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Harald Niederreiter
Harald G. Niederreiter (born June 7, 1944) is an Austrian mathematician known for his work in discrepancy theory, algebraic geometry, quasi-Monte Carlo methods, and cryptography. Education and career Niederreiter was born on June 7, 1944, in Vienna, and grew up in Salzburg... He began studying mathematics at the University of Vienna in 1963, and finished his doctorate there in 1969, with a thesis on discrepancy in compact abelian groups supervised by Edmund Hlawka. He began his academic career as an assistant professor at the University of Vienna, but soon moved to Southern Illinois University. During this period he also visited the University of Illinois at Urbana-Champaign, Institute for Advanced Study, and University of California, Los Angeles. In 1978 he moved again, becoming the head of a new mathematics department at the University of the West Indies in Jamaica. In 1981 he returned to Austria for a post at the Austrian Academy of Sciences, where from 1989 to 2000 he serve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jamaica
Jamaica is an island country in the Caribbean Sea and the West Indies. At , it is the third-largest island—after Cuba and Hispaniola—of the Greater Antilles and the Caribbean. Jamaica lies about south of Cuba, west of Hispaniola (the island containing Haiti and the Dominican Republic), and southeast of the Cayman Islands (a British Overseas Territories, British Overseas Territory). With million people, Jamaica is the third most populous English-speaking world, Anglophone country in the Americas and the fourth most populous country in the Caribbean. Kingston, Jamaica, Kingston is the country's capital and largest city. The indigenous Taíno peoples of the island gradually came under Spanish Empire, Spanish rule after the arrival of Christopher Columbus in 1494. Many of the indigenous people either were killed or died of diseases, after which the Spanish brought large numbers of Africans to Jamaica as slaves. The island remained a possession of Spain, under the name Colo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Public Key Cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions. Security of public-key cryptography depends on keeping the private key secret; the public key can be openly distributed without compromising security. There are many kinds of public-key cryptosystems, with different security goals, including digital signature, Diffie–Hellman key exchange, public-key key encapsulation, and public-key encryption. Public key algorithms are fundamental security primitives in modern cryptosystems, including applications and protocols that offer assurance of the confidentiality and authenticity of electronic communications and data storage. They underpin numerous Internet standards, such as Transport Layer Security (TLS), SSH, S/MIME, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stream Cipher
stream cipher is a symmetric key cipher where plaintext digits are combined with a pseudorandom cipher digit stream ( keystream). In a stream cipher, each plaintext digit is encrypted one at a time with the corresponding digit of the keystream, to give a digit of the ciphertext stream. Since encryption of each digit is dependent on the current state of the cipher, it is also known as ''state cipher''. In practice, a digit is typically a bit and the combining operation is an exclusive-or (XOR). The pseudorandom keystream is typically generated serially from a random seed value using digital shift registers. The seed value serves as the cryptographic key for decrypting the ciphertext stream. Stream ciphers represent a different approach to symmetric encryption from block ciphers. Block ciphers operate on large blocks of digits with a fixed, unvarying transformation. This distinction is not always clear-cut: in some modes of operation, a block cipher primitive is used in such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanisław Ulam, was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically. Monte Carlo methods are widely used in va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Number Generation
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols is generated that cannot be reasonably predicted better than by random chance. This means that the particular outcome sequence will contain some patterns detectable in hindsight but impossible to foresee. True random number generators can be ''Hardware random number generator, hardware random-number generators'' (HRNGs), wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model. This would be in contrast to so-called "random number generations" done by ''pseudorandom number generators'' (PRNGs), which generate numbers that only look random but are in fact predetermined—these generations can be reproduced simply by knowing the state of the PRNG. Various applications of randomness have led to the development of different methods for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delone Set
In the mathematical theory of metric spaces, -nets, -packings, -coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals. Definitions If is a metric space, and is a subset of , then the packing radius, , of is half of the smallest distance between distinct members of . Open balls of radius centered at the points of will all be disjoint from each other. The covering radius, , of is the smallest distance such that every point of is within distance of at least one point in ; that is, is the smallest radius such that closed balls of that radius centered at the points of have all of as their union. An -packing is a set of packing radius (equivalently, minimu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saudi Arabia
Saudi Arabia, officially the Kingdom of Saudi Arabia (KSA), is a country in West Asia. Located in the centre of the Middle East, it covers the bulk of the Arabian Peninsula and has a land area of about , making it the List of Asian countries by area, fifth-largest country in Asia, the largest in the Middle East, and the List of countries and dependencies by area, 12th-largest in the world. It is bordered by the Red Sea to the west; Jordan, Iraq, and Kuwait to the north; the Persian Gulf, Bahrain, Qatar and the United Arab Emirates to the east; Oman to the southeast; and Yemen to Saudi Arabia–Yemen border, the south. The Gulf of Aqaba in the northwest separates Saudi Arabia from Egypt and Israel. Saudi Arabia is the only country with a coastline along both the Red Sea and the Persian Gulf, and most of Geography of Saudi Arabia, its terrain consists of Arabian Desert, arid desert, lowland, steppe, and List of mountains in Saudi Arabia, mountains. The capital and List of cities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
