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Secret Sharing Using The Chinese Remainder Theorem
Secret sharing consists of recovering a secret ''S'' from a set of shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence equations, the solution is unique in some , with under some appropriate conditions on the congruences. Secret sharing can thus use the CRT to produce the shares presented in the congruence equations and the secret could be recovered by solving the system of congruences to get the unique solution, which will be the secret to recover. Secret sharing schemes: several types There are several types of secret sharing schemes. The most basic types are the so-called threshold schemes, where only the cardinality of the set of shares matters. In other words, given a secret ''S'', and ''n'' shares, any set of ''t'' shares is a set with the smallest cardinality from which the secret can be recovered, in the sense that any set of ''t-1'' shares is not enough to give ''S''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secret Sharing
Secret sharing (also called secret splitting) refers to methods for distributing a secret among a group, in such a way that no individual holds any intelligible information about the secret, but when a sufficient number of individuals combine their 'shares', the secret may be reconstructed. Whereas ''insecure'' secret sharing allows an attacker to gain more information with each share, ''secure'' secret sharing is 'all or nothing' (where 'all' means the necessary number of shares). In one type of secret sharing scheme there is one ''dealer'' and ''n'' ''players''. The dealer gives a share of the secret to the players, but only when specific conditions are fulfilled will the players be able to reconstruct the secret from their shares. The dealer accomplishes this by giving each player a share in such a way that any group of ''t'' (for ''threshold'') or more players can together reconstruct the secret but no group of fewer than ''t'' players can. Such a system is called a -threshol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout's Identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that , x, \le , b/d , and , y, \le , a/d , ; equality occurs only if one of and is a multiple of the other. As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Cramer
Ronald John Fitzgerald Cramer (born 3 February 1968 in Haarlem) is a professor at the Centrum Wiskunde & Informatica (CWI) in Amsterdam and the University of Leiden. He obtained his PhD from the University of Amsterdam in 1997. Prior to returning to the Netherlands he was at the University of Aarhus. He is best known for his work with Victor Shoup on chosen ciphertext secure encryption in the standard model, in particular the Cramer–Shoup encryption scheme. Cramer became a member of the Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ... in 2013. He is member of the advisory board of the German Center for Advanced Security Research Darmstadt CASED. References External links * * * {{DEFAULTSORT:Cramer, Ronald 1968 births Living ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Stein
Clifford Seth Stein (born December 14, 1965), a computer scientist, is a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Science. Stein is chair of the Industrial Engineering and Operations Research Department at Columbia University. Prior to joining Columbia, Stein was a professor at Dartmouth College in New Hampshire. Stein's research interests include the design and analysis of algorithms, combinatorial optimization, operations research, network algorithms, scheduling, algorithm engineering and computational biology. Stein has published many influential papers in the leading conferences and journals in his fields of research, and has occupied a variety of editorial positions including in the journals ''ACM Transactions on Algorithms'', ''Mathematical Programming'', ''Journal of Algorithms'', ''SIAM Journal on Discrete Mathematics'' and ''Operations Resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald L
Ronald is a masculine given name derived from the Old Norse ''Rögnvaldr'', Hanks; Hardcastle; Hodges (2006) p. 234; Hanks; Hodges (2003) § Ronald. or possibly from Old English '' Regenweald''. In some cases ''Ronald'' is an Anglicised form of the Gaelic ''Raghnall'', a name likewise derived from ''Rögnvaldr''. The latter name is composed of the Old Norse elements ''regin'' ("advice", "decision") and ''valdr'' ("ruler"). ''Ronald'' was originally used in England and Scotland, where Scandinavian influences were once substantial, although now the name is common throughout the English-speaking world. A short form of ''Ronald'' is ''Ron''. Pet forms of ''Ronald'' include ''Roni'' and ''Ronnie''. ''Ronalda'' and ''Rhonda'' are feminine forms of ''Ronald''. '' Rhona'', a modern name apparently only dating back to the late nineteenth century, may have originated as a feminine form of ''Ronald''. Hanks; Hardcastle; Hodges (2006) pp. 230, 408; Hanks; Hodges (2003) § Rhona. The names ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles E
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its depr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas H
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Illinois * Thomas, Indiana * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts, entertainment, and media * ''Thomas'' (Burton novel) 1969 novel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madhu Sudan
Madhu Sudan (born 12 September 1966) is an Indian-American computer scientist. He has been a Gordon McKay Professor of Computer Science at the Harvard John A. Paulson School of Engineering and Applied Sciences since 2015. Career He received his bachelor's degree in computer science from IIT Delhi in 1987 and his doctoral degree in computer science at the University of California, Berkeley in 1992. He was a research staff member at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York from 1992 to 1997 and moved to MIT after that. From 2009 to 2015 he was a permanent researcher at Microsoft Research New England before joining Harvard University in 2015. Research contribution and awards He was awarded the Rolf Nevanlinna Prize at the 24th International Congress of Mathematicians (ICM) in 2002. The prize recognizes outstanding work in the mathematical aspects of computer science. Sudan was honored for his work in advancing the theory of probabilistically che ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dana Ron
Dana Ron Goldreich ( he, דנה רון גולדרייך; b. 1964) is a computer scientist, a professor of electrical engineering at the Tel Aviv University, Israel. Prof. Ron is one of the pioneers of research in property testing, and a leading researcher in that area. Professional career Dana Ron obtained her B.A. (1987) and M.A. (1989) in computer science from the Hebrew University in Jerusalem. Her Ph.D. (1995), also from the Hebrew University, was in the area of machine learning. Between the years 1995-97 she was an NSF post-doctoral fellow at the Massachusetts Institute of Technology (MIT). She was a Bunting fellow in 1997/8, and the Radcliffe fellow at Harvard University in 2003/4. Her research interests include sublinear-time algorithms (in particular property testing), randomized algorithms, and computational learning theory. She is married to Oded Goldreich, who is also a computer scientist at the Weizmann Institute, and has collaborated with Goldreich on approximation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oded Goldreich
Oded Goldreich ( he, עודד גולדרייך; b. 1957) is a professor of Computer Science at the Faculty of Mathematics and Computer Science of Weizmann Institute of Science, Israel. His research interests lie within the theory of computation and are, specifically, the interplay of randomness and computation, the foundations of cryptography, and computational complexity theory. He won the Knuth Prize in 2017 and was selected in 2021 to receive the Israel Prize in mathematics. Biography Goldreich received a DSc in Computer Science at Technion in 1983 under Shimon Even. Goldreich has contributed to the development of pseudorandomness, zero knowledge proofs, secure function evaluation, property testing,Oded Goldreich, Shafi Goldwasser, and Dana Ron. 1998 Property Testing and its connection to Learning and Approximation. ''Journal of the ACM'', pages 653-750. and other areas in cryptography and computational complexity. Goldreich has also authored several books including: ''Fou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 : ax + by = \gcd(a, b). This is a certifying algorithm, because 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. 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 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'' modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
