Misplaced Pages

#829170

103-535: The National Institute of Standards and Technology (NIST) proposed DSA for use in their Digital Signature Standard (DSS) in 1991, and adopted it as FIPS 186 in 1994. Five revisions to the initial specification have been released. The newest specification is: FIPS 186-5 from February 2023. DSA is patented but NIST has made this patent available worldwide royalty-free. Specification FIPS 186-5 indicates DSA will no longer be approved for digital signature generation, but may be used to verify signatures generated prior to

206-405: A s i − 1 + b t i − 1 ) − ( a s i + b t i ) q i = ( a s i − 1 − a s i q i ) + ( b t i − 1 − b t i q i ) =

309-409: A s i + 1 + b t i + 1 . {\displaystyle r_{i+1}=r_{i-1}-r_{i}q_{i}=(as_{i-1}+bt_{i-1})-(as_{i}+bt_{i})q_{i}=(as_{i-1}-as_{i}q_{i})+(bt_{i-1}-bt_{i}q_{i})=as_{i+1}+bt_{i+1}.} Thus s k {\displaystyle s_{k}} and t k {\displaystyle t_{k}} are Bézout coefficients. Consider

412-512: A < b {\displaystyle a<b} , it can be seen that the s and t sequences for ( a , b ) under the EEA are, up to initial 0s and 1s, the t and s sequences for ( b , a ). The definitions then show that the ( a , b ) case reduces to the ( b , a ) case. So assume that a > b {\displaystyle a>b} without loss of generality. It can be seen that s 2 {\displaystyle s_{2}}

515-469: A , b ) | ≥ 2 | s k | and | t k + 1 | = | a gcd ( a , b ) | ≥ 2 | t k | . {\displaystyle |s_{k+1}|=\left|{\frac {b}{\gcd(a,b)}}\right|\geq 2|s_{k}|\qquad {\text{and}}\qquad |t_{k+1}|=\left|{\frac {a}{\gcd(a,b)}}\right|\geq 2|t_{k}|.} This, accompanied by

618-406: A , b ) {\displaystyle ax+by=\gcd(a,b)} , one can solve for y {\displaystyle y} given a , b , x , gcd ( a , b ) {\displaystyle a,b,x,\gcd(a,b)} . Thus, an optimization to the above algorithm is to compute only the s k {\displaystyle s_{k}} sequence (which yields

721-430: A , b ) ≠ min ( a , b ) {\displaystyle \gcd(a,b)\neq \min(a,b)} , then for 0 ≤ i ≤ k , {\displaystyle 0\leq i\leq k,} where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } denotes the integral part of x , that is the greatest integer not greater than x . This implies that

824-490: A / b ⁠ is in canonical simplified form if a and b are coprime and b is positive. This canonical simplified form can be obtained by replacing the three output lines of the preceding pseudo code by The proof of this algorithm relies on the fact that s and t are two coprime integers such that as + bt = 0 , and thus a b = − t s {\displaystyle {\frac {a}{b}}=-{\frac {t}{s}}} . To get

927-417: A and b are coprime, one gets 1 in the right-hand side of Bézout's inequality. Otherwise, one may get any non-zero constant. In computer algebra , the polynomials commonly have integer coefficients, and this way of normalizing the greatest common divisor introduces too many fractions to be convenient. The second way to normalize the greatest common divisor in the case of polynomials with integer coefficients

1030-409: A and b as input, consists of computing a sequence q 1 , … , q k {\displaystyle q_{1},\ldots ,q_{k}} of quotients and a sequence r 0 , … , r k + 1 {\displaystyle r_{0},\ldots ,r_{k+1}} of remainders such that It is the main property of Euclidean division that

1133-666: A and b . (Until this point, the proof is the same as that of the classical Euclidean algorithm.) As a = r 0 {\displaystyle a=r_{0}} and b = r 1 , {\displaystyle b=r_{1},} we have a s i + b t i = r i {\displaystyle as_{i}+bt_{i}=r_{i}} for i = 0 and 1. The relation follows by induction for all i > 1 {\displaystyle i>1} : r i + 1 = r i − 1 − r i q i = (

SECTION 10

#1732783705830

1236-526: 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 ,

1339-476: 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

1442-509: A Bézout's identity, this shows that s k + 1 {\displaystyle s_{k+1}} and t k + 1 {\displaystyle t_{k+1}} are coprime . The relation a s k + 1 + b t k + 1 = 0 {\displaystyle as_{k+1}+bt_{k+1}=0} that has been proved above and Euclid's lemma show that s k + 1 {\displaystyle s_{k+1}} divides b , that

1545-439: A Federal standard (FIPS 186) in 1994. Five revisions to the initial specification have been released: FIPS 186–1 in 1998, FIPS 186–2 in 2000, FIPS 186–3 in 2009, FIPS 186–4 in 2013, and FIPS 186–5 in 2023. Standard FIPS 186-5 forbids signing with DSA, while allowing verification of signatures generated prior to the implementation date of the standard as a document. It is to be replaced by newer signature schemes such as EdDSA . DSA

1648-582: 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

1751-694: 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

1854-488: 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

1957-407: A message, and such a signature can be verified by using the signer's corresponding public key. The digital signature provides message authentication (the receiver can verify the origin of the message), integrity (the receiver can verify that the message has not been modified since it was signed) and non-repudiation (the sender cannot falsely claim that they have not signed the message). In 1982,

2060-539: 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

2163-432: A set of parameters, the second phase computes the key pair for a single user: x {\displaystyle x} is the private key and y {\displaystyle y} is the public key. The signer should publish the public key y {\displaystyle y} . That is, they should send the key to the receiver via a reliable, but not necessarily secret, mechanism. The signer should keep

SECTION 20

#1732783705830

2266-845: A signature ( r , s ) {\displaystyle \left(r,s\right)} is a valid signature for a message m {\displaystyle m} as follows: The signature scheme is correct in the sense that the verifier will always accept genuine signatures. This can be shown as follows: First, since g = h ( p − 1 ) / q   mod   p {\textstyle g=h^{(p-1)/q}~{\text{mod}}~p} , it follows that g q ≡ h p − 1 ≡ 1 mod p {\textstyle g^{q}\equiv h^{p-1}\equiv 1\mod p} by Fermat's little theorem . Since g > 0 {\displaystyle g>0} and q {\displaystyle q}

2369-644: 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

2472-548: 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

2575-416: Is 1 and s 3 {\displaystyle s_{3}} (which exists by gcd ( a , b ) ≠ min ( a , b ) {\displaystyle \gcd(a,b)\neq \min(a,b)} ) is a negative integer. Thereafter, the s i {\displaystyle s_{i}} alternate in sign and strictly increase in magnitude, which follows inductively from

2678-458: Is a subresultant polynomial . In particular, if the input polynomials are coprime, then the Bézout's identity becomes where Res ⁡ ( a , b ) {\displaystyle \operatorname {Res} (a,b)} denotes the resultant of a and b . In this form of Bézout's identity, there is no denominator in the formula. If one divides everything by the resultant one gets

2781-401: Is a choice of algorithm parameters which may be shared between different users of the system, while the second phase computes a single key pair for one user. The algorithm parameters are ( p {\displaystyle p} , q {\displaystyle q} , g {\displaystyle g} ). These may be shared between different users of the system. Given

2884-479: 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

2987-400: Is chosen in order to subliminally leak information via signatures. For example, an offline private key could be leaked from a perfect offline device that only released innocent-looking signatures. Below is a list of cryptographic libraries that provide support for DSA: National Institute of Standards and Technology The National Institute of Standards and Technology ( NIST )

3090-487: Is covered by U.S. patent 5,231,668 , filed July 26, 1991 and now expired, and attributed to David W. Kravitz, a former NSA employee. This patent was given to "The United States of America as represented by the Secretary of Commerce , Washington, D.C.", and NIST has made this patent available worldwide royalty-free. Claus P. Schnorr claims that his U.S. patent 4,995,082 (also now expired) covered DSA; this claim

3193-521: Is disputed. In 1993, Dave Banisar managed to get confirmation, via a FOIA request, that the DSA algorithm hasn't been designed by the NIST, but by the NSA. OpenSSH announced that DSA is scheduled to be removed in 2025. The DSA algorithm involves four operations: key generation (which creates the key pair), key distribution, signing and signature verification. Key generation has two phases. The first phase

Digital Signature Algorithm - Misplaced Pages Continue

3296-649: 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,

3399-412: Is output, may have an incorrect sign. This is easy to correct at the end of the computation but has not been done here for simplifying the code. Similarly, if either a or b is zero and the other is negative, the greatest common divisor that is output is negative, and all the signs of the output must be changed. Finally, notice that in Bézout's identity, a x + b y = gcd (

3502-502: Is particularly useful when a and b are coprime . With that provision, x is the modular multiplicative inverse of a modulo b , and y is the modular multiplicative inverse of b modulo a . Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography . In particular,

3605-459: Is prime, g {\displaystyle g} must have order  q {\displaystyle q} . The signer computes Thus Since g {\displaystyle g} has order q {\displaystyle q} we have Finally, the correctness of DSA follows from With DSA, the entropy, secrecy, and uniqueness of the random signature value k {\displaystyle k} are critical. It

3708-575: 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

3811-597: Is so critical that violating any one of those three requirements can reveal the entire private key to an attacker. Using the same value twice (even while keeping k {\displaystyle k} secret), using a predictable value, or leaking even a few bits of k {\displaystyle k} in each of several signatures, is enough to reveal the private key x {\displaystyle x} . This issue affects both DSA and Elliptic Curve Digital Signature Algorithm ( ECDSA ) – in December 2010,

3914-683: Is that b = d s k + 1 {\displaystyle b=ds_{k+1}} for some integer d . Dividing by s k + 1 {\displaystyle s_{k+1}} the relation a s k + 1 + b t k + 1 = 0 {\displaystyle as_{k+1}+bt_{k+1}=0} gives a = − d t k + 1 . {\displaystyle a=-dt_{k+1}.} So, s k + 1 {\displaystyle s_{k+1}} and − t k + 1 {\displaystyle -t_{k+1}} are coprime integers that are

4017-557: Is that, in the Euclidean division and the algorithm, the inequality 0 ≤ r i + 1 < | r i | {\displaystyle 0\leq r_{i+1}<|r_{i}|} has to be replaced by an inequality on the degrees deg ⁡ r i + 1 < deg ⁡ r i . {\displaystyle \deg r_{i+1}<\deg r_{i}.} Otherwise, everything which precedes in this article remains

4120-484: Is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor. In mathematics, it is common to require that the greatest common divisor be a monic polynomial . To get this, it suffices to divide every element of the output by the leading coefficient of r k . {\displaystyle r_{k}.} This allows that, if

4223-688: Is the identity matrix and its determinant is one. The determinant of the rightmost matrix in the preceding formula is −1. It follows that the determinant of A i {\displaystyle A_{i}} is ( − 1 ) i − 1 . {\displaystyle (-1)^{i-1}.} In particular, for i = k + 1 , {\displaystyle i=k+1,} we have s k t k + 1 − t k s k + 1 = ( − 1 ) k . {\displaystyle s_{k}t_{k+1}-t_{k}s_{k+1}=(-1)^{k}.} Viewing this as

Digital Signature Algorithm - Misplaced Pages Continue

4326-408: Is the last non zero entry, 2 in the column "remainder". The computation stops at row 6, because the remainder in it is 0 . Bézout coefficients appear in the last two columns of the second-to-last row. In fact, it is easy to verify that −9 × 240 + 47 × 46 = 2 . Finally the last two entries 23 and −120 of the last row are, up to the sign, the quotients of the input 46 and 240 by

4429-595: Is the most computationally expensive part of the signing operation, but it may be computed before the message is known. Calculating the modular inverse k − 1 mod q {\displaystyle k^{-1}{\bmod {\,}}q} is the second most expensive part, and it may also be computed before the message is known. It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k q − 2 mod q {\displaystyle k^{q-2}{\bmod {\,}}q} . One can verify that

4532-425: Is to divide every output by the content of r k , {\displaystyle r_{k},} to get a primitive greatest common divisor. If the input polynomials are coprime, this normalisation also provides a greatest common divisor equal to 1. The drawback of this approach is that a lot of fractions should be computed and simplified during the computation. A third approach consists in extending

4635-438: Is zero; the greatest common divisor is then the last non zero remainder r k . {\displaystyle r_{k}.} The extended Euclidean algorithm proceeds similarly, but adds two other sequences, as follows The computation also stops when r k + 1 = 0 {\displaystyle r_{k+1}=0} and gives Moreover, if a and b are both positive and gcd (

4738-524: 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

4841-525: 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

4944-752: 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

5047-568: 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

5150-559: 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,

5253-737: 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,

SECTION 50

#1732783705830

5356-870: 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

5459-842: 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

5562-402: The modular integers and the algebraic field extensions . A notable instance of the latter case are the finite fields of non-prime order. If n is a positive integer, the ring Z / n Z may be identified with the set {0, 1, ..., n -1} of the remainders of Euclidean division by n , the addition and the multiplication consisting in taking the remainder by n of the result of

5665-700: 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

5768-406: 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,

5871-511: The Bézout coefficient x {\displaystyle x} ), and then compute y {\displaystyle y} at the end: However, in many cases this is not really an optimization: whereas the former algorithm is not susceptible to overflow when used with machine integers (that is, integers with a fixed upper bound of digits), the multiplication of old_s * a in computation of bezout_t can overflow, limiting this optimization to inputs which can be represented in less than half

5974-729: 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

6077-702: 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

6180-536: 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

6283-461: The NIST website. Extended Euclidean algorithm In arithmetic and computer programming , the extended Euclidean algorithm is an extension to the Euclidean algorithm , and computes, in addition to the greatest common divisor (gcd) of integers a and b , also the coefficients of Bézout's identity , which are integers x and y such that This is a certifying algorithm , because

SECTION 60

#1732783705830

6386-600: 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

6489-551: The U.S government solicited proposals for a public key signature standard. In August 1991 the National Institute of Standards and Technology (NIST) proposed DSA for use in their Digital Signature Standard (DSS). Initially there was significant criticism, especially from software companies that had already invested effort in developing digital signature software based on the RSA cryptosystem . Nevertheless, NIST adopted DSA as

6592-511: 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

6695-487: 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

6798-531: The addition and the multiplication of integers. An element a of Z / n Z has a multiplicative inverse (that is, it is a unit ) if it is coprime to n . In particular, if n is prime , a has a multiplicative inverse if it is not zero (modulo n ). Thus Z / n Z is a field if and only if n is prime. Bézout's identity asserts that a and n are coprime if and only if there exist integers s and t such that Reducing this identity modulo n gives Thus t , or, more exactly,

6901-391: The algorithm of subresultant pseudo-remainder sequences in a way that is similar to the extension of the Euclidean algorithm to the extended Euclidean algorithm. This allows that, when starting with polynomials with integer coefficients, all polynomials that are computed have integer coefficients. Moreover, every computed remainder r i {\displaystyle r_{i}}

7004-537: The algorithm stops eventually. As r i + 1 = r i − 1 − r i q i , {\displaystyle r_{i+1}=r_{i-1}-r_{i}q_{i},} the greatest common divisor is the same for ( r i − 1 , r i ) {\displaystyle (r_{i-1},r_{i})} and ( r i , r i + 1 ) . {\displaystyle (r_{i},r_{i+1}).} This shows that

7107-409: The canonical simplified form, it suffices to move the minus sign for having a positive denominator. If b divides a evenly, the algorithm executes only one iteration, and we have s = 1 at the end of the algorithm. It is the only case where the output is an integer. The extended Euclidean algorithm is the essential tool for computing multiplicative inverses in modular structures, typically

7210-404: The classical Bézout's identity, with an explicit common denominator for the rational numbers that appear in it. To implement the algorithm that is described above, one should first remark that only the two last values of the indexed variables are needed at each step. Thus, for saving memory, each indexed variable must be replaced by just two variables. For simplicity, the following algorithm (and

7313-414: The computation of the modular multiplicative inverse is an essential step in the derivation of key-pairs in the RSA public-key encryption method. The standard Euclidean algorithm proceeds by a succession of Euclidean divisions whose quotients are not used. Only the remainders are kept. For the extended algorithm, the successive quotients are used. More precisely, the standard Euclidean algorithm with

7416-522: 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

7519-459: The definitions and the fact that q i ≥ 1 {\displaystyle q_{i}\geq 1} for 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} , the case i = 1 {\displaystyle i=1} holds because a > b {\displaystyle a>b} . The same is true for the t i {\displaystyle t_{i}} after

7622-493: 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

7725-401: The end. This results in the pseudocode, in which the input n is an integer larger than 1. The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions . An important case, widely used in cryptography and coding theory , is that of finite fields of non-prime order. In fact, if p is a prime number, and q = p ,

7828-516: The fact that s k , t k {\displaystyle s_{k},t_{k}} are larger than or equal to in absolute value than any previous s i {\displaystyle s_{i}} or t i {\displaystyle t_{i}} respectively completed the proof. For univariate polynomials with coefficients in a field , everything works similarly, Euclidean division, Bézout's identity and extended Euclidean algorithm. The first difference

7931-701: The first few terms, for the same reason. Furthermore, it is easy to see that q k ≥ 2 {\displaystyle q_{k}\geq 2} (when a and b are both positive and gcd ( a , b ) ≠ min ( a , b ) {\displaystyle \gcd(a,b)\neq \min(a,b)} ). Thus, noticing that | s k + 1 | = | s k − 1 | + q k | s k | {\displaystyle |s_{k+1}|=|s_{k-1}|+q_{k}|s_{k}|} , we obtain | s k + 1 | = | b gcd (

8034-459: The gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials . The extended Euclidean algorithm

8137-483: The greatest common divisor 2 . As 0 ≤ r i + 1 < | r i | , {\displaystyle 0\leq r_{i+1}<|r_{i}|,} the sequence of the r i {\displaystyle r_{i}} is a decreasing sequence of nonnegative integers (from i = 2 on). Thus it must stop with some r k + 1 = 0. {\displaystyle r_{k+1}=0.} This proves that

8240-411: The greatest common divisor of the input a = r 0 , b = r 1 {\displaystyle a=r_{0},b=r_{1}} is the same as that of r k , r k + 1 = 0. {\displaystyle r_{k},r_{k+1}=0.} This proves that r k {\displaystyle r_{k}} is the greatest common divisor of

8343-525: The group fail0verflow announced the recovery of the ECDSA private key used by Sony to sign software for the PlayStation 3 game console. The attack was made possible because Sony failed to generate a new random k {\displaystyle k} for each signature. This issue can be prevented by deriving k {\displaystyle k} deterministically from the private key and

8446-411: The implementation date of that standard. The DSA works in the framework of public-key cryptosystems and is based on the algebraic properties of modular exponentiation , together with the discrete logarithm problem , which is considered to be computationally intractable. The algorithm uses a key pair consisting of a public key and a private key. The private key is used to generate a digital signature for

8549-482: The inequalities on the right define uniquely q i {\displaystyle q_{i}} and r i + 1 {\displaystyle r_{i+1}} from r i − 1 {\displaystyle r_{i-1}} and r i . {\displaystyle r_{i}.} The computation stops when one reaches a remainder r k + 1 {\displaystyle r_{k+1}} which

8652-701: The matrix A i = ( s i − 1 s i t i − 1 t i ) . {\displaystyle A_{i}={\begin{pmatrix}s_{i-1}&s_{i}\\t_{i-1}&t_{i}\end{pmatrix}}.} The recurrence relation may be rewritten in matrix form A i + 1 = A i ⋅ ( 0 1 1 − q i ) . {\displaystyle A_{i+1}=A_{i}\cdot {\begin{pmatrix}0&1\\1&-q_{i}\end{pmatrix}}.} The matrix A 1 {\displaystyle A_{1}}

8755-407: The maximal size. When using integers of unbounded size, the time needed for multiplication and division grows quadratically with the size of the integers. This implies that the "optimisation" replaces a sequence of multiplications/divisions of small integers by a single multiplication/division, which requires more computing time than the operations that it replaces, taken together. A fraction ⁠

8858-431: The message hash, as described by RFC   6979 . This ensures that k {\displaystyle k} is different for each H ( m ) {\displaystyle H(m)} and unpredictable for attackers who do not know the private key x {\displaystyle x} . In addition, malicious implementations of DSA and ECDSA can be created where k {\displaystyle k}

8961-409: 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

9064-429: 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

9167-404: The other algorithms in this article) uses parallel assignments . In a programming language which does not have this feature, the parallel assignments need to be simulated with an auxiliary variable. For example, the first one, is equivalent to and similarly for the other parallel assignments. This leads to the following code: The quotients of a and b by their greatest common divisor, which

9270-486: The pair of Bézout's coefficients provided by the extended Euclidean algorithm is the minimal pair of Bézout coefficients, as being the unique pair satisfying both above inequalities. Also it means that the algorithm can be done without integer overflow by a computer program using integers of a fixed size that is larger than that of a and b . The following table shows how the extended Euclidean algorithm proceeds with input 240 and 46 . The greatest common divisor

9373-495: The private key x {\displaystyle x} secret. A message m {\displaystyle m} is signed as follows: The signature is ( r , s ) {\displaystyle \left(r,s\right)} The calculation of k {\displaystyle k} and r {\displaystyle r} amounts to creating a new per-message key. The modular exponentiation in computing r {\displaystyle r}

9476-547: 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

9579-482: 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

9682-481: 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

9785-414: The quotients of a and b by a common factor, which is thus their greatest common divisor or its opposite . To prove the last assertion, assume that a and b are both positive and gcd ( a , b ) ≠ min ( a , b ) {\displaystyle \gcd(a,b)\neq \min(a,b)} . Then, a ≠ b {\displaystyle a\neq b} , and if

9888-488: The remainder of the division of t by n , is the multiplicative inverse of a modulo n . To adapt the extended Euclidean algorithm to this problem, one should remark that the Bézout coefficient of n is not needed, and thus does not need to be computed. Also, for getting a result which is positive and lower than n , one may use the fact that the integer t provided by the algorithm satisfies | t | < n . That is, if t < 0 , one must add n to it at

9991-434: The same, simply by replacing integers by polynomials. A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem. If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials ( s , t ) such that and A third difference

10094-431: 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

10197-415: 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

10300-523: 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

10403-553: 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

10506-719: 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

10609-619: 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

#829170