Deluxe Media Inc., also known simply as Deluxe and formerly Deluxe Entertainment Services Group, Inc., is an American multinational multimedia and entertainment service provisions company owned by Platinum Equity, founded in 1915 by Hungarian-born American film producer William Fox and headquartered in Burbank, California. The company services multiple clients in the film, television, digital content and advertising industries across the globe, and has been recognized with 10 Academy Awards for scientific and technical achievements, including developments in CinemaScope pictures (as part of 20th Century Fox) and more recently for a process of creating archival separations from digital image data. == History == Deluxe began as a film processing laboratory established in 1915 by William Fox under the name De Luxe as part of his eponymous film conglomerate corporation in Fort Lee, New Jersey. In 1916, Fox Film Corporation opened its studio in Hollywood on 13 acres at Sunset and Western. The first Deluxe film laboratory on the west coast was built on the south side of the lot (Fernwood and Serrano), and the laboratory was moved to the new Fox studios building on Manhattan's west side in 1919, where it remained for over 40 years. The "business manager" (later president) of the laboratory was Alan E. Freedman, who guided the company into the 1960s. In 1927, Fox (Deluxe) received a patent for sound-on-film, the Fox Movietone system. In 1927, "Sunrise: A Song of Two Humans," an early Movietone film, opened. Fox Movietone News, ran weekly in theaters until 1963. During the Great Depression, Fox Film Corporation encountered financial difficulties. Among the actions taken to maintain liquidity, Fox sold the laboratories in 1932 to Freedman, who renamed the operation Deluxe. Under Freedman's leadership, Deluxe added two more plants in Chicago and Toronto. In January 1934, Fox was granted an option to rebuy DeLuxe before December 31, 1938. On 31 May 1935, under Sidney Kent, Fox merged his film company with Twentieth Century Pictures to form The Twentieth Century-Fox Film Corporation following a bank-infused reorganisation. The merged company then exercised this option in July 1936, with Freedman remaining as president. In 1953, Deluxe developed the widescreen format CinemaScope. Titles included "There's No Business Like Show Business" (1954) and "The Seven Year Itch" (1955). Other innovations included the processing and sound striping of CinemaScope, and were patented and/or received Academy awards. In 1962 Freedman retired. In the 1960s, Deluxe closed its New York plant, followed by its plants in Chicago and Toronto, as motion picture production declined on the East Coast. In 1972, Deluxe began large volume videocassette production, with a billion by 1996. In 1990, The Rank Organisation acquired Deluxe from Fox. In 2000, Deluxe began large volume DVD production. In 2006, The Rank Organisation sold Deluxe Film Group to MacAndrews & Forbes, renamed Deluxe Entertainment Services Group. On 9 February 2012, Deluxe acquired Hong Kong–based visual effects and post-production company, Centro Digital Pictures, with its founder John Chu remaining as president while reporting to Alaric McAusland, managing director for Deluxe in Australia. In May 2014, Deluxe shut down its Los Angeles plant at Sunset & Western Studios complex, where other studios themselves were demolished way back in 1971. Also that same year, Deluxe closed the Hollywood film labs, and they gave thousands of orphaned film elements to the Academy Film Archive. The Deluxe Laboratories Collection at the Academy Film Archive consists of over 7,500 35mm and 16mm film elements of various motion pictures dating back to the early 1960s. On 22 April 2015, Deluxe and its longtime competitor, Technicolor S.A., announced that they had entered into a binding agreement to create a new joint venture known as Deluxe Technicolor Digital Cinema which will specialize in cinema mastering, distribution and management services. Deluxe got acquired on 4 September 2019 by creditors in a debt-for-equity swap to avoid bankruptcy. On 3 October 2019, Deluxe filed for bankruptcy, pending in the Southern District of New York. The same month on the 24th, the company received court approval to emerge from bankruptcy with a comprehensive restructuring plan. On July 1, 2020, Platinum Equity agreed to acquire the distribution division of Deluxe and re-unite with former CEO Cyril Drabinsky who would merge CineVizion, a film distribution company he founded after leaving Deluxe in 2016, into it. The companies Company 3 and Method Studios which formed the creative divisions of Deluxe were sold to Framestore in November 2020.
Coherent extrapolated volition
Coherent extrapolated volition (CEV) is a theoretical framework in the field of AI alignment describing an approach by which an artificial superintelligence (ASI) would act on a benevolent supposition of what humans would want if they were more knowledgeable, more rational, had more time to think, and had matured together as a society, as opposed to humanity's current individual or collective preferences. It was proposed by Eliezer Yudkowsky in 2004 as part of his work on friendly AI. == Concept == CEV proposes that an advanced AI system should derive its goals by extrapolating the idealized volition of humanity. This means aggregating and projecting human preferences into a coherent utility function that reflects what people would desire under ideal epistemic and moral conditions. The aim is to ensure that AI systems are aligned with humanity's true interests, rather than with transient or poorly informed preferences. In poetic terms, our coherent extrapolated volition is our wish if we knew more, thought faster, were more the people we wished we were, had grown up farther together; where the extrapolation converges rather than diverges, where our wishes cohere rather than interfere; extrapolated as we wish that extrapolated, interpreted as we wish that interpreted. == Debate == Yudkowsky and Nick Bostrom note that CEV has several interesting properties. It is designed to be humane and self-correcting, by capturing the source of human values instead of trying to list them. It avoids the difficulty of laying down an explicit, fixed list of rules. It encapsulates moral growth, preventing flawed current moral beliefs from getting locked in. It limits the influence that a small group of programmers can have on what the ASI would value, thus also reducing the incentives to build ASI first. And it keeps humanity in charge of its destiny. CEV also faces significant theoretical and practical challenges. Bostrom notes that CEV has "a number of free parameters that could be specified in various ways, yielding different versions of the proposal." One such parameter is the extrapolation base (whose extrapolated volition is taken into account). For example, whether it should include people with severe dementia, patients in a vegetative state, foetuses, or embryos. He also notes that if CEV's extrapolation base only includes humans, there is a risk that the result would be ungenerous toward other animals and digital minds. One possible solution would be to include a mechanism to expand CEV's extrapolation base. == Variants and alternatives == A proposed theoretical alternative to CEV is to rely on an artificial superintelligence's superior cognitive capabilities to figure out what is morally right, and let it act accordingly. It is also possible to combine both techniques, for instance with the ASI following CEV except when it is morally impermissible. In another review, a philosophical analysis explores CEV through the lens of social trust in autonomous systems. Drawing on Anthony Giddens' concept of "active trust", the author proposes an evolution of CEV into "Coherent, Extrapolated and Clustered Volition" (CECV). This formulation aims to better reflect the moral preferences of diverse cultural groups, thus offering a more pragmatic ethical framework for designing AI systems that earn public trust while accommodating societal diversity.
Novell File Reporter
Novell File Reporter (NFR) is software that allows network administrators to identify files stored on the network and generates reports regarding the size of individual files, file type, when files were last accessed, and where duplicates exist. Additionally, the File Reporter tracks storage volume capacity and usage. It is a component of the Novell File Management Suite. == How it works == Novell File Reporter examines and reports on terabytes of data via a central reporting engine (NFR Engine) and distributed agents (NFR Agents). The NFR Engine schedules the scans of file instances conducted by NFR Agents, processes and compiles the scans for reporting purposes, and provides report information to the user interface. In addition to the standard reports it can generate, the NFR Engine can also produce "trigger reports" in response to specific events (a server volume crossing a capacity threshold, for example). Accordingly, the NFR Engine monitors the data gathered by the NFR Agents in order to identify these "triggers." The NFR Engine when working in either eDirectory or Active Directory connects to the directory via a Directory Services Interface (DSI) and thus can monitor and check file permissions.
Paper data storage
Paper data storage refers to the use of paper as a data storage device. This includes writing, illustrating, and the use of data that can be interpreted by a machine or is the result of the functioning of a machine. A defining feature of paper data storage is the ability of humans to produce it with only simple tools and interpret it visually. Though now mostly obsolete, paper was once an important form of computer data storage as both paper tape and punch cards were a common staple of working with computers before the 1980s. == History == Before paper was used for storing data, it had been used in several applications for storing instructions to specify a machine's operation. The earliest use of paper to store instructions for a machine was the work of Basile Bouchon who, in 1725, used punched paper rolls to control textile looms. This technology was later developed into the wildly successful Jacquard loom. The 19th century saw several other uses of paper for controlling machines. In 1846, telegrams could be prerecorded on punched tape and rapidly transmitted using Alexander Bain's automatic telegraph. Several inventors took the concept of a mechanical organ and used paper to represent the music. In the late 1880s Herman Hollerith invented the recording of data on a medium that could then be read by a machine. Prior uses of machine readable media, above, had been for control (automatons, piano rolls, looms, ...), not data. "After some initial trials with paper tape, he settled on punched cards..." Hollerith's method was used in the 1890 census. Hollerith's company eventually became the core of IBM. Other technologies were also developed that allowed machines to work with marks on paper instead of punched holes. This technology was widely used for tabulating votes and grading standardized tests. Banks used magnetic ink on checks, supporting MICR scanning. In an early electronic computing device, the Atanasoff–Berry Computer, electric sparks were used to singe small holes in paper cards to represent binary data. The altered dielectric constant of the paper at the location of the holes could then be used to read the binary data back into the machine by means of electric sparks of lower voltage than the sparks used to create the holes. This form of paper data storage was never made reliable and was not used in any subsequent machine. == Modern techniques == === 1D barcodes === Barcodes make it possible for any object that was to be sold or transported to have some computer readable information securely attached to it. Universal Product Code barcodes, first used in 1974, are ubiquitous today. Some people recommend a width of at least 3 pixels for each minimum-width gap and each minimum-width bar for 1D barcodes. The density is about 50 bits per linear inch (about 2 bit/mm). === 2D barcodes === 2D barcodes allow to store much more data on paper, up to 2.9 kbyte per barcode. It is recommended to have a width of at least 4 pixels—e.g., a 4 × 4 pixel = 16 pixel module. == Limits == The limits of data storage depend on the technology to write and read such data. The theoretical limits assume a scanner that can perfectly reproduce the printed image at its printing resolution, and a program which can accurately interpret such an image. For example, an 8 in × 10 in (200 mm × 250 mm) 600 dpi black-and-white image contains 3.43 MiB of data, as does a 300 dpi CMYK printed image. A 2,400 ppi True color (24-bit) image contains about 1.29 GiB of information; printing an image maintaining this data would require a printing resolution of about 120,000 dpi in black and white, or 60,000 dpi with CMYK dots.
Knowledge organization system
Knowledge organization system (KOS), concept system, or concept scheme is the generic term used in knowledge organization (KO) for the selection of concepts with an indication of selected semantic relations. Despite their differences in type, coverage, and application, all KOS aim to support the organization of knowledge and information to facilitate their management and retrieval. KOS vary in complexity from simple sorted lists to complex relational networks. They represent both structural and functional features, and serve to eliminate ambiguity, control synonyms, establish relationships, and present properties. From their origins in library and information science (LIS), KOS have been applied to other domains and disciplines within science and industry, although scholarly research and debate remain primarily within the KO field. Challenges of KOS include ambiguity of terminology, repercussions of biased systems, and potential obsolescence. KOS can be expressed in RDF and RDFS as per the Simple Knowledge Organization System (SKOS) recommendation by W3C, which aims to enable the sharing and linking of KOS via the Web. One of the largest collections of KOS is the BARTOC registry. == Types == While different schema of KOS have been proposed, most are generally arranged in terms of the complexity of their construction and maintenance. Some scholars argue that organizing KOS on a spectrum oversimplifies the shared characteristics among them, and may even result in a non-ideal structure being chosen. The following types are not exhaustive, and are often not mutually-exclusive in practice. === Term lists === Term lists are the least structured form of KOS. They include lists, glossaries, dictionaries, and synonym rings. Authority files and gazetteers may also be considered term lists, however other scholars categorize them and directories as "metadata-like models". Examples include the Union List of Artist Names name authority file and the GeoNames gazetteer. === Categorization and classification === KOS that emphasize specific (and often hierarchical) structures include subject headings, taxonomies, categorization schema, and classification schema & systems. Despite inconsistent use of the terms "categorization" and "classification" in some literature, categorization is generally loosely-assembled grouping schema and may include attributes that are not mutually exclusive (or having fuzzy boundaries), while classification is related to the arrangement of non-overlapping and mutually-exclusive classes. Classification schema may be universal (such as Dewey Decimal Classification and Information Coding Classification) or domain-specific (such as the National Library of Medicine Classification). === Relationship models === The types of KOS with greatest complexity and which utilize connections between concepts include thesauri, semantic networks, and ontologies. One of the most prominent examples of a semantic network is WordNet. === Others === Certain structures proposed to be considered types of KOS—but are not consistently included in schema—include folksonomies, topic maps, web directory structures, publication organization systems, and bibliometric maps. Some KOS organize other KOS themselves—for instance, PeriodO is a gazetteer of periodization categories. == Applications == Some early KOS were developed as a support system for abstracting and indexing services to be used by specially-trained searchers. With the growth of information digitization, usability became increasingly accessible, and more complex structures were developed. Prominent examples of KOS outside of LIS include organism taxonomy in biology, the periodic table of elements in chemistry, SIC and NAICS classification systems for industry & business, and AGROVOC agricultural controlled vocabulary. == Challenges == The study and design of KOS is an ongoing topic of discussion among KO scholars. === Terminology === [There is] a serious lack of vocabulary control in the literature on controlled vocabulary. Inconsistency of terminology within the study of KOS is a common issue. For instance, "ontology" is used for both a specific type of KOS as well as a generic term for any KOS. The terms "taxonomy", "classification", and "categorization" are also sometimes used interchangeably. === Bias === As knowledge can be historically and culturally biased, scholars have also discussed how KOS themselves can perpetuate harmful practices or stereotypes. For example, a number of concerns and criticisms about the classification of mental disorders in the Diagnostic and Statistical Manual of Mental Disorders have been raised, contributing to ongoing revisions. Ethical and intentional design approaches have been proposed for multi-perspective KOS in efforts to mitigate bias and other harmful practices. === Obsolescence === The possible obsolescence of the thesaurus and other simpler KOS has been the topic of debate, especially in the face of increasingly complex ontologies, the growing usage of "Google-like retrieval systems", and the move of KO theory and research away from LIS and toward computer science. Supporters of thesauri argue its continued usefulness for metadata enrichment, vocabulary mapping, and web services, as well as its usage in specific domains such as corporate intranets and digital image libraries.
VoxForge
VoxForge is a free speech corpus and acoustic model repository for open source speech recognition engines. VoxForge was set up to collect transcribed speech to create a free GPL speech corpus in order to be uses with open source speech recognition engines. The speech audio files will be 'compiled' into acoustic models for use with open source speech recognition engines such as Julius, ISIP, and Sphinx and HTK (note: HTK has distribution restrictions). VoxForge has used LibriVox as a source of audio data since 2007.
Berlekamp–Rabin algorithm
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a