othman (Salahaddin University-Erbil)