CRYPTOGRAPH Y
General cryptography information
Cryptography is the study of secure communication techniques that allow only the sender and
intended recipient of a message to view its contents.
Cryptography is built upon a framework of four information security objectives / goals:
Confidentiality; a service used to keep the content from all but those authorized to have it.
There are many approaches, ranging from physical protection to mathematical algorithms.
Data integrity; a service which addresses the unauthorized alteration of data. The system
should be able to detect data manipulation, such as insertion/deletion/substitution.
Authentication; a service related to identification. Two parties entering into a communication
should identify each other (entity authentication), and information sent over a channel should
be authenticated to origin, date, content, time, etc. (data origin authentication).
Non-repudiation; a service which prevents an entity from denying previous commitments or
actions. A procedure involving a trusted third party is needed to resolve such disputes.
Cryptographic tools (primitives) should be evaluated according to the following criteria:
• Level of security → the number of operations required to defeat the intended objective
• Functionality → which primitives are most effective for a given objective
• Methods of operation → applying primitives in various ways will provide different functionlty
• Performance → efficiency of a primitive in a particular mode of operation
• Ease of implementation → the ease of realizing the primitive in a practical instantion
Mathematical background of cryptography
Functions
Basic mathematical concepts that are useful to understand in regards to cryptography, are functions
(a.k.a. mappings or transformations).
• A function is defined by rule f, the domain set ‘X’, and the codomain set ‘Y’; f: X → Y.
• The image of x is an element y ∈ Y for which f(x) = y, and the preimage of y is an element x
∈ X for which f(x) = y.
You can view the domain X as the encrypted message and the
codomain Y as the decrypted message. The arrows (f) correspond with
the mapping used for encryption and decryption. Therefore, often only
the domain X and the rule f are given, like in this example.
, *de functie/mapping is kijken hoe vaak 11 in x2 past, en de restwaarde is dan y. Als 11 nul keer helemaal in x 2 past, noteer je
voor y gewoon de uitkomst van x2 .
Eigenlijk is f(4) hetzelfde als “16
mod 11” (hetzelfde als 5). Je noemt
dit modulo arithmetics.
There are different types of functions.
➔ Injective (1 – 1): each element in codomain Y is the image of at
most one element in the domain X.
➔ Surjective (onto): each element in the codomain Y is the image
of at least one element in the domain X.
➔ Bijection (f -1 ): one-to-one function such that Im(f) = Y (each domain
element is connected to one codomain element).
Each bijection ‘f’ has an inverse function ‘g’, which you see here. In
cryptography, bijections are used to encrypt messages and inverse
bijections are used to decrypt messages.
➔ One-way function: a function ‘f’ from X to Y where it is easy to compute the y-value
(f(x)) for each x-value, but hard to compute the x-value for each y-value.
A one-way function is said to be trapdoor if some extra information is given that
makes it easier/feasible to find the x-value for each y-value. An example of trapdoor
information is a value that you can fill in for the element ‘n’ that is in the function.
One-way trapdoor functions are fundamental for public key encryption.
Special functions
Permutations are functions which are often used in various cryptographic constructs. A permutation
‘p’ is a bijection from a set ‘S’ to itself. You just randomly define that p(1) = 3 etcetera, and note it
like this. The top row is the domain, and the bottom row is the image under mapping p.
The inverse function of a permutation, is swapping the two
rows, and then reordering the numbers of the top row.
A function is an involution if f = f-1.
General cryptography information
Cryptography is the study of secure communication techniques that allow only the sender and
intended recipient of a message to view its contents.
Cryptography is built upon a framework of four information security objectives / goals:
Confidentiality; a service used to keep the content from all but those authorized to have it.
There are many approaches, ranging from physical protection to mathematical algorithms.
Data integrity; a service which addresses the unauthorized alteration of data. The system
should be able to detect data manipulation, such as insertion/deletion/substitution.
Authentication; a service related to identification. Two parties entering into a communication
should identify each other (entity authentication), and information sent over a channel should
be authenticated to origin, date, content, time, etc. (data origin authentication).
Non-repudiation; a service which prevents an entity from denying previous commitments or
actions. A procedure involving a trusted third party is needed to resolve such disputes.
Cryptographic tools (primitives) should be evaluated according to the following criteria:
• Level of security → the number of operations required to defeat the intended objective
• Functionality → which primitives are most effective for a given objective
• Methods of operation → applying primitives in various ways will provide different functionlty
• Performance → efficiency of a primitive in a particular mode of operation
• Ease of implementation → the ease of realizing the primitive in a practical instantion
Mathematical background of cryptography
Functions
Basic mathematical concepts that are useful to understand in regards to cryptography, are functions
(a.k.a. mappings or transformations).
• A function is defined by rule f, the domain set ‘X’, and the codomain set ‘Y’; f: X → Y.
• The image of x is an element y ∈ Y for which f(x) = y, and the preimage of y is an element x
∈ X for which f(x) = y.
You can view the domain X as the encrypted message and the
codomain Y as the decrypted message. The arrows (f) correspond with
the mapping used for encryption and decryption. Therefore, often only
the domain X and the rule f are given, like in this example.
, *de functie/mapping is kijken hoe vaak 11 in x2 past, en de restwaarde is dan y. Als 11 nul keer helemaal in x 2 past, noteer je
voor y gewoon de uitkomst van x2 .
Eigenlijk is f(4) hetzelfde als “16
mod 11” (hetzelfde als 5). Je noemt
dit modulo arithmetics.
There are different types of functions.
➔ Injective (1 – 1): each element in codomain Y is the image of at
most one element in the domain X.
➔ Surjective (onto): each element in the codomain Y is the image
of at least one element in the domain X.
➔ Bijection (f -1 ): one-to-one function such that Im(f) = Y (each domain
element is connected to one codomain element).
Each bijection ‘f’ has an inverse function ‘g’, which you see here. In
cryptography, bijections are used to encrypt messages and inverse
bijections are used to decrypt messages.
➔ One-way function: a function ‘f’ from X to Y where it is easy to compute the y-value
(f(x)) for each x-value, but hard to compute the x-value for each y-value.
A one-way function is said to be trapdoor if some extra information is given that
makes it easier/feasible to find the x-value for each y-value. An example of trapdoor
information is a value that you can fill in for the element ‘n’ that is in the function.
One-way trapdoor functions are fundamental for public key encryption.
Special functions
Permutations are functions which are often used in various cryptographic constructs. A permutation
‘p’ is a bijection from a set ‘S’ to itself. You just randomly define that p(1) = 3 etcetera, and note it
like this. The top row is the domain, and the bottom row is the image under mapping p.
The inverse function of a permutation, is swapping the two
rows, and then reordering the numbers of the top row.
A function is an involution if f = f-1.
