Rademacher complexity

In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. == Definitions == === Rademacher complexity of a set === Given a set A ⊆ R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows: Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} where σ 1 , σ 2 , … , σ m {\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{m}} are independent random variables drawn from the Rademacher distribution i.e. Pr ( σ i = + 1 ) = Pr ( σ i = − 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i ∈ { 1 , 2 , … , m } {\displaystyle i\in \{1,2,\dots ,m\}} , and a = ( a 1 , … , a m ) ∈ A {\displaystyle a=(a_{1},\ldots ,a_{m})\in A} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displaystyle A} is symmetric this makes no difference. === Rademacher complexity of a function class === Let S = { z 1 , z 2 , … , z m } ⊆ Z {\displaystyle S=\{z_{1},z_{2},\dots ,z_{m}\}\subseteq Z} be a sample of points and consider a function class F {\displaystyle {\mathcal {F}}} of real-valued functions over Z {\displaystyle Z} . Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S ⁡ ( F ) = 1 m E σ [ sup f ∈ F | ∑ i = 1 m σ i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right|\right]} This can also be written using the previous definition: Rad S ⁡ ( F ) = Rad ⁡ ( F ∘ S ) {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)} where F ∘ S {\displaystyle {\mathcal {F}}\circ S} denotes function composition, i.e.: F ∘ S := { ( f ( z 1 ) , … , f ( z m ) ) ∣ f ∈ F } {\displaystyle {\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad ¯ m ( F ) = sup S = { z 1 , … , z m } Rad S ⁡ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distribution over Z {\displaystyle Z} . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is: Rad P , m ⁡ ( F ) := E S ∼ P m [ Rad S ⁡ ( F ) ] {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]} where the above expectation is taken over an identically independently distributed (i.i.d.) sample S = ( z 1 , z 2 , … , z m ) {\displaystyle S=(z_{1},z_{2},\dots ,z_{m})} generated according to P {\displaystyle P} . == Intuition == The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of σ i {\displaystyle \sigma _{i}} under the expectation, so that this quantity in the sum is maximized. The Rademacher complexity of a set A {\displaystyle A} can be rewritten as Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] = 1 m 2 m ∑ σ ∈ { − 1 / m , + 1 / m } m [ sup a ∈ A ⟨ σ , a ⟩ ] . {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]={\frac {1}{{\sqrt {m}}2^{m}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}}\left[\sup _{a\in A}\langle \sigma ,a\rangle \right].} Each term in the summation is the farthest distance of the set A {\displaystyle A} from the origin, along a unit-length direction σ {\displaystyle \sigma } . The directions are along the vertices of a hypercube. Thus, we can also write it as Rad ⁡ ( A ) = 1 2 m 1 2 m − 1 ∑ σ ∈ { − 1 / m , + 1 / m } m / { − 1 , + 1 } [ sup a ∈ A ⟨ σ , a ⟩ − inf a ∈ A ⟨ σ , a ⟩ ] {\displaystyle \operatorname {Rad} (A)={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\in A}\langle \sigma ,a\rangle -\inf _{a\in A}\langle \sigma ,a\rangle \right]} Here, the set { − 1 / m , + 1 / m } m / { − 1 , + 1 } {\displaystyle \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}} denotes half of the vertices of a hypercube, selected so that each diagonal has exactly one vertex selected. In words, this states that 2 m Rad ⁡ ( A ) {\displaystyle 2{\sqrt {m}}\operatorname {Rad} (A)} is precisely the average width of the set A {\displaystyle A} along all diagonal directions of a hypercube. == Examples == A singleton set has 0 width in any direction, so it has Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it has Rademacher complexity 1 / 4 {\displaystyle 1/4} . The unit cube [ 0 , 1 ] m {\displaystyle [0,1]^{m}} has constant width m {\displaystyle {\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq 1\}} has constant width 2 / m {\displaystyle 2/{\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / m {\displaystyle 1/m} . == Using the Rademacher complexity == The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. === Bounding the representativeness === In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probability distribution from which the samples are drawn. Denote by H {\displaystyle H} the set of hypotheses (potential classifiers) and denote by F {\displaystyle {\mathcal {F}}} the corresponding set of error functions, i.e., for every hypothesis h ∈ H {\displaystyle h\in H} , there is a function f h ∈ F {\displaystyle f_{h}\in F} , that maps each training sample (features,label) to the error of the classifier h {\displaystyle h} (note in this case hypothesis and classifier are used interchangeably). For example, in the case that h {\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle h} correctly classifies a sample and 1 else. We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z ∼ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} – the expected error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m ∑ i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})} – the estimated error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the sample S {\displaystyle S} . The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P ⁡ ( F , S ) := sup f ∈ F ( L P ( f ) − L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then Rad P , m ⁡ ( F ) − ln ⁡ 2 2 m ≤ E S ∼ P m [ Rep P ⁡ ( F , S ) ] ≤ 2 Rad P , m ⁡ ( F ) {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}})-{\sqrt {\frac {\ln 2}{2m}}}\leq \mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}({\

Mobile simulator

A mobile simulator is a software application for a personal computer which creates a virtual machine version of a mobile device, such as a mobile phone, iPhone, other smartphone, or calculator, on the computer. This may sometimes also be termed an emulator. The mobile simulator allows the user to use features and run applications on the virtual mobile on their computer as though it was the actual mobile device. A mobile simulator lets you test a website and determine how well it performs on various types of mobile devices. A good simulator tests mobile content quickly on multiple browsers and emulates several device profiles simultaneously. This allows analysis of mobile content in real-time, locate errors in code, view rendering in an environment that simulates the mobile browser, and optimize the site for performance. Mobile simulators may be developed using programming languages such as Java, .NET and JavaScript.

Protecting Kids From Social Media Act

Protecting Kids on Social Media Act or HB 1891 is an American law that was introduced by William Lamberth of Sumner County, Tennessee and was signed into law by Tennessee's governor on May 2, 2024. The bill requires social media websites such as X, YouTube, TikTok, Facebook and others to verify the age of users and if those users are under 18, they must have parental consent. == Progress == The law passed the Tennessee State Legislature with little opposition: the bill had only two no votes in the House from Aftyn Behn and Vincent B. Dixie, and it had zero no votes in the Senate. == Bill summary == Every social media company must verify the age of new users after the law takes effect, and if the user had created an account before the law took effect, they must verify the age of the person attempting to access the account within 14 days. If the new user or the user who originally owned an account is under 18 years of age, they must get parental consent and the third party or social media company must not retain the data from the age verification process or obtaining parental consent. Parents who are account holders of those under 18 can view the privacy settings, set daily time restrictions, and implement breaks during which the minor cannot access the account. The law is enforced by the Attorney General of Tennessee and went into effect on January 1, 2025. == Lawsuit == On October 3, 2024, the trade association NetChoice filed a lawsuit against Tennessee Attorney General Jonathan Skrmetti in the Middle District Court of Tennessee, claiming that the law violates the First Amendment. The Judge for the case is William L. Campbell Jr. An initial case management conference was originally scheduled for December 4, 2024, however it was delayed because of the Supreme Court case United States v. Skrmetti, recommending that the conference be delayed after January 20, 2025. On February 14, 2025, Judge Eli Richardson denied NetChoice's motion for a temporary restraining order because it would disrupt the status quo of the case.

Contrast set learning

Contrast set learning is a form of association rule learning that seeks to identify meaningful differences between separate groups by reverse-engineering the key predictors that identify for each particular group. For example, given a set of attributes for a pool of students (labeled by degree type), a contrast set learner would identify the contrasting features between students seeking bachelor's degrees and those working toward PhD degrees. == Overview == A common practice in data mining is to classify, to look at the attributes of an object or situation and make a guess at what category the observed item belongs to. As new evidence is examined (typically by feeding a training set to a learning algorithm), these guesses are refined and improved. Contrast set learning works in the opposite direction. While classifiers read a collection of data and collect information that is used to place new data into a series of discrete categories, contrast set learning takes the category that an item belongs to and attempts to reverse engineer the statistical evidence that identifies an item as a member of a class. That is, contrast set learners seek rules associating attribute values with changes to the class distribution. They seek to identify the key predictors that contrast one classification from another. For example, an aerospace engineer might record data on test launches of a new rocket. Measurements would be taken at regular intervals throughout the launch, noting factors such as the trajectory of the rocket, operating temperatures, external pressures, and so on. If the rocket launch fails after a number of successful tests, the engineer could use contrast set learning to distinguish between the successful and failed tests. A contrast set learner will produce a set of association rules that, when applied, will indicate the key predictors of each failed tests versus the successful ones (the temperature was too high, the wind pressure was too high, etc.). Contrast set learning is a form of association rule learning. Association rule learners typically offer rules linking attributes commonly occurring together in a training set (for instance, people who are enrolled in four-year programs and take a full course load tend to also live near campus). Instead of finding rules that describe the current situation, contrast set learners seek rules that differ meaningfully in their distribution across groups (and thus, can be used as predictors for those groups). For example, a contrast set learner could ask, “What are the key identifiers of a person with a bachelor's degree or a person with a PhD, and how do people with PhD's and bachelor’s degrees differ?” Standard classifier algorithms, such as C4.5, have no concept of class importance (that is, they do not know if a class is "good" or "bad"). Such learners cannot bias or filter their predictions towards certain desired classes. As the goal of contrast set learning is to discover meaningful differences between groups, it is useful to be able to target the learned rules towards certain classifications. Several contrast set learners, such as MINWAL or the family of TAR algorithms, assign weights to each class in order to focus the learned theories toward outcomes that are of interest to a particular audience. Thus, contrast set learning can be thought of as a form of weighted class learning. === Example: Supermarket Purchases === The differences between standard classification, association rule learning, and contrast set learning can be illustrated with a simple supermarket metaphor. In the following small dataset, each row is a supermarket transaction and each "1" indicates that the item was purchased (a "0" indicates that the item was not purchased): Given this data, Association rule learning may discover that customers that buy onions and potatoes together are likely to also purchase hamburger meat. Classification may discover that customers that bought onions, potatoes, and hamburger meats were purchasing items for a cookout. Contrast set learning may discover that the major difference between customers shopping for a cookout and those shopping for an anniversary dinner are that customers acquiring items for a cookout purchase onions, potatoes, and hamburger meat (and do not purchase foie gras or champagne). == Treatment learning == Treatment learning is a form of weighted contrast-set learning that takes a single desirable group and contrasts it against the remaining undesirable groups (the level of desirability is represented by weighted classes). The resulting "treatment" suggests a set of rules that, when applied, will lead to the desired outcome. Treatment learning differs from standard contrast set learning through the following constraints: Rather than seeking the differences between all groups, treatment learning specifies a particular group to focus on, applies a weight to this desired grouping, and lumps the remaining groups into one "undesired" category. Treatment learning has a stated focus on minimal theories. In practice, treatment are limited to a maximum of four constraints (i.e., rather than stating all of the reasons that a rocket differs from a skateboard, a treatment learner will state one to four major differences that predict for rockets at a high level of statistical significance). This focus on simplicity is an important goal for treatment learners. Treatment learning seeks the smallest change that has the greatest impact on the class distribution. Conceptually, treatment learners explore all possible subsets of the range of values for all attributes. Such a search is often infeasible in practice, so treatment learning often focuses instead on quickly pruning and ignoring attribute ranges that, when applied, lead to a class distribution where the desired class is in the minority. === Example: Boston housing data === The following example demonstrates the output of the treatment learner TAR3 on a dataset of housing data from the city of Boston (a nontrivial public dataset with over 500 examples). In this dataset, a number of factors are collected for each house, and each house is classified according to its quality (low, medium-low, medium-high, and high). The desired class is set to "high", and all other classes are lumped together as undesirable. The output of the treatment learner is as follows: Baseline class distribution: low: 29% medlow: 29% medhigh: 21% high: 21% Suggested Treatment: [PTRATIO=[12.6..16), RM=[6.7..9.78)] New class distribution: low: 0% medlow: 0% medhigh: 3% high: 97% With no applied treatments (rules), the desired class represents only 21% of the class distribution. However, if one filters the data set for houses with 6.7 to 9.78 rooms and a neighborhood parent-teacher ratio of 12.6 to 16, then 97% of the remaining examples fall into the desired class (high-quality houses). == Algorithms == There are a number of algorithms that perform contrast set learning. The following subsections describe two examples. === STUCCO === The STUCCO contrast set learner treats the task of learning from contrast sets as a tree search problem where the root node of the tree is an empty contrast set. Children are added by specializing the set with additional items picked through a canonical ordering of attributes (to avoid visiting the same nodes twice). Children are formed by appending terms that follow all existing terms in a given ordering. The formed tree is searched in a breadth-first manner. Given the nodes at each level, the dataset is scanned and the support is counted for each group. Each node is then examined to determine if it is significant and large, if it should be pruned, and if new children should be generated. After all significant contrast sets are located, a post-processor selects a subset to show to the user - the low order, simpler results are shown first, followed by the higher order results which are "surprising and significantly different." The support calculation comes from testing a null hypothesis that the contrast set support is equal across all groups (i.e., that contrast set support is independent of group membership). The support count for each group is a frequency value that can be analyzed in a contingency table where each row represents the truth value of the contrast set and each column variable indicates the group membership frequency. If there is a difference in proportions between the contrast set frequencies and those of the null hypothesis, the algorithm must then determine if the differences in proportions represent a relation between variables or if it can be attributed to random causes. This can be determined through a chi-square test comparing the observed frequency count to the expected count. Nodes are pruned from the tree when all specializations of the node can never lead to a significant and large contrast set. The decision to prune is based on: The minimum deviation size: The maximum difference between the support

Cover-coding

Cover-coding is a technique for obscuring the data that is transmitted over an insecure link, to reduce the risks of snooping. An example of cover-coding would be for the sender to perform a bitwise XOR (exclusive OR) of the original data with a password or random number which is known to both sender and receiver. The resulting cover-coded data is then transmitted from sender to the receiver, who uncovers the original data by performing a further bitwise XOR (exclusive OR) operation on the received data using the same password or random number. ISO 18000-6C (EPC Class 1 Generation 2) RFID tags protect some operations with a cover code. The reader requests a random number from the tag, and the tag responds with a new random number. The reader then encrypts future communications with this number, using bitwise XOR, to the data it sends. Cover coding is secure if the tag signal can't be intercepted and the random number is not re-used. Compared to the loud transmissions from the reader, tag backscatter is much weaker and difficult -- but not impossible -- to intercept.

Mojito (framework)

Mojito is an environment agnostic, Model-View-Controller (MVC) web application framework. It was designed by Yahoo. == Features == Mojito supports agile development of web applications. Mojito has built-in support for unit testing, Internationalization, syntax and coding convention checks. Both server and client components are written in JavaScript. Mojito allows developers designing web applications to leverage the utilities of both configuration and MVC framework. Mojito is capable of running on both JavaScript-enabled web browsers and servers using Node.js because they both utilize JavaScript. Mojito applications mainly consist of two components: JSON Configuration files: these define relationships between code components, assets, routing paths, and framework defaults and are available at the application and mojit level. Directories: these reflect MVC architecture and are used to separate resources such as assets, libraries, middleware, etc. == Architecture == In Mojito, both server and "client" side scripting is done in JavaScript, allowing it to run on both client and server thereby breaking the "front-end back-end barrier." It has both client and server runtimes. === Server runtime === This block houses operations needed by server side components. Services include: Routing rules, HTTP Server, config loader and disk-based loader. === Client runtime === This block houses operations called upon while running client sides components. Services include local storage/cache access and JSON based /URL based loader === Core === Core function can be accessed on client or server. Services include Registry, Dispatcher, Front controller, Resource store. === Container === mojit object comes into the picture. This container also include the services used by mojits. API and Mojito services are the blocks which caters to services needed for execution of mojits. === API (Action Context) === Mojito services are a customizable service block. It offers mojits a range of services which might be needed by mojit to carry out certain actions. These services can be availed at both client and server side. Reusable services can be created and aggregated to the core here. == Mojits == Mojits are the modules of a Mojito application. An application consists of one or more mojits. A mojit encompasses a Model, Views and a Controller defined by JSON configuration files. It includes a View factory where views are created according to the model and a View cache that holds frequently requested views to aid performance. === Application Architecture === A Mojito application is a set of mojits facilitated by configurable JSON files which define the code for model, view and controller. This MVC structure works with API block and Mojito services, and can be deployed at both client and server side. While the application is deployed at client side, it can call server-side modules using binders. Binders are mojit codes that let mojits request services from each other. Mojit Proxy acts as an intermediary between binders and mojit's API (application context) block and other mojits. Controllers are command-issuing units of mojits. Models mirror the core logic and hold data. Applications can have multiple models. They can be centrally accessed from controllers. View files are created in accordance with controllers and models, and are marked-up before they are sent to users as output. === Application Directory Structure === Directory structure of a Mojito application with one mojit: [mojito_app]/ |-- application.json |-- assets/ | `-- favicon.icon |-- yui_modules/ | `-- .{affinity}.js |-- index.js |-- mojits/ | `-- [mojit_name | |-- assets/ | |-- yui_modules/ | | `-- .{affinity}.js | |-- binders/ | | `-- {view_name}.js | |-- controller.{affinity}.js | |-- defaults.json | |-- definition.json | |-- lang/ | | `-- {mojit_name}_{lang}.js | |-- models/ | | `-- {model_name}.{affinity}.js | |-- tests/ | | |-- yui_modules/ | | | `-- {module_name}.{affinity}-tests.js | | |-- controller.{affinity}-tests.js | | `-- models/ | | `-- {model_name}.{affinity}-tests.js | `-- views/ | |-- {view_name}.{view_engine}.html | `-- {view_name}.{device}.{view_engine}.html |-- package.json |-- routes.json (deprecated) |-- server.js == Model, View and Controller == The Model hosts data, which is accessed by the Controller and presented to the View. Controller also handles any client requests for data, in which case controller fetches data from the model and passes the data to the client. All three components are clustered in the mojit. Mojits are physically illustrated by directory structures and an application can have multiple mojits. Every mojit can have one controller, one or more views and zero or more models. === Model === The model it represents the application data and is independent of view or controller. Model contains code to manipulate the data. They are found in the models directory of each mojit. Functions include: Storing information for access by controller. Validation and error handling. Metadata required by the view === Controller === The controller acts like a connecting agent between model and view. It supplies input to Model and after fetching data from model, passes it to View. Functions include Redirection Monitors authentication Web safety Encoding === View === The view acts as a presentation filter by highlighting some model attributes and suppressing others. A view can be understood as a visual permutation of the model. The view renders data received from controller and displays it to the end user.

Cryptosystem

In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term cipher (sometimes cypher) is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term cryptosystem is most often used when the key generation algorithm is important. For this reason, the term cryptosystem is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques. == Formal definition == Mathematically, a cryptosystem or encryption scheme can be defined as a tuple ( P , C , K , E , D ) {\displaystyle ({\mathcal {P}},{\mathcal {C}},{\mathcal {K}},{\mathcal {E}},{\mathcal {D}})} with the following properties. P {\displaystyle {\mathcal {P}}} is a set called the "plaintext space". Its elements are called plaintexts. C {\displaystyle {\mathcal {C}}} is a set called the "ciphertext space". Its elements are called ciphertexts. K {\displaystyle {\mathcal {K}}} is a set called the "key space". Its elements are called keys. E = { E k : k ∈ K } {\displaystyle {\mathcal {E}}=\{E_{k}:k\in {\mathcal {K}}\}} is a set of functions E k : P → C {\displaystyle E_{k}:{\mathcal {P}}\rightarrow {\mathcal {C}}} . Its elements are called "encryption functions". D = { D k : k ∈ K } {\displaystyle {\mathcal {D}}=\{D_{k}:k\in {\mathcal {K}}\}} is a set of functions D k : C → P {\displaystyle D_{k}:{\mathcal {C}}\rightarrow {\mathcal {P}}} . Its elements are called "decryption functions". For each e ∈ K {\displaystyle e\in {\mathcal {K}}} , there is d ∈ K {\displaystyle d\in {\mathcal {K}}} such that D d ( E e ( p ) ) = p {\displaystyle D_{d}(E_{e}(p))=p} for all p ∈ P {\displaystyle p\in {\mathcal {P}}} . Note; typically this definition is modified in order to distinguish an encryption scheme as being either a symmetric-key or public-key type of cryptosystem. == Examples == A classical example of a cryptosystem is the Caesar cipher. A more contemporary example is the RSA cryptosystem. Another example of a cryptosystem is the Advanced Encryption Standard (AES). AES is a widely used symmetric encryption algorithm that has become the standard for securing data in various applications. Paillier cryptosystem is another example used to preserve and maintain privacy and sensitive information. It is featured in electronic voting, electronic lotteries and electronic auctions.