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Don Coppersmith
Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis. He also improved the quantum Fourier transform discovered by Peter Shor in the same year (1994). He has also worked on algorithms for computing discrete logarithms, the cryptanalysis of RSA, methods for rapid matrix multiplication (see Coppersmith–Winograd algorithm) and IBM's MARS cipher. Don is also a co-designer of the SEAL and Scream ciphers. In 1972, Coppersmith obtained a bachelor's degree in mathematics at the Massachusetts Institute of Technology, and a Masters and Ph.D. in mathematics from Harvard University in 1975 and 1977 respectively. He was a Putnam Fellow each year from 1968–1971, becoming the first four-time Putnam Fellow in history. In 1998, he started ''Ponder This'', an online monthly column on mathematical puz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territories, nine Minor Outlying Islands, and 326 Indian reservations. The United States is also in free association with three Pacific Island sovereign states: the Federated States of Micronesia, the Marshall Islands, and the Republic of Palau. It is the world's third-largest country by both land and total area. It shares land borders with Canada to its north and with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations. With a population of over 333 million, it is the most populous country in the Americas and the third most populous in the world. The national capital of the United States is Washington, D.C. and its most populous city and principal financial center is New York City. Paleo-Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b''''k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ''a'' is an integer ''k'' such that . In number theory, the more commonly used term is index: we can write ''x'' = ind''r'' ''a'' (mod ''m'') (read "the index of ''a'' to the base ''r'' modulo ''m''") for ''r''''x'' ≡ ''a'' (mod ''m'') if ''r'' is a primitive root of ''m'' and gcd(''a'',''m'') = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Definition Let ''G'' be any group. Denote its group operation by mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coppersmith Method
The Coppersmith method, proposed by Don Coppersmith, is a method to find small integer zeroes of univariate or bivariate polynomials modulo a given integer. The method uses the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) to find a polynomial that has the same zeroes as the target polynomial but smaller coefficients. In cryptography, the Coppersmith method is mainly used in attacks on RSA when parts of the secret key are known and forms a base for Coppersmith's attack. Approach Coppersmith's approach is a reduction of solving modular polynomial equations to solving polynomials over the integers. Let F(x) = x^n+a_x^+\ldots +a_1x+a_0 and assume that F(x_0)\equiv 0 \pmod for some integer , x_0, < M^. Coppersmith’s algorithm can be used to find this integer solution . Finding roots over is easy using, e.g., |
Coppersmith's Attack
Coppersmith's attack describes a class of cryptographic attacks on the public-key cryptosystem RSA based on the Coppersmith method. Particular applications of the Coppersmith method for attacking RSA include cases when the public exponent ''e'' is small or when partial knowledge of a prime factor of the secret key is available. RSA basics The public key in the RSA system is a tuple of integers (N, e), where ''N'' is the product of two primes ''p'' and ''q''. The secret key is given by an integer ''d'' satisfying ed \equiv 1 \pmod; equivalently, the secret key may be given by d_p \equiv d \pmod and d_q \equiv d \pmod if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA. Encryption of a message ''M'' produces the ciphertext C \equiv M^e \pmod, which can be decrypted using d by computing C^d \equiv M \pmod. Low public exponent attack In order to reduce encryption or signature verification time, it is useful to use a small public exponent (e). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Puzzle
Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions. Mathematical puzzles require mathematics to solve them. Logic puzzles are a common type of mathematical puzzle. Conway's Game of Life and fractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set, the rules of the puzzle determine all subsequent changes and moves. Many of the puzzles are well known because they were discussed by Martin Gardner in his "Mathematical Games" column in Scientific American. Mathematical puzzles are sometimes used to motivate students in teaching elementary school math problem solving techniques.Kulkarni, DEnjoying Math: Learning Problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the '' American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR. Mission and Vision The mission of the MAA is to advance the understanding of mathematics and its impact on our world. We envision a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Lowell Putnam Mathematical Competition
The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual list of mathematics competitions, mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regardless of the students' nationalities). It awards a scholarship and cash prizes ranging from $250 to $2,500 for the top students and $5,000 to $25,000 for the top schools, plus one of the top five individual scorers (designated as ''#Putnam_Fellows, Putnam Fellows'') is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship), the top 100 individual scorers have their names mentioned in the American Mathematical Monthly (alphabetically ordered within rank), and the names and addresses of the top 500 contestants are mailed to all participating institutions. It is widely considered to be the most prestigious university-level mathematics competition in the world, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scream (cipher)
The Scream cipher is a word-based stream cipher developed by Shai Halevi, Don Coppersmith and Charanjit Jutla from IBM. The cipher is designed as a software efficient stream cipher. The authors describe the goal of the cipher to be a more secure version of the SEAL cipher. The general design of Scream is close to the design of SEAL with block cipher-like round functions. There are two versions of Scream. One of them, ''Scream-F'', reuses the S-boxes from the Advanced Encryption Standard The Advanced Encryption Standard (AES), also known by its original name Rijndael (), is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001. AES is a varian ... (AES) block cipher, while the other, ''Scream'', internally generates new, key-dependent S-boxes as part of the initialization phase. The round function is also based on the AES-round function, but is narrower, 64 bits instead of 128 bits. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SEAL (cipher)
In cryptography, SEAL (Software-Optimized Encryption Algorithm) is a stream cipher optimised for machines with a 32-bit word size and plenty of RAM with a reported performance of around 4 cycles per byte. SEAL is actually a pseudorandom function family in that it can easily generate arbitrary portions of the keystream without having to start from the beginning. This makes it particularly well suited for applications like encrypting hard drives. The first version was published by Phillip Rogaway and Don Coppersmith Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysi ... in 1994. The current version, published in 1997, is 3.0. SEAL, covered by two patents in the United States, both of which are assigned to IBM. References * "Software-efficient pseudorandom function and the use there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MARS (cryptography)
MARS is a block cipher that was IBM's submission to the Advanced Encryption Standard process. MARS was selected as an AES finalist in August 1999, after the AES2 conference in March 1999, where it was voted as the fifth and last finalist algorithm. The MARS design team included Don Coppersmith, who had been involved in the creation of the previous Data Encryption Standard (DES) twenty years earlier. The project was specifically designed to resist future advances in cryptography by adopting a layered, compartmentalized approach. IBM's official report stated that MARS and Serpent were the only two finalists to implement any form of safety net with regard to would-be advances in cryptographic mathematics. The Twofish team made a similar statement about its cipher. MARS has a 128-bit block size and a variable key size of between 128 and 448 bits (in 32-bit increments). Unlike most block ciphers, MARS has a heterogeneous structure: several rounds of a cryptographic core are "jac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
