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Arjen Lenstra
Arjen Klaas Lenstra (born 2 March 1956, in Groningen) is a Dutch mathematician, cryptographer and computational number theorist. He is currently a professor at the École Polytechnique Fédérale de Lausanne (EPFL) where he heads of the Laboratory for Cryptologic Algorithms. Career He studied mathematics at the University of Amsterdam. He is currently a professor at the EPFL (Lausanne), in the Laboratory for Cryptologic Algorithms, and previously worked for Citibank and Bell Labs. Research Lenstra is active in cryptography and computational number theory, especially in areas such as integer factorization. With Mark Manasse, he was the first to seek volunteers over the internet for a large scale volunteer computing project. Such projects became more common after the Factorization of RSA-129 which was a high publicity distributed factoring success led by Lenstra along with Derek Atkins, Michael Graff and Paul Leyland. He was also a leader in the successful factorizations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Netherlands
) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Netherlands , established_title2 = Act of Abjuration , established_date2 = 26 July 1581 , established_title3 = Peace of Münster , established_date3 = 30 January 1648 , established_title4 = Kingdom established , established_date4 = 16 March 1815 , established_title5 = Liberation Day , established_date5 = 5 May 1945 , established_title6 = Kingdom Charter , established_date6 = 15 December 1954 , established_title7 = Caribbean reorganisation , established_date7 = 10 October 2010 , official_languages = Dutch , languages_type = Regional languages , languages_sub = yes , languages = , languages2_type = Recognised languages , languages2_sub = yes , languages2 = , demonym = Dutch , capital = Amsterdam , largest_city = capital , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number ( RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eindhoven University Of Technology
The Eindhoven University of Technology ( nl, Technische Universiteit Eindhoven), abbr. TU/e, is a public technical university in the Netherlands, located in the city of Eindhoven. In 2020–21, around 14,000 students were enrolled in its BSc and MSc programs and around 1350 students were enrolled in its PhD and PDEng programs. In 2021, the TU/e employed around 3900 people. Eindhoven University of Technology has been ranked in the top 200 universities in three major ranking systems. The 2019 QS World University Rankings place Eindhoven 99th in the world, 34th in Europe, and 3rd in the Netherlands. TU/e is the Dutch member of thEuroTech Universities Alliance a strategic partnership of universities of science & technology in Europe: Technical University of Denmark (DTU), École Polytechnique Fédérale de Lausanne (EPFL), École Polytechnique (L’X), The Technion, Eindhoven University of Technology (TU/e), and Technical University of Munich (TUM). History The Eindhoven Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xiaoyun Wang
Wang Xiaoyun (; born 1966) is a Chinese cryptographer, mathematician, and computer scientist. She is a professor in the Department of Mathematics and System Science of Shandong University and an academician of the Chinese Academy of Sciences. Early life and education Wang was born in Zhucheng, Shandong Province. She gained bachelor (1987), master (1990) and doctorate (1993) degrees at Shandong University, and subsequently lectured in the mathematics department from 1993. Her doctoral advisor was Pan Chengdong. Wang was appointed assistant professor in 1995, and full professor in 2001. She became the Chen Ning Yang Professor of the Center for Advanced Study, Tsinghua University in 2005. Career and research At the rump session of CRYPTO 2004, she and co-authors demonstrated collision attacks against MD5, SHA-0 and other related hash functions (a collision occurs when two distinct messages result in the same hash function output). They received a standing ovation for their w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptosystem
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term '' cipher'' (sometimes ''cypher'') is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term ''cryptosystem'' is most often used when the key generation algorithm is important. For this reason, the term ''cryptosystem'' is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques. Formal definition Mathematically, a cryptosystem or encryption scheme can be defined as a tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Lovász
László Lovász (; born March 9, 1948) is a Hungarian mathematician and professor emeritus at Eötvös Loránd University, best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He was the president of the International Mathematical Union from 2007 to 2010 and the president of the Hungarian Academy of Sciences from 2014 to 2020. In graph theory, Lovász's notable contributions include the proofs of Kneser's conjecture and the Lovász local lemma, as well as the formulation of the Erdős–Faber–Lovász conjecture. He is also one of the eponymous authors of the LLL lattice reduction algorithm. Early life and education Lovász was born on March 9, 1948, in Budapest, Hungary. Lovász attended the Fazekas Mihály Gimnázium in Budapest. He won three gold medals (1964–1966) and one silver medal (1963) at the International Mathematical Olympiad. He also participated in a Hungarian game show about math prodigies. Pau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Willem Lenstra
Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987 he was appointed to the faculty of the University of California, Berkeley; starting in 1998, he divided his time between Berkeley and the University of Leiden, until 2003, when he retired from Berkeley to take a full-time position at Leiden. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI). Hendrik Lenstra was the Chairman of the Program Committee of the International Congress of Mathematicians in 2010. Scientific contributions Lenstra has worked principally in computational number theory. He is well known for: * Co-discovering of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (in 1982); * Developing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization Of Polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2''k'' + 1 is prime and ''k'' > 0, then ''k'' must be a power of 2, so 2''k'' + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are ''F''0 = 3, ''F''1 = 5, ''F''2 = 17, ''F''3 = 257, and ''F''4 = 65537 ; heuristics suggest that there are no more. Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for ''n'' ≥ 2. Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( \left(\sqrt + o(1)\right)(\ln n)^(\ln \ln n)^\right) =L_n\left .html"_;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in_L-notation.html" ;"title="">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation">">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation), where is the natural logarithm. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Leyland
Paul Leyland is a British number theorist who has studied integer factorization and primality testing. He has contributed to the factorization of RSA-129, RSA-140, and RSA-155, as well as potential factorial primes as large as 400! + 1. He has also studied Cunningham numbers, Cullen numbers, Woodall numbers, etc., and numbers of the form x^y + y^x, which are now called Leyland numbers. He was involved with the NFSNet project to use distributed computing on the Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ... from 2005 to 2008. References External linksPaul Leyland's home page Living people Year of birth missing (living people) Number theorists 20th-century British mathematicians 21st-century British mathematicians {{UK-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derek Atkins
Derek A Atkins is a computer scientist specializing in computer security. He studied electrical engineering and computer science at the Massachusetts Institute of Technology. In June 2014, he became the chief technology officer of SecureRF. Atkins is an author or co-author of several computer security books, including ''Internet Security Professional Reference'' and ''Reputational Risk: A Question of Trust''. In April 1994, he co-authored the paper The Magic Words are Squeamish Ossifrage detailing the factorization of RSA-129."The Assault on 114,381,625,757,888,867,669,235,779,976,146,612,010,218,296,271,242,362,562,561,842,935,706,935,245,733,897,830,597,123,563,958,705,058,989,075,147,599,290,026,879,543,541" By Gina Kolata ''The New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
