Mathematical Cryptography

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The course was developed within the framework of the project "ALMA - Advanced Learning Multimedia Alliance", a Digital Education Hub funded by the European Union – Next Generation EU, National Recovery and Resilience Plan (NRRP), Mission 4, Component 1, Investment 3.4 "Advanced university ...

Mathematical Cryptography

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Lingua

Inglese

Tipologia

Online

Modalità

Autoapprendimento

APERTURA

30 maggio 2026

ORE DI FORMAZIONE

Certificazione

Attestato

Accesso

Gratuito

THE COURSE IS NOW OPEN

Descrizione

The MOOC offers an overview of the theoretical/mathematical foundations necessary to allow for a critical study of cryptographic protocols. These protocols are currently used in many applications: authentication, digital commerce, digital signatures, etc.

Module Release Timeline:

May 30, 2026: Access to Modules 1, 2 & 3
June 5, 2026: Access to Modules 4, 5 & 6
June 12, 2026: Access to Modules 7, 8 & 9
TBD: Access to Module 10 and Module 11

Docenti

Carlo Mariconda

Laureato in Matematica presso l’Università di Padova nel 1987, ha conseguito il Ph.D. in Analisi Funzionale e Applicazioni presso la SISSA-ISAS di Trieste nel 1992. Professore ordinario di Analisi Matematica presso l’Università di Padova dove insegna a ingegneri e matematici. Il campo di ricerca scientifica è il Calcolo delle Variazioni.

Dal 2011 coordina la commissione Nuove Tecnologie per la Didattica del Dipartimento di Matematica “Tullio Levi-Civita”. Ha realizzato 8 Mooc su varie piattaforme nazionali (EduOpen, Federica.eu) e internazionali (Futurelearn). Il Mooc “Matematica di Base”, realizzato per gli esami di ammissione nelle università italiane tramite il TOLC del CISIA è seguito da più di 140.000 studenti.

Advisor per l’e-learning e la didattica innovativa per l’ateneo patavino dal 2016 al 2021, per il quale ricopre attualmente il ruolo di Advisor per la didattica digitale. Partecipante e co-organizzatore del progetto Teaching4Learning https://www.unipd.it/teaching4learning, il cui scopo è il miglioramento e la modernizzazione nella didattica.

Nel 2017 ha sviluppato, assieme al Prof. Alberto Tonolo, la BoardOnAir™, una lavagna di vetro innovativa che permette di realizzare video in autonomia, frontalmente agli studenti, inserendovi immagini, video o commenti a penna senza bisogno di postproduzione.

Nel 2024-25 ha partecipato alla stesura delle linee guida Unipd per la didattica blended e online e per l'utilizzo della intelligenza artificiale generativa nella didattica.

Course outline

Mathematical Cryptography

Platform guide: How to navigate the MOOC

Welcome to the Course

Getting to know you

Introduction to Module 1

Classical cryptography: monoalphabetic substitution ciphers

Modern cryptography: Polyalphabetic ciphers

Cipher Machines

Modern cryptography

Exercises of the module

Open discussion

Introduction to Module 2

Divisibility

Greatest Common Divisor

Extended Euclidean algorithm

Coprime numbers

The Linear Diophantine equation au+bv=c - The general case

Prime numbers

Exercises of the module

Open discussion

Introduction to Module 3

Classes modulo m

Invertibility modulo m

Linear congruences

The Chinese Remainder Theorem

Euler’s phi function

Exercises of the module

Open discussion

Introduction to module 4

The Fast Powering Algorithm

Fermat’s little Theorem

Fermat’s Little Theorem: Applications

Exercises of the module

Open discussion

Introduction to module 5

Order and primitive roots modulo a prime

Primitive roots: a test

Number of primitive roots

Exercises of the module

Open discussion

Introduction to module 6

Symmetric ciphers

Examples of symmetric ciphers

Asymmetric ciphers

Exercises of the module

Open discussion

Introduction to the Module 7

The Discrete Logarithm Problem (DLP)

Diffie-Hellman key exchange

Computational complexity

The “collision” Babystep-Giantsep algorithm

Pohlig-Hellman Algorithm

Complexity of the Pohlig-Hellman Algorithm

Exercises of the module

Open discussion

Introduction to Module 8

The ElGamal Public key

Euler’s Formula and Theorem

The Rivest - Shamir - Adleman (RSA)Public Key Cryptosystem

Security issues of RSA Public Key Cryptosystem Computing modular e-th roots

Exercises of the module

Open discussion

Introduction to Module 9

Public key Digital signatures

RSA type Digital Signature

Elgamal type Digital Signature

Exercises of the module

Open discussion

Introduction to Module 10

Prime number Theorem

Fermat (non) Primality test

Fermat’s alternative Theorem

Strong Fermat (non) Primality test

Exercises of the module

Open discussion

Introduction to Module 11

Trial Division

Factorisation via difference of squares

Strong witnesses and factorization

Pollard’s p-1 factorization algorithm

Exercises of the module

Open discussion

Go further

Course evaluation

Documents of the course

1.1 Classical cryptography: monoalphabetic substitution ciphers - Part 1

1.1 Classical cryptography: monoalphabetic substitution ciphers - Part 2

1.2 Modern cryptography: Polyalphabetic ciphers

1.3 Cipher Machines - Part 1

1.3 Cipher Machines - Part 2

1.4 Modern cryptography - Part 1

1.4 Modern cryptography - Part 2

2.1 Divisibility - Part 1

2.1 Divisibility - Part 2

2.2. Greatest Common Divisor

2.3. Extended Euclidean algorithm

2.4. Coprime numbers - Part 1

2.4. Coprime numbers - Part 2

2.5. The Linear Diophantine equation au+bv=c - The general case

2.6. Prime numbers - Part 1

2.6. Prime numbers - Part 2

3.1 Classes modulo m - Part 1

3.1 Classes modulo m - Part 2

3.2 Invertibility modulo m

3.3 Linear congruences

3.4 The Chinese Remainder Theorem - Part 1

3.4 The Chinese Remainder Theorem - Part 2

3.5 Euler’s phi function - Part 1

3.5 Euler’s phi function - Part 2

3.5 Euler’s phi function - Part 3

4.1 The Fast Powering Algorithm - Part 1

4.1 The Fast Powering Algorithm - Part 2

4.2 Fermat’s little Theorem

4.3 Fermat’s Little Theorem: Applications - Part 1

4.3 Fermat’s Little Theorem: Applications - Part 2

5.1 Order and primitive roots modulo a prime - Part 1

5.1 Order and primitive roots modulo a prime - Part 2

5.2 Primitive roots: a test

5.3 Number of primitive roots

6.1 Symmetric ciphers

6.2 Examples of symmetric ciphers - Part 1

6.2 Examples of symmetric ciphers - Part 2

6.3 Asymmetric ciphers

7.1 The Discrete Logarithm Problem (DLP)

7.2 Diffie-Hellman key exchange

7.3 Computational complexity - Part 1

7.3 Computational complexity - Part 2

7.4 The “collision” Babystep-Giantsep algorithm - Part 1

7.4 The “collision” Babystep-Giantsep algorithm - Part 2

7.5 Pohlig-Hellman Algorithm - Part 1

7.5 Pohlig-Hellman Algorithm - Part 2

7.6 Complexity of the Pohlig-Hellman Algorithm - Part 1

7.6 Complexity of the Pohlig-Hellman Algorithm - Part 2

8.1 The ElGamal Public key

8.2 Euler’s Formula and Theorem - Part 1

8.2 Euler’s Formula and Theorem - Part 2

8.3 The Rivest - Shamir - Adleman (RSA) Public Key Cryptosystem

8.4 Security issues of RSA Public Key Cryptosystem Computing modular e-th roots

9.1 Public key Digital signatures

9.2 RSA type Digital Signature

9.3 Elgamal type Digital Signature

10.1 Prime number Theorem

10.2 Fermat (non) Primality test

10.3 Fermat’s alternative Theorem

10.4 Strong Fermat (non) Primality test - Part 1

10.4 Strong Fermat (non) Primality test - Part 2

11.1 Trial Division

11.2 Factorization via difference of squares - Part 1

11.2 Factorization via difference of squares - Part 2

11.3 Strong witnesses and factorization

11.4 Pollard’s p-1 factorization algorithm - Part 1

11.4 Pollard’s p-1 factorization algorithm - Part 2

Risultati di apprendimento attesi

The overall goal is to provide the mathematical foundations of cryptography.

Prerequisites

Topics covered in the Algebra courses (congruences, groups and cyclic groups, finite fields) and Calculus I (differential and integral calculus, numerical series) in the Mathematics degree program.

THE COURSE IS NOW OPEN

Destinatari

STEM master's students, professionals and enthusiasts with a mathematical background for understanding cryptography.

Accesso e Attestato

Pass with at least 70% of the questions on the final test. Online test with at least 30 multiple-choice questions.

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Mathematical Cryptography

Mathematical Cryptography

THE COURSE IS NOW OPEN

Descrizione

Docenti

Carlo Mariconda

Course outline

Mathematical Cryptography

Course introduction

Platform guide: How to navigate the MOOC

Welcome to the Course

Getting to know you

Why MATHS came in Cryptography

Introduction to Module 1

Classical cryptography: monoalphabetic substitution ciphers

Modern cryptography: Polyalphabetic ciphers

Cipher Machines

Modern cryptography

Exercises of the module

Open discussion

Divisions

Introduction to Module 2

Divisibility

Greatest Common Divisor

Extended Euclidean algorithm

Coprime numbers

The Linear Diophantine equation au+bv=c - The general case

Prime numbers

Exercises of the module

Open discussion

Introduction to modular algebra

Introduction to Module 3

Classes modulo m

Invertibility modulo m

Linear congruences

The Chinese Remainder Theorem

Euler’s phi function

Exercises of the module

Open discussion

Powers modulo m and Fermat little theorem

Introduction to module 4

The Fast Powering Algorithm

Fermat’s little Theorem

Fermat’s Little Theorem: Applications

Exercises of the module

Open discussion

Order and primitive roots modulo a prime

Introduction to module 5

Order and primitive roots modulo a prime

Primitive roots: a test

Number of primitive roots

Exercises of the module

Open discussion

Introduction to Cryptography

Introduction to module 6

Symmetric ciphers

Examples of symmetric ciphers

Asymmetric ciphers

Exercises of the module

Open discussion

Discrete Logarithms and Diffi e-Hellman key exchange

Introduction to the Module 7

The Discrete Logarithm Problem (DLP)

Diffie-Hellman key exchange

Computational complexity

The “collision” Babystep-Giantsep algorithm

Pohlig-Hellman Algorithm

Complexity of the Pohlig-Hellman Algorithm

Exercises of the module

Open discussion

Public key cryptosystems ElGamal - RSA

Introduction to Module 8

The ElGamal Public key

Euler’s Formula and Theorem

The Rivest - Shamir - Adleman (RSA)Public Key Cryptosystem

Security issues of RSA Public Key Cryptosystem Computing modular e-th roots

Exercises of the module

Open discussion

Digital Signatures

Introduction to Module 9