NumbermadaECTS3.0LevelbasicOverviewThis course introduces the students into the fundamental mathematical notions and results of elementary number theory used in data communication, and gives some applications.
A data communication model
Problems of data communication
Number Theory for Public Key Cryptography
- Arithmetic in Z
- Modular arithmetic
- Extended euclidean algorithm
- Linear diophantic equations in two variables
- Square-and-multiply algorithm
- Complexity
Group Theory
- Properties of groups
- Examples of groups: Z/nZ, Z/nZ* etc.
- Subgroups and cosets
- Homomorphisms of groups
RSA and Diffie-Hellman
- Key generation
- Encryption and decryption
- Key-agreement protocol
- Man-in-the-middle attack
Codes
- Examples of Codes
- Error-recognizing and error-correcting codes
- Linear codes
- Syndrome decoding
- Applications
(The order and emphasis of the topics are left to the lecturers.)
Learning objectivesNumber Theory
The students understand the fundamental notions of elementary number theory. They are able to compute with residue classes. They understand the extended euclidean algorithm and the square-and-multiply algorithm for modular exponentiation. They are able to apply these algorithms and to explain their complexities. They are able to solve linear diophantic equations in two variables.
Group Theory
The students are able to explain what a group is, know some examples of groups and can calculate in the groups Z/nZ und Z/nZ*. They understand the notions of cyclic groups, generating elements, the order of a group and the order of an element. They understand subgroups and cosets, and know that the order of a subgroup of a finite group always divides the order of the group. In simple cases, they are able to determine the subgroups of a group. They can explain what a group homomorphism is and know some examples.
RSA and Diffie-Hellman
The students are able to generate RSA-keys and to encrypt and decrypt messages. They know some precautions for the key generation.
They know the Diffie-Hellman protocol and are able to use this protocol.
They understand the man-in-the-middle attack
Codes
The students know what (n,M,d)-codes are and can explain their parameters.
They know some specific codes and are able to use these codes for error-correction. In simple cases, they are able to decode linear codes by syndrome decoding.
Previous knowledgeNumber Theory
The students understand the fundamental notions of elementary number theory. They are able to compute with residue classes. They understand the extended euclidean algorithm and the square-and-multiply algorithm for modular exponentiation. They are able to apply these algorithms and to explain their complexities. They are able to solve linear diophantic equations in two variables.
Group Theory
The students are able to explain what a group is, know some examples of groups and can calculate in the groups Z/nZ und Z/nZ*. They understand the notions of cyclic groups, generating elements, the order of a group and the order of an element. They understand subgroups and cosets, and know that the order of a subgroup of a finite group always divides the order of the group. In simple cases, they are able to determine the subgroups of a group. They can explain what a group homomorphism is and know some examples.
RSA and Diffie-Hellman
The students are able to generate RSA-keys and to encrypt and decrypt messages. They know some precautions for the key generation.
They know the Diffie-Hellman protocol and are able to use this protocol.
They understand the man-in-the-middle attack
Codes
The students know what (n,M,d)-codes are and can explain their parameters.
They know some specific codes and are able to use these codes for error-correction. In simple cases, they are able to decode linear codes by syndrome decoding.
Exam formatContinuous assessment grade with final written exam 
