Hochschule f¨ ur Technik und Wirtschaft
Studiengang Kommunikationsinformatik (Master) Prof. Dr.–Ing. Damian Weber
Handout – Definitions
1 Set With One Operation
LetG be a set with|G| ≥1.
A binary operation◦on G is a map
◦:G×G−→G.
2 Semigroup
(G,◦) is called asemigroupif◦ is associative:
(a◦b)◦c=a◦(b◦c) ∀a, b, c∈G.
3 Monoid
A semigroup (G,◦) is called amonoidif there exists a neutral element ewith a◦e=e◦a=a ∀a∈G.
4 Group
A monoid (G,◦) is called agroup if, for each a∈G, there exists an inverse element a−1 with
a−1◦a=e.
5 Abelian Group
A group (G,◦) is called anabelian group orcommutative groupif a◦b=b◦a ∀a, b∈G.
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6 Ring: Two Operations
LetR be a set with |R| ≥1.
Let ,,+” be an operation which makes (R,+) an abelian group. The neutral element ofR is called 0.
Let ,,·” be an operation which makes (R,·) an semigroup.
Then R is called aringifa(b+c) =ab+acand (b+c)a=ba+cafor all a, b, c∈R.
The ringR is called commutative ifab=bafor all a, b∈R.
The ringR is called ring with identity if 1∈R witha·1 = 1·a=afor all a∈R.
7 Field: Two Operations
LetF be a set with{0,1} ∈F.
LetF be a commutative ring with identity.
Let (F\ {0},·) be an abelian group.
Then F is called afield.
8 Protocols
A protocol is a finite sequence of instructions describing the interaction between two or more entities. This compounds the data format of messages between the entities and the handling of error cases.
Properties:
• Each entity must know the protocol steps.
• The steps must be deterministic for every situation.
Notation:A sends message mtoB A−→B:m
If the messagem is encrypted by a key k, we write {m}k.
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