Given organizations' increasing dependency on information technology (IT) to run their operations, business continuity planning (and its subset IT service continuity planning) covers the entire organization, while disaster recovery focuses on IT. Auditing documents covering an organization's business continuity and disaster recovery (BCDR) plans provides a third-party validation to stakeholders that the documentation is complete and does not contain material misrepresentations. == Overview == Often used together, the terms business continuity (BC) and disaster recovery (DR) are very different. BC refers to the ability of a business to continue critical functions and business processes after the occurrence of a disaster, whereas DR refers specifically to the IT functions of the business, albeit a subset of BC. == Metrics == The primary objective is to protect the organization in the event that all or part of its operations and/or computer services are rendered partially or completely unusable. === DR metrics === Minimizing downtime and data loss during disaster recovery is typically measured in terms of two key concepts: Recovery time objective (RTO), time until a system is completely up and running Recovery point objective (RPO), a measure of the ability to recover files by specifying a point in time the backup copy will restore to. == The auditor's role == Role of the Internal Auditor in Auditing a Disaster Recovery Plan (DRP): 1. Governance & Oversight - Confirm roles, responsibilities, and oversight are defined, and DRP aligns with risk appetite and continuity strategy. 2. Risk Assessment & BIA - Verify risk and impact assessments identify critical systems and define RTO/RPO. 3. Plan Design & Documentation - Ensure the DRP is current, complete, and includes key recovery procedures. 4. Testing & Validation - Confirm regular DRP testing occurs and results are used to improve the plan. 5. Backup & Recovery - Assess backup frequency and recovery capabilities against RTO/RPO targets. 6. Communication & Training - Verify staff are trained and communication protocols are in place for crises. 7. Maintenance & Improvement - Ensure the DRP is regularly updated and lessons learned are integrated. == Documentation == === Disaster recovery plan === A disaster recovery plan (DRP) is a documented process or set of procedures to execute an organization's disaster recovery processes and recover and protect a business IT infrastructure in the event of a disaster. It is "a comprehensive statement of consistent actions to be taken before, during and after a disaster". The disaster could be natural, environmental or man-made. Man-made disasters could be intentional (for example, an act of a terrorist) or unintentional (that is, accidental, such as the breakage of a man-made dam or even "fat fingers" - or errant commands entered - on a computer system). ==== Types of plans ==== Although there is no one-size-fits-all plan, there are three basic strategies: prevention, including proper backups, having surge protectors and generators detection, a byproduct of routine inspections, which may discover new (potential) threats correction The latter may include securing proper insurance policies, and holding a "lessons learned" brainstorming session. ==== Best practices ==== To maximize their effectiveness, DRPs are most effective when updated frequently, and should: be an integral part of all business analysis processes, be revisited at every major corporate acquisition, at every new product launch and at every new system development milestone. be thoroughly tested, not just unpracticed bureaucratic documentation Adequate records need to be retained by the organization. The auditor examines records, billings, and contracts to verify that records are being kept. One such record is a current list of the organization's hardware and software vendors. Such list is made and periodically updated to reflect changing business practices and as part of an IT asset management system. Copies of it are stored on and off site and are made available or accessible to those who require them. An auditor tests the procedures used to meet this objective and determine their effectiveness. === Relationship to BCPs === Disaster recovery is a subset of business continuity. Where DRP encompasses the policies, tools and procedures to enable recovery of data following a catastrophic event, BCP involves keeping all aspects of a business functioning regardless of potential disruptive events. As such, a business continuity plan is a comprehensive organizational strategy that includes the DRP as well as threat prevention, detection, recovery, and resumption of operations should a data breach or other disaster event occur. Therefore, BCP consists of five component plans: Business resumption plan Occupant emergency plan Continuity of operations plan Incident management plan Disaster recovery plan The first three components (business resumption, occupant emergency, and continuity of operations plans) do not deal with the IT infrastructure. The incident management plan (IMP) does deal with the IT infrastructure, but since it establishes structure and procedures to address cyber attacks against an organization's IT systems, it generally does not represent an agent for activating the DRP; thus DRP is the only BCP component of active interest to IT. == Testing == The overall categorization of tests are functional- and discussion-based. Types of tests include: tabletop exercises, checklists, simulations, parallel processing (testing recovery site while primary site is in operation), and full interruption (fail over) tests. These apply to both BC and DR. == Benefits == Like every insurance plan, there are benefits that can be obtained from proper business continuity planning, including: Studies have shown a correlation between higher spending on auditing fees and lower rates of Incidents. Minimizing risk of delays Guaranteeing the reliability of standby systems (even automating the failure detection and recovery in certain scenarios) Providing a standard for testing the plan Minimizing decision-making during a disaster Reducing potential legal liabilities Lowering unnecessarily stressful work environment === Planning and testing methodology === According to Geoffrey H. Wold of the Disaster Recovery Journal, the entire process involved in developing a Disaster Recovery Plan consists of 10 steps: Performing a risk assessment: The planning committee prepares a risk analysis and a business impact analysis (BIA) that includes a range of possible disasters. Each functional area of the organization is analyzed to determine potential consequences. Traditionally, fire has posed the greatest threat. A thorough plan provides for "worst case" situations, such as destruction of the main building. Establishing priorities for processing and operations: Critical needs of each department are evaluated and prioritized. Written agreements for alternatives selected are prepared, with details specifying duration, termination conditions, system testing, cost, any special security procedures, procedure for the notification of system changes, hours of operation, the specific hardware and other equipment required for processing, personnel requirements, definition of the circumstances constituting an emergency, process to negotiate service extensions, guarantee of compatibility, availability, non-mainframe resource requirements, priorities, and other contractual issues. Collecting data: This includes various lists (employee backup position listing, critical telephone numbers list, master call list, master vendor list, notification checklist), inventories (communications equipment, documentation, office equipment, forms, insurance policies, workgroup and data center computer hardware, microcomputer hardware and software, office supply, off-site storage location equipment, telephones, etc.), distribution register, software and data files backup/retention schedules, temporary location specifications, any other such lists, materials, inventories, and documentation. Pre-formatted forms are often used to facilitate the data gathering process. Organizing and documenting a written plan Developing testing criteria and procedures: reasons for testing include Determining the feasibility and compatibility of backup facilities and procedures. Identifying areas in the plan that need modification. Providing training to the team managers and team members. Demonstrating the ability of the organization to recover. Providing motivation for maintaining and updating the disaster recovery plan. Testing the plan: An initial "dry run" of the plan is performed by conducting a structured walk-through test. An actual test-run must be performed. Problems are corrected. Initial testing can be plan is done in sections and after normal business hours to minimize disruptions. Subsequent tests occur during normal business hours. === Caveats/controversie
Circular thresholding
Circular thresholding is an algorithm for automatic image threshold selection in image processing. Most threshold selection algorithms assume that the values (e.g. intensities) lie on a linear scale. However, some quantities such as hue and orientation are a circular quantity, and therefore require circular thresholding algorithms. The example shows that the standard linear version of Otsu's method when applied to the hue channel of an image of blood cells fails to correctly segment the large white blood cells (leukocytes). In contrast the white blood cells are correctly segmented by the circular version of Otsu's method. == Methods == There are a relatively small number of circular image threshold selection algorithms. The following examples are all based on Otsu's method for linear histograms: (Tseng, Li and Tung 1995) smooth the circular histogram, and apply Otsu's method. The histogram is cyclically rotated so that the selected threshold is shifted to zero. Otsu's method and histogram rotation are applied iteratively until several heuristics involving class size, threshold location, and class variance are satisfied. (Wu et al. 2006) smooth the circular histogram until it contains only two peaks. The histogram is cyclically rotated so that the midpoint between the peaks is shifted to zero. Otsu's method and histogram rotation are applied iteratively until convergence of the threshold. (Lai and Rosin 2014) applied Otsu's method to the circular histogram. For the two class circular thresholding task they showed that, for a histogram with an even number of bins, the optimal solution for Otsu's criterion of within-class variance is obtained when the histogram is split into two halves. Therefore the optimal solution can be efficiently obtained in linear rather than quadratic time. == References and further reading == D.-C. Tseng, Y.-F. Li, and C.-T. Tung, Circular histogram thresholding for color image segmentation in Proc. Int. Conf. Document Anal. Recognit., 1995, pp. 673–676. J. Wu, P. Zeng, Y. Zhou, and C. Olivier, A novel color image segmentation method and its application to white blood cell image analysis in Proc. Int. Conf. Signal Process., vol. 2. 2006, pp. 16–20. Y.K. Lai, P.L. Rosin, Efficient Circular Thresholding, IEEE Trans. on Image Processing 23(3), 992–1001 (2014). doi:10.1109/TIP.2013.2297014
Kurzsignale
The Short Signal Code, also known as the Short Signal Book (German: Kurzsignalbuch), was a short code system used by the Kriegsmarine (German Navy) during World War II to minimize the transmission duration of messages. == Description == The transmission of radio messages had the potential risks of revealing the submarine's presence and direction; if decoded the content was also revealed. Submarines need to provide information, mostly in standard form (position of convoy to attack and of submarine, weather information), to their bases. Initially Morse code transmissions could be used. To inhibit detection, the duration of messages needed to be minimised; for this, Kurzsignale short-coding was used. To prevent interception, messages needed to be encrypted by the Enigma machine. To shorten transmission even further, the message could be sent by a fast machine instead of a human radio operator. For example, the Kurier system – not implemented in time – decreased the time to send a Morse dot from around 50 milliseconds for a human to 1 millisecond. == Short Signal book == The Kurzsignale code was intended to shorten transmission time to below the time required to get a directional fix. It was not primarily intended to hide signal contents; protection was intended to be achieved by encoding with the Enigma machine. A copy of the Kurzsignale code book was captured from German submarine U-110 on 9 May 1941. In August 1941, Dönitz began addressing U-boats by the names of their commanders, instead of boat numbers. The method of defining U-boat meeting points in the Short Signal Book was regarded as compromised, so a method was defined by B-Dienst cryptanalysts to disguise their positions on the Kriegsmarine German Naval Grid System (German:Gradnetzmeldeverfahren) was introduced and used until the end of the war == Radio direction finding == Aware of the danger presented by radio direction finding (RDF), the Kriegsmarine developed various systems to speed up broadcast. The Kurzsignale code system condensed messages into short codes consisting of short sequences for common terms such as "convoy location" so that additional descriptions would not be needed in the message. The resulting Kurzsignal was then encoded with the Enigma machine and subsequently transmitted as rapidly as possible, typically taking about 20 seconds. Typical length of an information or weather signal was about 25 characters. Conventional RDF needed about a minute to fix the bearing of a radio signal, and the Kurzsignale protected against this. However, the huff-duff system which was in use by the Allies could cope with these short transmissions. The fully automated burst transmission Kurier system, in testing from August 1944, could send a Kurzsignal in not more than 460 milliseconds; this was short enough to prevent location even by huff-duff and, if deployed, would have been a serious setback for Allied anti-submarine and code-breaking activities. By late 1944 the Kurier program was a top priority, but the war ended before the system was operational. == Short Weather cipher == A similar coding system was used for weather reports from U-boats, the Wetterkurzschlüssel (Short Weather Cipher). Code books were captured from U-559 on 30 October 1942.
Embedded analytics
Embedded analytics enables organisations to integrate analytics capabilities into their own, often software as a service, applications, portals, or websites. This differs from embedded software and web analytics (also commonly known as product analytics). This integration typically provides contextual insights, quickly, easily and conveniently accessible since these insights should be present on the web page right next to the other, operational, parts of the host application. Insights are provided through interactive data visualisations, such as charts, diagrams, filters, gauges, maps and tables often in combination as dashboards embedded within the system. This setup enables easier, in-depth data analysis without the need to switch and log in between multiple applications. Embedded analytics is also known as customer facing analytics. Embedded analytics is the integration of analytic capabilities into a host, typically browser-based, business-to-business, software as a service, application. These analytic capabilities would typically be relevant and contextual to the use-case of the host application. == History == The term "embedded analytics" was first used by Howard Dresner: consultant, author, former Gartner analyst and inventor of the term "business intelligence" said Howard Dresner while he was working for Hyperion Solutions, a company that Oracle bought in 2007. Oracle started then to use the term "embedded analytics" at their press release for Oracle Rapid Planning on 2009 . == Considerations with embedded analytics == When evaluating embedding analytics, consideration would normally be given to integration at various levels, these would likely include: security integration, data integration, application logic integration, business rules integration, and user experience integration. This is in contrast to traditional BI, which expects users to leave their workflow applications to look at data insights in a separate set of tools. This immediacy makes embedded analytics much more intuitive and likely to be valued by users. A December 2016 report from Nucleus Research found that using BI tools, which require toggling between applications, can take up as much as 1–2 hours of an employee's time each week, whereas embedded analytics eliminate the need to toggle between apps.
G.hn
Gigabit Home Networking (G.hn) is a specification for wired home networking that supports speeds up to 2 Gbit/s and operates over four types of legacy wires: telephone wiring, coaxial cables, power lines and plastic optical fiber. Some benefits of a multi-wire standard are lower equipment development costs and lower deployment costs for service providers (by allowing customer self-install). == History == G.hn was developed under the International Telecommunication Union's Telecommunication Standardization sector (the ITU-T) and promoted by the HomeGrid Forum and several other organizations. ITU-T Recommendation (the ITU's term for standard) G.9960, which received approval on October 9, 2009, specified the physical layers and the architecture of G.hn. The Data Link Layer (Recommendation G.9961) was approved on June 11, 2010. Prominent organizations, including CEPca, HomePNA, and UPA, who were creators of some of these interfaces, rallied behind the latest version of the standard, emphasizing its potential and significance in the home networking domain. Moreover, the ITU-T extended the technology with multiple input, multiple output (MIMO) technology to increase data rates and signaling distance. This new feature was approved in March 2012 under G.9963 Recommendation. The development and promotion of G.hn have been significantly supported by the HomeGrid Forum and several other organizations. The technology was not only designed to address home-networking challenges but also found applications beyond this initial scope, showcasing its versatility and potential in the networking domain. == Technical specifications == === Technical overview === G.hn specifies a single physical layer based on fast Fourier transform (FFT) orthogonal frequency-division multiplexing (OFDM) modulation and low-density parity-check code (LDPC) forward error correction (FEC) code. G.hn includes the capability to notch specific frequency bands to avoid interference with amateur radio bands and other licensed radio services. G.hn includes mechanisms to avoid interference with legacy home networking technologies and also with other wireline systems such as VDSL2 or other types of DSL used to access the home. OFDM systems split the transmitted signal into multiple orthogonal sub-carriers. In G.hn each one of the sub-carriers is modulated using QAM. The maximum QAM constellation supported by G.hn is 4096-QAM (12-bit QAM). The G.hn media access control is based on a time division multiple access (TDMA) architecture, in which a "domain master" schedules Transmission Opportunities (TXOPs) that can be used by one or more devices in the "domain". There are two types of TXOPs: Contention-Free Transmission Opportunities (CFTXOP), which have a fixed duration and are allocated to a specific pair of transmitter and receiver. CFTXOP are used for implementing TDMA Channel Access for specific applications that require quality of service (QoS) guarantees. Shared Transmission Opportunities (STXOP), which are shared among multiple devices in the network. STXOP are divided into Time Slots (TS). There are two types of TS: Contention-Free Time Slots (CFTS), which are used for implementing "implicit" token passing Channel Access. In G.hn, a series of consecutive CFTS is allocated to a number of devices. The allocation is performed by the "domain master" and broadcast to all nodes in the network. There are pre-defined rules that specify which device can transmit after another device has finished using the channel. As all devices know "who is next", there is no need to explicitly send a "token" between devices. The process of "passing the token" is implicit and ensures that there are no collisions during Channel access. Contention-Based Time Slots (CBTS), which are used for implementing CSMA/CARP Channel Access. In general, CSMA systems cannot completely avoid collisions, so CBTS are only useful for applications that do not have strict Quality of Service requirements. ==== Optimization for each medium ==== Although most elements of G.hn are common for all three media supported by the standard (power lines, phone lines and coaxial cable), G.hn includes media-specific optimizations for each media. Some of these media-specific parameters include: OFDM Carrier Spacing: 195.31 kHz in coaxial, 48.82 kHz in phone lines, 24.41 kHz in power lines. FEC Rates: G.hn's FEC can operate with code rates 1/2, 2/3, 5/6, 16/18 and 20/21. Although these rates are not media specific, it is expected that the higher code rates will be used in cleaner media (such as coaxial) while the lower code rates will be used in noisy environments such as power lines. Automatic repeat request (ARQ) mechanisms: G.hn supports operation both with and without ARQ (re-transmission). Although this is not media specific, it is expected that ARQ-less operation is sometimes appropriate for cleaner media (such as coaxial) while ARQ operation is appropriate for noisy environments such as power lines. Power levels and frequency bands: G.hn defines different power masks for each medium. MIMO support: Recommendation G.9963 includes provisions for transmitting G.hn signals over multiple AC wires (phase, neutral, ground), if they are physically available. In July 2016, G.9963 was updated to include MIMO support over twisted pairs. ==== Security ==== G.hn uses the Advanced Encryption Standard (AES) encryption algorithm (with a 128-bit key length) using the CCMP protocol to ensure confidentiality and message integrity. Authentication and key exchange is done following ITU-T Recommendation X.1035. G.hn specifies point-to-point security inside a domain, which means that each pair of transmitter and receiver uses a unique encryption key which is not shared by other devices in the same domain. For example, if node Alice sends data to node Bob, node Eve (in the same domain as Alice and Bob) will not be able to easily eavesdrop their communication. G.hn supports the concept of relays, in which one device can receive a message from one node and deliver it to another node farther away in the same domain. Relaying becomes critical for applications with complex network topologies that need to cover large distances, such as those found in industrial or utility applications. While a relay can read the source and target addresses, it cannot read the message's content due to its body being end-to-end-encrypted. ==== Profiles ==== The G.hn architecture includes the concept of profiles. Profiles are intended to address G.hn nodes with significantly different levels of complexity. In G.hn the higher complexity profiles are proper supersets of lower complexity profiles, so that devices based on different profiles can interoperate with each other. Examples of G.hn devices based on high complexity profiles are Residential Gateways or Set-Top Boxes. Examples of G.hn devices based on low complexity profiles are home automation, home security and smart grid devices. ==== Technical parameters ==== The chart depicts a summary of the crucial technical specifications of the G.hn standard. Many of these technical elements are consistent across different physical media, with variations seen in areas such as Tone Spacing and frequency ranges. This uniformity is essential as it allows silicon manufacturers to produce a singular chip capable of implementing all three media types, leading to cost savings. Presently, G.hn chipsets are compatible with all three media types. This compatibility allows system manufacturers to create devices that can adjust to any wiring type simply by modifying a software configuration in the equipment. === Spectrum === The G.hn spectrum depends on the medium as shown in the diagram below: === Protocol stack === G.hn specifies the physical layer and the data link layer, according to the OSI model. The G.hn Data Link Layer (Recommendation G.9961) is divided into three sub-layers: The Application Protocol Convergence (APC) Layer, which accepts frames (usually in Ethernet format) from the upper layer (Application Entity) and encapsulates them into G.hn APC protocol data units (APDUs). The maximum payload of each APDU is 214 bytes. The logical link control (LLC), which is responsible for encryption, aggregation, segmentation and automatic repeat-request. This sub-layer is also responsible for "relaying" of APDUs between nodes that may not be able to communicate through a direct connection. The medium access control (MAC), which schedules channel access. The G.hn physical layer (Recommendation G.9960) is divided into three sub-layers: The Physical Coding Sub-layer (PCS), responsible for generating PHY headers. The Physical Medium Attachment (PMA), responsible for scrambling and forward error correction coding/decoding. The Physical Medium Dependent (PMD), responsible for bit-loading and OFDM modulation. The interface between the Application Entity and the Data Link Layer is called A-interface. The interface between the Data Link Layer and the ph
Yorba (software)
Yorba is a web-based personal information management platform for finding, monitoring, or deleting online accounts and subscriptions. Yorba is a participating member of Consumer Reports’ Data Rights Protocol (DRP) consortium that develops open technical standards for exercising consumer data rights under laws including the California Consumer Privacy Act. == History == Yorba began as a research project around 2021. It was founded by Chris Zeunstrom (CEO), Nolan Cabeje (CDO) and David Schmudde (CTO). Zeunstrom says he began developing Yorba after growing frustrated with managing numerous email accounts, noting overloaded inboxes create distraction and potential security vulnerabilities. Yorba’s early development was also influenced by security issues he encountered at a previous company, which had been affected by data breaches at a time when such incidents were becoming increasingly common. In 2023, Yorba launched a private beta as a public benefit corporation funded through a give-back model operated by Zeunstrom's New York-based design firm, Ruca. In January 2024, Yorba entered public beta and reported over 1,000 users, including 160 premium subscribers. At the time of the public beta launch, Yorba integrated with Gmail and announced plans to expand compatibility to other online services and cloud storage providers. In September 2024, Yorba completed conformance testing under the Data Rights Protocol, an initiative developed by Consumer Reports, to establish a standard and open-source framework for securely transmitting consumer data rights requests under laws like the California Consumer Privacy Act. Yorba was named among twelve participating companies that implemented the protocol alongside OneTrust and Consumer Reports’ own Permission Slip app. Yorba was one of nine startups selected as 2025 finalist in the Santander X Global Awards international entrepreneurship competition. == Features == Yorba scans user inbox history data to identify online accounts, mailing lists, and possible data breaches. It uses natural language processing and machine learning to identify a user's accounts, services, and subscriptions. The platform prompts password resets for compromised accounts and locates unused accounts. The platform also supports mailing list management by identifying and helping users unsubscribe from newsletters. Paid subscribers can locate and cancel recurring charges. Yorba links with financial institutions in the U.S., Canada, and EU via Plaid Inc. to detect recurring charges and delete unwanted subscriptions. == Privacy and Ethics == Yorba's founder has openly criticized dark patterns that make canceling services difficult, citing personal frustration with inbox clutter as part of his inspiration for Yorba. Yorba offers privacy policy analysis in partnership with Amsterdam-based nonprofit Terms of Service; Didn’t Read, assigning grades based on invasiveness or ethical concerns. As of 2024, the company described its pricing as designed to cover operational costs and sustain the platform without outside investment.
Symmetric Boolean function
In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input. For this reason they are also known as Boolean counting functions. There are 2n+1 symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map n+1 elements to two elements, f : { 0 , 1 , . . . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems. == Special cases == A number of special cases are recognized: Majority function: their value is 1 on input vectors with more than n/2 ones Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k All-equal and not-all-equal function: their values is 1 when the inputs do (not) all have the same value Exact-count functions: their value is 1 on input vectors with k ones for a fixed k One-hot or 1-in-n function: their value is 1 on input vectors with exactly one one One-cold function: their value is 1 on input vectors with exactly one zero Congruence functions: their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m Parity function: their value is 1 if the input vector has odd number of ones The n-ary versions of AND, OR, XOR, NAND, NOR and XNOR are also symmetric Boolean functions. == Properties == In the following, f k {\displaystyle f_{k}} denotes the value of the function f : { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}} when applied to an input vector of weight k {\displaystyle k} . === Weight === The weight of the function can be calculated from its value vector: | f | = ∑ k = 0 n ( n k ) f k {\displaystyle |f|=\sum _{k=0}^{n}{\binom {n}{k}}f_{k}} === Algebraic normal form === The algebraic normal form either contains all monomials of certain order m {\displaystyle m} , or none of them; i.e. the Möbius transform f ^ {\displaystyle {\hat {f}}} of the function is also a symmetric function. It can thus also be described by a simple (n+1) bit vector, the ANF vector f ^ m {\displaystyle {\hat {f}}_{m}} . The ANF and value vectors are related by a Möbius relation: f ^ m = ⨁ k 2 ⊆ m 2 f k {\displaystyle {\hat {f}}_{m}=\bigoplus _{k_{2}\subseteq m_{2}}f_{k}} where k 2 ⊆ m 2 {\displaystyle k_{2}\subseteq m_{2}} denotes all the weights k whose base-2 representation is covered by the base-2 representation of m (a consequence of Lucas’ theorem). Effectively, an n-variable symmetric Boolean function corresponds to a log(n)-variable ordinary Boolean function acting on the base-2 representation of the input weight. For example, for three-variable functions: f ^ 0 = f 0 f ^ 1 = f 0 ⊕ f 1 f ^ 2 = f 0 ⊕ f 2 f ^ 3 = f 0 ⊕ f 1 ⊕ f 2 ⊕ f 3 {\displaystyle {\begin{array}{lcl}{\hat {f}}_{0}&=&f_{0}\\{\hat {f}}_{1}&=&f_{0}\oplus f_{1}\\{\hat {f}}_{2}&=&f_{0}\oplus f_{2}\\{\hat {f}}_{3}&=&f_{0}\oplus f_{1}\oplus f_{2}\oplus f_{3}\end{array}}} So the three variable majority function with value vector (0, 0, 1, 1) has ANF vector (0, 0, 1, 0), i.e.: Maj ( x , y , z ) = x y ⊕ x z ⊕ y z {\displaystyle {\text{Maj}}(x,y,z)=xy\oplus xz\oplus yz} === Unit hypercube polynomial === The coefficients of the real polynomial agreeing with the function on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} are given by: f m ∗ = ∑ k = 0 m ( − 1 ) | k | + | m | ( m k ) f k {\displaystyle f_{m}^{}=\sum _{k=0}^{m}(-1)^{|k|+|m|}{\binom {m}{k}}f_{k}} For example, the three variable majority function polynomial has coefficients (0, 0, 1, -2): Maj ( x , y , z ) = ( x y + x z + y z ) − 2 ( x y z ) {\displaystyle {\text{Maj}}(x,y,z)=(xy+xz+yz)-2(xyz)} == Examples ==