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67-511: In 2024 NIST announced the Module-Lattice-Based Digital Signature Standard for post-quantum cryptography. In 1996, Miklós Ajtai introduced the first lattice-based cryptographic construction whose security could be based on the hardness of well-studied lattice problems, and Cynthia Dwork showed that a certain average-case lattice problem, known as short integer solutions (SIS), is at least as hard to solve as

134-431: A i b i : a i ∈ Z } . {\displaystyle L={\Big \{}\sum a_{i}\mathbf {b} _{i}:a_{i}\in \mathbb {Z} {\Big \}}.} For example, Z n {\displaystyle \mathbb {Z} ^{n}} is a lattice, generated by the standard basis for R n {\displaystyle \mathbb {R} ^{n}} . Crucially,

201-534: A hat to emphasize their status as unit vectors ( standard unit vectors ). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},}

268-522: A metrology agency, the Bureau of Standards was directed by Herbert Hoover to set up divisions to develop commercial standards for materials and products. Some of these standards were for products intended for government use, but product standards also affected private-sector consumption. Quality standards were developed for products including some types of clothing, automobile brake systems and headlamps, antifreeze , and electrical safety. During World War I ,

335-470: A neutron science user facility: the NIST Center for Neutron Research (NCNR). The NCNR provides scientists access to a variety of neutron scattering instruments, which they use in many research fields (materials science, fuel cells, biotechnology, etc.). The SURF III Synchrotron Ultraviolet Radiation Facility is a source of synchrotron radiation , in continuous operation since 1961. SURF III now serves as

402-426: A ring R , which are zero except for a finite number of indices , if we interpret 1 as 1 R , the unit in R . The existence of other 'standard' bases has become a topic of interest in algebraic geometry , beginning with work of Hodge from 1943 on Grassmannians . It is now a part of representation theory called standard monomial theory . The idea of standard basis in the universal enveloping algebra of

469-464: A worst-case lattice problem. She then showed a cryptographic hash function whose security is equivalent to the computational hardness of SIS. In 1998, Jeffrey Hoffstein , Jill Pipher , and Joseph H. Silverman introduced a lattice-based public-key encryption scheme, known as NTRU . However, their scheme is not known to be at least as hard as solving a worst-case lattice problem. The first lattice-based public-key encryption scheme whose security

536-569: A NIST team as part of a DARPA competition. In September 2013, both The Guardian and The New York Times reported that NIST allowed the National Security Agency (NSA) to insert a cryptographically secure pseudorandom number generator called Dual EC DRBG into NIST standard SP 800-90 that had a kleptographic backdoor that the NSA can use to covertly predict the future outputs of this pseudorandom number generator thereby allowing

603-665: A combination of vacuum tubes and solid-state diode logic. About the same time the Standards Western Automatic Computer , was built at the Los Angeles office of the NBS by Harry Huskey and used for research there. A mobile version, DYSEAC , was built for the Signal Corps in 1954. Due to a changing mission, the "National Bureau of Standards" became the "National Institute of Standards and Technology" in 1988. Following

670-485: A draft of the CSF 2.0 for public comment through November 4, 2023. NIST decided to update the framework to make it more applicable to small and medium size enterprises that use the framework, as well as to accommodate the constantly changing nature of cybersecurity. In August 2024, NIST released a final set of encryption tools designed to withstand the attack of a quantum computer. These post-quantum encryption standards secure

737-410: A part of Libgcrypt , according to Falco Strenzke. Lattice-based cryptographic constructions hold a great promise for public-key post-quantum cryptography . Indeed, the main alternative forms of public-key cryptography are schemes based on the hardness of factoring and related problems and schemes based on the hardness of the discrete logarithm and related problems . However, both factoring and

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804-533: A program to provide metrology services for United States scientific and commercial users. A laboratory site was constructed in Washington, DC , and instruments were acquired from the national physical laboratories of Europe. In addition to weights and measures, the Bureau developed instruments for electrical units and for measurement of light. In 1905 a meeting was called that would be the first "National Conference on Weights and Measures". Initially conceived as purely

871-706: A selection of lattice-based schemes for the purpose of digital signatures. CRYSTALS-Dilithium or simply Dilithium is built upon module-LWE and module-SIS. Dilithium was selected by the NIST as the basis for a digital signature standard. According to a message from Ray Perlner, writing on behalf of the NIST PQC team, the NIST module-LWE signing standard is to be based on version 3.1 of the Dilithium specification. NIST's changes on Dilithium 3.1 intend to support additional randomness in signing (hedged signing) and other improvements. Dilithium

938-644: A user-accessible cleanroom nanomanufacturing facility. This "NanoFab" is equipped with tools for lithographic patterning and imaging (e.g., electron microscopes and atomic force microscopes ). NIST has seven standing committees: As part of its mission, NIST supplies industry, academia, government, and other users with over 1,300 Standard Reference Materials (SRMs). These artifacts are certified as having specific characteristics or component content, used as calibration standards for measuring equipment and procedures, quality control benchmarks for industrial processes, and experimental control samples. NIST publishes

1005-543: A wide range of electronic information, from confidential email messages to e-commerce transactions that propel the modern economy. Four scientific researchers at NIST have been awarded Nobel Prizes for work in physics : William Daniel Phillips in 1997, Eric Allin Cornell in 2001, John Lewis Hall in 2005 and David Jeffrey Wineland in 2012, which is the largest number for any US government laboratory not accounting for ubiquitous government contracts to state institutions and

1072-469: Is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical science laboratory programs that include nanoscale science and technology , engineering , information technology , neutron research, material measurement, and physical measurement. From 1901 to 1988, the agency

1139-502: Is any set and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta , equal to zero whenever i ≠ j and equal to 1 if i = j . This family is the canonical basis of the R -module ( free module ) R ( I ) {\displaystyle R^{(I)}} of all families f = ( f i ) {\displaystyle f=(f_{i})} from I into

1206-1070: Is commonly called monomial basis . For matrices M m × n {\displaystyle {\mathcal {M}}_{m\times n}} , the standard basis consists of the m × n -matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices e 11 = ( 1 0 0 0 ) , e 12 = ( 0 1 0 0 ) , e 21 = ( 0 0 1 0 ) , e 22 = ( 0 0 0 1 ) . {\displaystyle \mathbf {e} _{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\quad \mathbf {e} _{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {e} _{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad \mathbf {e} _{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.} By definition,

1273-459: Is in fact hard in this regime. This section presents selected lattice-based schemes, grouped by primitive. Selected schemes for the purpose of encryption: Selected schemes for the purpose of homomorphic encryption : Selected lattice-based cryptographic schemes for the purpose of hashing: Selected schemes for the purpose of key exchange, also called key establishment, key encapsulation and key encapsulation mechanism (KEM): This section lists

1340-632: Is now the Handbook 44 since 1918 and began publication under the current name in 1949. The 2010 edition conforms to the concept of the primary use of the SI (metric) measurements recommended by the Omnibus Foreign Trade and Competitiveness Act of 1988 . NIST is developing government-wide identity document standards for federal employees and contractors to prevent unauthorized persons from gaining access to government buildings and computer systems. In 2002,

1407-575: Is providing practical guidance and tools to better prepare facility owners, contractors, architects, engineers, emergency responders, and regulatory authorities to respond to future disasters. The investigation portion of the response plan was completed with the release of the final report on 7 World Trade Center on November 20, 2008. The final report on the WTC Towers—including 30 recommendations for improving building and occupant safety—was released on October 26, 2005. NIST works in conjunction with

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1474-434: Is the shortest vector problem (SVP or sometimes GapSVP), which asks us to approximate the minimal Euclidean length of a non-zero lattice vector. This problem is thought to be hard to solve efficiently, even with approximation factors that are polynomial in n {\displaystyle n} , and even with a quantum computer. Many (though not all) lattice-based cryptographic constructions are known to be secure if SVP

1541-495: The worst-case hardness of certain lattice problems. I.e., if there exists an algorithm that can efficiently break the cryptographic scheme with non-negligible probability, then there exists an efficient algorithm that solves a certain lattice problem on any input. However, for the practical lattice-based constructions (such as schemes based on NTRU and even schemes based on LWE with efficient parameters), meaningful reduction-based guarantees of security are not known. Assessments of

1608-513: The Biden administration began plans to create a U.S. AI Safety Institute within NIST to coordinate AI safety matters. According to The Washington Post , NIST is considered "notoriously underfunded and understaffed", which could present an obstacle to these efforts. NIST, known between 1901 and 1988 as the National Bureau of Standards (NBS), is a measurement standards laboratory , also known as

1675-512: The Constitution of the United States , ratified in 1789, granted these powers to the new Congress: "The Congress shall have power ... To coin money, regulate the value thereof, and of foreign coin, and fix the standard of weights and measures". In January 1790, President George Washington , in his first annual message to Congress , said, "Uniformity in the currency, weights, and measures of

1742-729: The Handbook 44 each year after the annual meeting of the National Conference on Weights and Measures (NCWM). Each edition is developed through cooperation of the Committee on Specifications and Tolerances of the NCWM and the Weights and Measures Division (WMD) of NIST. The purpose of the book is a partial fulfillment of the statutory responsibility for "cooperation with the states in securing uniformity of weights and measures laws and methods of inspection". NIST has been publishing various forms of what

1809-545: The National Construction Safety Team Act mandated NIST to conduct an investigation into the collapse of the World Trade Center buildings 1 and 2 and the 47-story 7 World Trade Center. The "World Trade Center Collapse Investigation", directed by lead investigator Shyam Sunder, covered three aspects, including a technical building and fire safety investigation to study the factors contributing to

1876-543: The National Medal of Science has been awarded to NIST researchers Cahn (1998) and Wineland (2007). Other notable people who have worked at NBS or NIST include: Since 1989, the director of NIST has been a Presidential appointee and is confirmed by the United States Senate , and since that year the average tenure of NIST directors has fallen from 11 years to 2 years in duration. Since the 2011 reorganization of NIST,

1943-717: The September 11, 2001 attacks, under the National Construction Safety Team Act (NCST), NIST conducted the official investigation into the collapse of the World Trade Center buildings. Following the 2021 Surfside condominium building collapse , NIST sent engineers to the site to investigate the cause of the collapse. In 2019, NIST launched a program named NIST on a Chip to decrease the size of instruments from lab machines to chip size. Applications include aircraft testing, communication with satellites for navigation purposes, and temperature and pressure. In 2023,

2010-812: The Technical Guidelines Development Committee of the Election Assistance Commission to develop the Voluntary Voting System Guidelines for voting machines and other election technology. In February 2014 NIST published the NIST Cybersecurity Framework that serves as voluntary guidance for organizations to manage and reduce cybersecurity risk. It was later amended and Version 1.1 was published in April 2018. Executive Order 13800, Strengthening

2077-806: The Treaty of the Meter , which established the International Bureau of Weights and Measures under the control of an international committee elected by the General Conference on Weights and Measures . NIST is headquartered in Gaithersburg, Maryland , and operates a facility in Boulder, Colorado , which was dedicated by President Eisenhower in 1954. NIST's activities are organized into laboratory programs and extramural programs. Effective October 1, 2010, NIST

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2144-667: The proximity fuze and the standardized airframe used originally for Project Pigeon , and shortly afterwards the autonomously radar-guided Bat anti-ship guided bomb and the Kingfisher family of torpedo-carrying missiles. In 1948, financed by the United States Air Force, the Bureau began design and construction of SEAC , the Standards Eastern Automatic Computer. The computer went into operation in May 1950 using

2211-619: The scalars v x {\displaystyle v_{x}} ,  v y {\displaystyle v_{y}} ,  v z {\displaystyle v_{z}} being the scalar components of the vector v . In the n - dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the standard basis consists of n distinct vectors { e i : 1 ≤ i ≤ n } , {\displaystyle \{\mathbf {e} _{i}:1\leq i\leq n\},} where e i denotes

2278-406: The Bureau worked on multiple problems related to war production, even operating its own facility to produce optical glass when European supplies were cut off. Between the wars, Harry Diamond of the Bureau developed a blind approach radio aircraft landing system. During World War II, military research and development was carried out, including development of radio propagation forecast methods,

2345-693: The Cybersecurity of Federal Networks and Critical Infrastructure , made the Framework mandatory for U.S. federal government agencies. An extension to the NIST Cybersecurity Framework is the Cybersecurity Maturity Model (CMMC) which was introduced in 2019 (though the origin of CMMC began with Executive Order 13556). It emphasizes the importance of implementing Zero-trust architecture (ZTA) which focuses on protecting resources over

2412-692: The EC-DRBG algorithm from the NIST SP 800-90 standard. In addition to these journals, NIST (and the National Bureau of Standards before it) has a robust technical reports publishing arm. NIST technical reports are published in several dozen series, which cover a wide range of topics, from computer technology to construction to aspects of standardization including weights, measures and reference data. In addition to technical reports, NIST scientists publish many journal and conference papers each year; an database of these, along with more recent technical reports, can be found on

2479-531: The NIST cryptography process because of its recognized expertise. NIST is also required by statute to consult with the NSA." Recognizing the concerns expressed, the agency reopened the public comment period for the SP800-90 publications, promising that "if vulnerabilities are found in these or any other NIST standards, we will work with the cryptographic community to address them as quickly as possible". Due to public concern of this cryptovirology attack, NIST rescinded

2546-441: The NIST website. Standard basis In mathematics , the standard basis (also called natural basis or canonical basis ) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in

2613-587: The National Metrological Institute (NMI), which is a non-regulatory agency of the United States Department of Commerce . The institute's official mission is to: Promote U.S. innovation and industrial competitiveness by advancing measurement science , standards , and technology in ways that enhance economic security and improve our quality of life . NIST had an operating budget for fiscal year 2007 (October 1, 2006 – September 30, 2007) of about $ 843.3 million. NIST's 2009 budget

2680-508: The US national standard for source-based radiometry throughout the generalized optical spectrum. All NASA -borne, extreme-ultraviolet observation instruments have been calibrated at SURF since the 1970s, and SURF is used for the measurement and characterization of systems for extreme ultraviolet lithography . The Center for Nanoscale Science and Technology (CNST) performs research in nanotechnology , both through internal research efforts and by running

2747-481: The United States is an object of great importance, and will, I am persuaded, be duly attended to." On October 25, 1791, Washington again appealed Congress: A uniformity of the weights and measures of the country is among the important objects submitted to you by the Constitution and if it can be derived from a standard at once invariable and universal, must be no less honorable to the public council than conducive to

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2814-795: The axes of the Cartesian coordinate system , so the basis with these vectors does not meet the definition of standard basis. There is a standard basis also for the ring of polynomials in n indeterminates over a field , namely the monomials . All of the preceding are special cases of the indexed family ( e i ) i ∈ I = ( ( δ i j ) j ∈ I ) i ∈ I {\displaystyle {(e_{i})}_{i\in I}=((\delta _{ij})_{j\in I})_{i\in I}} where I {\displaystyle I}

2881-495: The basis for a lattice is not unique. For example, the vectors ( 3 , 1 , 4 ) {\displaystyle (3,1,4)} , ( 1 , 5 , 9 ) {\displaystyle (1,5,9)} , and ( 2 , − 1 , 0 ) {\displaystyle (2,-1,0)} form an alternative basis for Z 3 {\displaystyle \mathbb {Z} ^{3}} . The most important lattice-based computational problem

2948-476: The case of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} formed by the pairs ( x , y ) of real numbers , the standard basis is formed by the vectors e x = ( 1 , 0 ) , e y = ( 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1).} Similarly,

3015-509: The country. NIST publishes the Handbook 44 that provides the "Specifications, tolerances, and other technical requirements for weighing and measuring devices". The Congress of 1866 made use of the metric system in commerce a legally protected activity through the passage of Metric Act of 1866 . On May 20, 1875, 17 out of 20 countries signed a document known as the Metric Convention or

3082-488: The director also holds the title of Under Secretary of Commerce for Standards and Technology. Fifteen individuals have officially held the position (in addition to four acting directors who have served on a temporary basis). NIST holds patents on behalf of the Federal government of the United States , with at least one of them being custodial to protect public domain use, such as one for a Chip-scale atomic clock , developed by

3149-407: The discrete logarithm problem are known to be solvable in polynomial time on a quantum computer . Furthermore, algorithms for factorization tend to yield algorithms for discrete logarithm, and conversely. This further motivates the study of constructions based on alternative assumptions, such as the hardness of lattice problems. Many lattice-based cryptographic schemes are known to be secure assuming

3216-601: The first fully homomorphic encryption scheme, which was based on a lattice problem. In linear algebra , a lattice L ⊂ R n {\displaystyle L\subset \mathbb {R} ^{n}} is the set of all integer linear combinations of vectors from a basis { b 1 , … , b n } {\displaystyle \{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}} of R n {\displaystyle \mathbb {R} ^{n}} . In other words, L = { ∑

3283-409: The national physical laboratory for the United States. Southard had previously sponsored a bill for metric conversion of the United States. President Theodore Roosevelt appointed Samuel W. Stratton as the first director. The budget for the first year of operation was $ 40,000. The Bureau took custody of the copies of the kilogram and meter bars that were the standards for US measures, and set up

3350-429: The network perimeter. ZTA utilizes zero trust principles which include "never trust, always verify", "assume breach" and "least privileged access" to safeguard users, assets, and resources. Since ZTA holds no implicit trust to users within the network perimeter, authentication and authorization are performed at every stage of a digital transaction. This reduces the risk of unauthorized access to resources. NIST released

3417-532: The private sector. All four were recognized for their work related to laser cooling of atoms, which is directly related to the development and advancement of the atomic clock. In 2011, Dan Shechtman was awarded the Nobel Prize in chemistry for his work on quasicrystals in the Metallurgy Division from 1982 to 1984. In addition, John Werner Cahn was awarded the 2011 Kyoto Prize for Materials Science, and

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3484-476: The probable cause of the collapses of the WTC Towers (WTC 1 and 2) and WTC 7. NIST also established a research and development program to provide the technical basis for improved building and fire codes, standards, and practices, and a dissemination and technical assistance program to engage leaders of the construction and building community in implementing proposed changes to practices, standards, and codes. NIST also

3551-488: The provable security results for such systems do not provide any meaningful concrete security for practical values of the parameters. For many cryptographic primitives, the only known constructions are based on lattices or closely related objects. These primitives include fully homomorphic encryption , indistinguishability obfuscation , cryptographic multilinear maps , and functional encryption . NIST The National Institute of Standards and Technology ( NIST )

3618-477: The public convenience. In 1821, President John Quincy Adams declared, "Weights and measures may be ranked among the necessities of life to every individual of human society.". Nevertheless, it was not until 1838 that the United States government adopted a uniform set of standards. From 1830 until 1901, the role of overseeing weights and measures was carried out by the Office of Standard Weights and Measures, which

3685-421: The security levels provided by reduction arguments from hard problems - based on recommended parameter sizes, standard estimates of the computational complexity of the hard problems, and detailed examination of the steps in the reductions - are called concrete security and sometimes practice-oriented provable security . Authors who have investigated concrete security for lattice-based cryptosystems have found that

3752-489: The standard basis for the three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} is formed by vectors e x = ( 1 , 0 , 0 ) , e y = ( 0 , 1 , 0 ) , e z = ( 0 , 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).} Here

3819-762: The standard basis is a sequence of orthogonal unit vectors . In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e. , v 1 = ( 3 2 , 1 2 ) {\displaystyle v_{1}=\left({{\sqrt {3}} \over 2},{1 \over 2}\right)\,} v 2 = ( 1 2 , − 3 2 ) {\displaystyle v_{2}=\left({1 \over 2},{-{\sqrt {3}} \over 2}\right)\,} are also orthogonal unit vectors, but they are not aligned with

3886-431: The standard by NSA). NIST responded to the allegations, stating that "NIST works to publish the strongest cryptographic standards possible" and that it uses "a transparent, public process to rigorously vet our recommended standards". The agency stated that "there has been some confusion about the standards development process and the role of different organizations in it...The National Security Agency (NSA) participates in

3953-415: The surreptitious decryption of data. Both papers report that the NSA worked covertly to get its own version of SP 800-90 approved for worldwide use in 2006. The whistle-blowing document states that "eventually, NSA became the sole editor". The reports confirm suspicions and technical grounds publicly raised by cryptographers in 2007 that the EC-DRBG could contain a kleptographic backdoor (perhaps placed in

4020-421: The vector e x points in the x direction, the vector e y points in the y direction, and the vector e z points in the z direction. There are several common notations for standard-basis vectors, including { e x ,  e y ,  e z }, { e 1 ,  e 2 ,  e 3 }, { i ,  j ,  k }, and { x ,  y ,  z }. These vectors are sometimes written with

4087-417: The vector with a 1 in the i th coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces , whose definition involves coefficients , such as polynomials and matrices . In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and

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4154-515: Was $ 992 million, and it also received $ 610 million as part of the American Recovery and Reinvestment Act . NIST employs about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 1,800 NIST associates (guest researchers and engineers from American companies and foreign countries) complement the staff. In addition, NIST partners with 1,400 manufacturing specialists and staff at nearly 350 affiliated centers around

4221-541: Was named the National Bureau of Standards . The Articles of Confederation , ratified by the colonies in 1781, provided: The United States in Congress assembled shall also have the sole and exclusive right and power of regulating the alloy and value of coin struck by their own authority, or by that of the respective states—fixing the standards of weights and measures throughout the United States. Article 1, section 8, of

4288-405: Was one of the two digital signature schemes initially chosen by the NIST in their post-quantum cryptography process, the other one being SPHINCS⁺, which is not based on lattices but on hashes. In August 2023, NIST published FIPS 204 (Initial Public Draft), and started calling Dilithium as Module-Lattice-Based Digital Signature Algorithm (ML-DSA). As of October 2023, ML-DSA was being implemented as

4355-690: Was part of the Survey of the Coast—renamed the United States Coast Survey in 1836 and the United States Coast and Geodetic Survey in 1878—in the United States Department of the Treasury . In 1901, in response to a bill proposed by Congressman James H. Southard (R, Ohio), the National Bureau of Standards was founded with the mandate to provide standard weights and measures, and to serve as

4422-444: Was proven under worst-case hardness assumptions was introduced by Oded Regev in 2005, together with the learning with errors problem (LWE). Since then, much follow-up work has focused on improving Regev's security proof and improving the efficiency of the original scheme. Much more work has been devoted to constructing additional cryptographic primitives based on LWE and related problems. For example, in 2009, Craig Gentry introduced

4489-619: Was realigned by reducing the number of NIST laboratory units from ten to six. NIST Laboratories include: Extramural programs include: NIST's Boulder laboratories are best known for NIST‑F1 , which houses an atomic clock . NIST‑F1 serves as the source of the nation's official time. From its measurement of the natural resonance frequency of cesium —which defines the second —NIST broadcasts time signals via longwave radio station WWVB near Fort Collins , Colorado, and shortwave radio stations WWV and WWVH , located near Fort Collins and Kekaha, Hawaii , respectively. NIST also operates

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