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Peter Montgomery (mathematician)
Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research. He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in applications of elliptic curves to integer factorization and other problems, and the Montgomery ladder, which is used to protect against side-channel attacks in elliptic curve cryptography. Education and career Montgomery began his undergraduate career at the University of California, Riverside, in 1965 and transferred to Berkeley in 1967, earning a BA in mathematics in 1969 and an MA in mathematics in 1971, He joined the System Development Corporation (SDC) in 1972, where he worked for many years as a programmer implementing algorithms for the CDC 7600 and PDP series of computers, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
San Francisco
San Francisco (; Spanish language, Spanish for "Francis of Assisi, Saint Francis"), officially the City and County of San Francisco, is the commercial, financial, and cultural center of Northern California. The city proper is the List of California cities by population, fourth most populous in California and List of United States cities by population, 17th most populous in the United States, with 815,201 residents as of 2021. It covers a land area of , at the end of the San Francisco Peninsula, making it the second most densely populated large U.S. city after New York City, and the County statistics of the United States, fifth most densely populated U.S. county, behind only four of the five New York City boroughs. Among the 91 U.S. cities proper with over 250,000 residents, San Francisco was ranked first by per capita income (at $160,749) and sixth by aggregate income as of 2021. Colloquial nicknames for San Francisco include ''SF'', ''San Fran'', ''The '', ''Frisco'', and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security ( data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synonymo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve Method
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra. Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors. , it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor ''p'' rather than by the size of the number ''n'' to be factored. Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RSA Numbers
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007. RSA Laboratories (which is an acronym of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. , the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, people ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montgomery Multiplication
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced in 1985 by the American mathematician Peter L. Montgomery.Martin Kochanski"Montgomery Multiplication" a colloquial explanation. Montgomery modular multiplication relies on a special representation of numbers called Montgomery form. The algorithm uses the Montgomery forms of and to efficiently compute the Montgomery form of . The efficiency comes from avoiding expensive division operations. Classical modular multiplication reduces the double-width product using division by and keeping only the remainder. This division requires quotient digit estimation and correction. The Montgomery form, in contrast, depends on a constant which is coprime to , and the only division necessary in Montgomery multiplication is division by . The constant can be chosen so that division by is easy, si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Programmed Data Processor
Programmed Data Processor (PDP), referred to by some customers, media and authors as "Programmable Data Processor," is a term used by the Digital Equipment Corporation from 1957 to 1990 for several lines of minicomputers. The name 'PDP' intentionally avoids the use of the term 'computer'. At the time of the first PDPs, computers had a reputation of being large, complicated, and expensive machines. The venture capitalists behind Digital (especially Georges Doriot) would not support Digital's attempting to build a 'computer' and the term 'minicomputer' had not yet been coined. So instead, Digital used their existing line of logic modules to build a ''Programmed Data Processor'' and aimed it at a market that could not afford the larger computers. The various PDP machines can generally be grouped into families based on word length. Series Members of the PDP series include: ;PDP-1: The original PDP, an 18-bit 4-rack machine used in early time-sharing operating system work, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CDC 7600
The CDC 7600 was the Seymour Cray-designed successor to the CDC 6600, extending Control Data's dominance of the supercomputer field into the 1970s. The 7600 ran at 36.4 MHz (27.5 ns clock cycle) and had a 65 Kword primary memory (with a 60-bit word size) using magnetic core and variable-size (up to 512 Kword) secondary memory (depending on site). It was generally about ten times as fast as the CDC 6600 and could deliver about 10 MFLOPS on hand-compiled code, with a peak of 36 MFLOPS.Gordon BellA Seymour Cray Perspective. In addition, in benchmark tests in early 1970 it was shown to be slightly faster than its IBM rival, the IBM System/360, Model 195. When the system was released in 1967, it sold for around $5 million in base configurations, and considerably more as options and features were added. Among the 7600's notable state-of-the-art contributions, beyond extensive pipelining, was the physical C-shape, which both reduced floor space and dramatically increased pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Of Arts
A Master of Arts ( la, Magister Artium or ''Artium Magister''; abbreviated MA, M.A., AM, or A.M.) is the holder of a master's degree awarded by universities in many countries. The degree is usually contrasted with that of Master of Science. Those admitted to the degree have typically studied subjects within the scope of the humanities and social sciences, such as history, literature, languages, linguistics, public administration, political science, communication studies, law or diplomacy; however, different universities have different conventions and may also offer the degree for fields typically considered within the natural sciences and mathematics. The degree can be conferred in respect of completing courses and passing examinations, research, or a combination of the two. The degree of Master of Arts traces its origins to the teaching license or of the University of Paris, designed to produce "masters" who were graduate teachers of their subjects. Europe Czech Republic a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bachelor Of Arts
Bachelor of arts (BA or AB; from the Latin ', ', or ') is a bachelor's degree awarded for an undergraduate program in the arts, or, in some cases, other disciplines. A Bachelor of Arts degree course is generally completed in three or four years, depending on the country and institution. * Degree attainment typically takes four years in Afghanistan, Armenia, Azerbaijan, Bangladesh, Brazil, Brunei, China, Egypt, Ghana, Greece, Georgia, Hong Kong, Indonesia, Iran, Iraq, Ireland, Japan, Kazakhstan, Kenya, Kuwait, Latvia, Lebanon, Lithuania, Mexico, Malaysia, Mongolia, Myanmar, Nepal, Netherlands, Nigeria, Pakistan, the Philippines, Qatar, Russia, Saudi Arabia, Scotland, Serbia, South Korea, Spain, Sri Lanka, Taiwan, Thailand, Turkey, Ukraine, the United States and Zambia. * Degree attainment typically takes three years in Albania, Australia, Bosnia and Herzegovina, the Caribbean, Iceland, India, Israel, Italy, New Zealand, Norway, South Africa, Switzerland, the Canadian province of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of California, Riverside
The University of California, Riverside (UCR or UC Riverside) is a public land-grant research university in Riverside, California. It is one of the ten campuses of the University of California system. The main campus sits on in a suburban district of Riverside with a branch campus of in Palm Desert. In 1907, the predecessor to UCR was founded as the UC Citrus Experiment Station, Riverside which pioneered research in biological pest control and the use of growth regulators responsible for extending the citrus growing season in California from four to nine months. Some of the world's most important research collections on citrus diversity and entomology, as well as science fiction and photography, are located at Riverside. UCR's undergraduate College of Letters and Science opened in 1954. The Regents of the University of California declared UCR a general campus of the system in 1959, and graduate students were admitted in 1961. To accommodate an enrollment of 21,000 stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve Cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.Commercial National Security Algorithm Suite and Quantum Computing FAQ U.S. National Security Agency, January 2016. Elliptic curves are applicable for , s, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Side-channel Attack
In computer security, a side-channel attack is any attack based on extra information that can be gathered because of the fundamental way a computer protocol or algorithm is implemented, rather than flaws in the design of the protocol or algorithm itself (e.g. flaws found in a cryptanalysis of a cryptographic algorithm) or minor, but potentially devastating, mistakes or oversights in the implementation. (Cryptanalysis also includes searching for side-channel attacks.) Timing information, power consumption, electromagnetic leaks, and sound are examples of extra information which could be exploited to facilitate side-channel attacks. Some side-channel attacks require technical knowledge of the internal operation of the system, although others such as differential power analysis are effective as black-box attacks. The rise of Web 2.0 applications and software-as-a-service has also significantly raised the possibility of side-channel attacks on the web, even when transmissions betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
