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Cryptography (C)

Credits Dept.
7.5 (6.0 ECTS) MAT


Person in charge:  (-)

General goals

Upon finishing this subject, students should:
-  Be familiar with the basic principles (especially those related to public key cryptography) underlying the most important techniques in cryptography: symmetrical and asymmetrical coding-decoding and the digital signature.
-  Understand the most important algorithms behind each of the relevant techniques, especially those used in the most commonly-accepted standards.
-  Be familiar with some of the applications of cryptography.

Specific goals


  1. Secret key cryptography. Flow and block encryption. DES and AES.
  2. Public key cryptography. The integer factorising problem.

    The discrete logarithm problem.
  3. Digital signatures.
  4. Cryptography protocols.


  1. Characteristics of secret key algorithms. Extracting the general characteristics of a given secret key algorithm. Understand and programme a given public key algorithm.
  2. Characteristics of public key algorithms. Knowing how to classify a digital signature algorithm depending on the problem it is based on. Understanding and programming a given public key algorithm.
  3. Knowing how to classify a digital signature algorithm depending on the problem it is based on. Understanding and programming a given public key algorithm.
  4. Understanding the workings of a given cryptographic protocol and finding its strengths and weaknesses.
  5. Telling the difference between sound cryptography and "snake-oil".


  1. Critical, logical-mathematical reasoning.
  2. Ability to create and use models of reality.
  3. Ability to design and carry out experiments and analyse the results.
  4. Ability to design a test to check the validity of the response after being given the answer.


Estimated time (hours):

T P L Alt Ext. L Stu A. time
Theory Problems Laboratory Other activities External Laboratory Study Additional time

1. Basic concepts
T      P      L      Alt    Ext. L Stu    A. time Total 
3,0 0 0 0 0 3,0 0 6,0
- Cryptology, Cryptography, and Cryptanalysis.

- Classic cryptography and modern cryptography.

- Basic techniques: encryption-decryption and signature.

- Private key cryptography and public key cryptography.

- The mathematical bases of cryptography.

2. Modern secret key techniques
T      P      L      Alt    Ext. L Stu    A. time Total 
9,0 0 12,0 0 12,0 9,0 0 42,0

- Block encryption, flow encryption.

- The Data Encryption Standard: Description, History, Standardisation, Cryptanalysis.

- The Advanced Encryption Standard: Description, Standardisation.

- Operation modes for block-encrypted systems.

3. Public key encryption systems
T      P      L      Alt    Ext. L Stu    A. time Total 
18,0 0 12,0 0 12,0 18,0 0 60,0
- Multi-precision arithmetic operations. Euclidean algorithms.

- Congruences, multiplication group, modular arithmetic, modular exponential, Chinese Remainder Theorem.

- Calculation of square roots.

- Prime numbers, probabilistic criteria of primeness, random generation of prime numbers.

- Factorising integers, current state of the problem and the outlook in this field.

- Concepts behind the unidirectional trap-door function.

- Power function trap-door.

- Discrete exponential function and the discrete algorithm problem. Finite body variants.

- The knapsack problem.

- RSA cryptosystem (Rivest, Shamir, Adleman).

- ElGamal cryptosystem.

- Diffie-Hellman system for distributing keys.

- Knapsack cryptosystem. Shamir"s cryptanalysis.

4. Digital signatures
T      P      L      Alt    Ext. L Stu    A. time Total 
3,0 0 4,0 0 4,0 3,0 0 14,0

- Cryptographic hash functions. Secure Hash Standard.

- Digital signatures: RSA and DSA

- Public key certificates.

- Certifying authorities.


5. Cryptographic protocols
T      P      L      Alt    Ext. L Stu    A. time Total 
12,0 0 2,0 0 2,0 12,0 0 28,0

- Encryption and decryption transformations. Mixed private key - public key techniques.

- Identification schemes and protocols.

- SSL.

- SET.

- Micro-payments.

- Shared secrets.

- Electronic voting.

- Watermarks.

Others aspects of cryptography:

- Standardisations. Bodies involved.

- Patents.

- Policy aspects. Government control.

- The Telecommunications Act.

- The Digital Signature Act.

- International Laws.

Total per kind T      P      L      Alt    Ext. L Stu    A. time Total 
45,0 0 30,0 0 30,0 45,0 0 150,0
Avaluation additional hours 0
Total work hours for student 150,0

Docent Methodolgy

Theory classes and exercises.

The nature of the theme under consideration will determine the scheduling of problems and theory.

Lab classes will consolidate students" knowledge.

Evaluation Methodgy

The grade (out of 10) is calculated by adding the following:

* Lab exercise (4 points).

* Final exam (6 points).

Basic Bibliography

  • José Pastor Franco, Miguel Ángel Sarasa López, José Luis Salazar Riaño Criptografía digital: fundamentos y aplicaciones, Publicaciones Universitarias Universidad de Zaragoza, 2001.
  • Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone Handbook of applied cryptography, CRC Press,, 1997.
  • Mollin, Richard A. RSA and public-key cryptography, Chapman & Hall/CRC , 2003.
  • William Stallings Cryptography and network security: principles and practice, Prentice Hall, 2010.
  • PAAR, Christof , PRENEEL, Bart Understanding Cryptography, Springer, 2010.

Complementary Bibliography

  • Ross Anderson Security Engineering, Wiley (, .
  • Gustavus J. Simmons, [editors] Contemporary cryptology : the science of information integrity, IEEE Press, 1992.
  • Douglas R. Stinson Cryptography : theory and practice, Chapman & Hall / CRC, 2002.
  • Katzenbeisser, S., Petitcolas, F.A.P. Information hiding techniques for steganography and digital watermaking, Artech House, 2000.
  • Rolf Oppliger Security technologies for the World Wide Web, Artech House, 2003.
  • Ingemar J. Cox, Matthew L. Miller, Jeffrey A. Bloom Digital watermarking, Morgan Kaufmann, 2002.
  • O'Mahony, D., Peirce, M., Tewari, H. Electronic Payment Systems for E-commerce, Artech House, 2001.
  • Salomaa, A. Public-key Cryptography, Springer-Verlag, 1990.
  • Neal Koblitz Algebraic aspects of cryptography, Springer,, 1998.
  • Koblitz, N. A A course in Number Theory and Cryptography , Springer-Verlag, 1994.
  • Ian Blake, Gadiel Seroussi & Nigel Smart Elliptic curves in cryptography, Cambridge University Press, 1999.
  • Manuel José Lucena López Criptografía y seguridad en computadores, , 2003.
  • Jorge Ramió Aguirre Aplicaciones criptográficas : libro guía de la asignatura de Seguridad Informática, Escuela Universitaria de Informática. Universidad Politécnica de Madrid, 1998.
  • Song Y. Yan Number theory for computing, Springer, 2002.
  • BALDONI, M.W., CILIBERTO, C., PIACENTINI CATTANEO, G.M. Elementary Number Theory, Cryptography and Codes, Springer, 2009.
  • VAUDENAY, Serge A classical Introduction to Cryptography, Springer, 2006.
  • DELFS, Hans; KNEBL, Helmut Introduction to Cryptography, Springer, 2007.
  • Bruce Schneier Applied cryptography : protocols, algorithms, and source code in C, John Wiley & Sons, 1996.

Web links


Previous capacities

Students are recommended to take this course a few terms before completing the Selection Stage.


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