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Pollard's Lambda Method
In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist John M. Pollard, in the same paper as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime ''p'', it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group. Algorithm Suppose G is a finite cyclic group of order n which is generated by the element \alpha, and we seek to find the discrete logarithm x of the element \beta to the base \alpha. In other words, one seeks x \in Z_n such that \alpha^x = \beta. The lambda algorithm allows one to search for x in some interval ,\ldots,bsubset Z_n. One may search the entire range of poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Number Theory
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry. Computational number theory has applications to cryptography, including RSA, elliptic curve cryptography and post-quantum cryptography, and is used to investigate conjectures and open problems in number theory, including the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the ABC conjecture, the modularity conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program. Software packages * Magma computer algebra system * SageMath * Number Theory Library * PARI/GP * Fast Library for Number Theory Further reading * Michael E. Pohst (1993): ''Computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Letter
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as well as consonants. In Archaic Greece, Archaic and early Classical Greece, Classical times, the Greek alphabet existed in Archaic Greek alphabets, many local variants, but, by the end of the 4th century BC, the Ionia, Ionic-based Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard throughout the Greek-speaking world and is the version that is still used for Greek writing today. The letter case, uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , , , , , , , The Greek alphabet is the ancestor of several scripts, such as the Latin script, Latin, Gothic alphabet, Gothic, Coptic script, Coptic, and Cyrillic scripts. Throughout antiquity, Greek had only a single uppercas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theoretic Algorithms
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a ''numeral'' is not clearly distinguished from the ''numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Association
The Mathematical Association is a professional society concerned with mathematics education in the UK. History It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1897. It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of the death of Sir Isaac Newton, attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal, Sir Horace Lamb, and G. H. Hardy. In 1951, Mary Cartwright became the first female president of the Mathematical Association. In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university, a teaching award that was examined was the Diploma of the Mathematical Association, later known as the Diploma in Mathematical Education of the Mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journal was established in 1894 by Edward Mann Langley as the successor to the ''Reports of the Association for the Improvement of Geometrical Teaching''. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors-in-chief The following persons are or have been editor-in-chief: Abstracting and indexing The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c ..., and zbMA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Association For Cryptologic Research
The International Association for Cryptologic Research (IACR) is a non-profit scientific organization that furthers research in cryptology and related fields. The IACR was organized at the initiative of David Chaum at the CRYPTO '82 conference. Activities The IACR organizes and sponsors three annual flagship conferences, four area conferences in specific sub-areas of cryptography, and one symposium: * Crypto (flagship) * Eurocrypt (flagship) * Asiacrypt (flagship) * Fast Software Encryption (FSE) * Public Key Cryptography (PKC) * Cryptographic Hardware and Embedded Systems (CHES) * Theory of Cryptography (TCC) * Real World Crypto Symposium (RWC) Several other conferences and workshops are held in cooperation with the IACR. Starting in 2015, selected summer schools will be officially sponsored by the IACR. CRYPTO '83 was the first conference officially sponsored by the IACR. The IACR publishes the '' Journal of Cryptology'', in addition to the proceedings of its conference and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Cryptology
The ''Journal of Cryptology'' () is a scientific journal in the field of cryptology and cryptography. The journal is published quarterly by the International Association for Cryptologic Research. Its editor-in-chief is Vincent Rijmen Vincent Rijmen (; born 16 October 1970) is a Belgium, Belgian cryptographer and one of the two designers of the Rijndael, the Advanced Encryption Standard. Rijmen is also the co-designer of the WHIRLPOOL cryptographic hash function, and the block .... Springer, retrieved 2022-05-09. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American, Inc
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which study the physical world, and the social sciences, which study individuals and societies. While referred to as the formal sciences, the study of logic, mathematics, and theoretical computer science are typically regarded as separate because they rely on deductive reasoning instead of the scientific method as their main methodology. Meanwhile, applied sciences are disciplines that use scientific knowledge for practical purposes, such as engineering and medicine. The history of science spans the majority of the historical record, with the earliest identifiable predecessors to modern science dating to the Bronze Age in Egypt and Mesopotamia (). Their contributions to mathematics, astronomy, and medicine entered and shaped the Greek natural philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the '' Journal Citation Reports'', the journal has a 2020 impact factor of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Academic journals established in 1943 American Mathematical Society academic journals Bimonthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow Table
A rainbow table is a precomputed table for caching the outputs of a cryptographic hash function, usually for cracking password hashes. Passwords are typically stored not in plain text form, but as hash values. If such a database of hashed passwords falls into the hands of attackers, they can use a precomputed rainbow table to recover the plaintext passwords. A common defense against this attack is to compute the hashes using a key derivation function that adds a "salt" to each password before hashing it, with different passwords receiving different salts, which are stored in plain text along with the hash. Rainbow tables are a practical example of a space–time tradeoff: they use less computer processing time and more storage than a brute-force attack which calculates a hash on every attempt, but more processing time and less storage than a simple table that stores the hash of every possible password. Rainbow tables were invented by Philippe Oechslin as an application of an ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal Count
The Kruskal count (also known as Kruskal's principle, Dynkin–Kruskal count, Dynkin's counting trick, Dynkin's card trick, coupling card trick or shift coupling) is a probabilistic concept originally demonstrated by the Russian mathematician Evgenii Borisovich Dynkin in the 1950s or 1960s discussing coupling (probability), coupling effects and rediscovered as a card trick by the American mathematician Martin David Kruskal in the early 1970s as a side-product while working on another problem. It was published by Kruskal's friend Martin Gardner and magician Karl Fulves in 1975. This is related to a similar trick published by magician Alexander F. Kraus in 1957 as ''Sum total'' and later called ''Kraus principle''. Besides uses as a card trick, the underlying phenomenon has applications in cryptography, code breaking, software tamper protection, code self-synchronization, control-flow resynchronization, design of variable-length codes and variable-length instruction sets, web navi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
