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Optimal Solutions For The Rubik's Cube
Optimal solutions for the Rubik's Cube are solutions that are the shortest in some sense. There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second and more popular is to count the number of outer-layer twists, called "face turns". A move to turn an outer layer two quarter (90°) turns in the same direction would be counted as two moves in the quarter turn metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric"). It means that the length of an optimal solution in HTM ≤ the length of an optimal solution in QTM. The maximal number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximal number of quarter turns is 26. These numbers are also the Distance (graph theory), diameters of the corresponding Cayley graphs of the Rubik's Cube group. In STM (slice turn metric) the minimal number of turns is unknown, lower bound being 18 and upper bound being 20. A randomly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scramble
Scramble, Scrambled, or Scrambling may refer to: Arts and entertainment Film and television * ''Scramble'' (film), a 1970 British children's sports drama * ''Scrambled'' (film), a 2023 American comedy-drama * ''Scrambled!'', a British children's TV programme 2014–2021 * "Scrambled" (''Law & Order''), a 1998 TV episode Music * ''Scramble'' (album), by the Coathangers, 2009 * ''Scrambles'' (album), by Bomb the Music Industry!, 2009 * "Scramble" (song), by Yui Horie; opening theme of the anime series ''School Rumble'', 2004 Other media * Scramble (comics), a Marvel Comics supervillain * ''Scramble'' (video game), a 1981 arcade game Codes and language * Scrambler, in telecommunications, a device that encodes a message at the transmitter to make the message unintelligible * Scrambling (linguistics), variation of word order Sports * Scramble (golf), a team play format in golf * Scrambling, a method of ascending rocky faces and ridges * Motorcycle scrambling, or motocross, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard E
Richard is a male given name. It originates, via Old French, from compound of the words descending from Proto-Germanic language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include " Richie", " Dick", " Dickon", " Dickie", " Rich", " Rick", "Rico (name), Rico", " Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Schroeppel
Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois. His research has included magic squares, elliptic curves, and cryptography. In 1964, Schroeppel won first place in the United States among over 225,000 high school students in the Annual High School Mathematics Examination, a contest sponsored by the Mathematical Association of America and the Society of Actuaries. In both 1966 and 1967, Schroeppel scored among the top 5 in the U.S. in the William Lowell Putnam Mathematical Competition. In 1973 he discovered that there are 275,305,224 normal magic squares of order 5. In 1998–1999 he designed the Hasty Pudding Cipher, which was a candidate for the Advanced Encryption Standard, and he is one of the designers of the SANDstorm hash, a submission to the NIST SHA-3 competition. Among other contributions, Schroeppel was the first to recognize the sub-exponential running time of certain integer factoring algorithms. While not entirely rigorous, his proof t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gábor Tardos
Gábor Tardos (born 11 July 1964) is a Hungarian mathematician, currently a professor at Central European University and previously a Canada Research Chair at Simon Fraser University. He works mainly in combinatorics and computer science. He is the younger brother of Éva Tardos. Education and career Gábor Tardos received his PhD in Mathematics from Eötvös University, Budapest in 1988. His counsellors were László Babai and Péter Pálfy. He held postdoctoral posts at the University of Chicago, Rutgers University, University of Toronto and the Princeton Institute for Advanced Study. From 2005 to 2013, he served as a Canada Research Chair of discrete and computational geometry at Simon Fraser University. He then returned to Budapest to the Alfréd Rényi Institute of Mathematics where he has served as a research fellow since 1991. Mathematical results Tardos started with a result in universal algebra: he exhibited a maximal clone of order-preserving operations that is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shahar Mozes
Shahar Mozes () is an Israeli mathematician. Mozes received in 1991, his doctorate from the Hebrew University of Jerusalem with thesis ''Actions of Cartan subgroups'' under the supervision of Hillel Fürstenberg. (doctoral dissertation) At the Hebrew University of Jerusalem, Mozes became in 1993 a senior lecturer, in 1996 associate professor, and in 2002 a full professor. Moses does research on Lie groups and discrete subgroups of Lie groups, geometric group theory, ergodic theory, and aperiodic tilings. His collaborators include Jean Bourgain, Alex Eskin, Elon Lindenstrauss, Gregory Margulis, and Hee Oh. In 2000 Mozes received the Erdős Prize. In 1998 he was an invited speaker with talk ''Products of trees, lattices and simple groups '' at the International Congress of Mathematicians (ICM) in Berlin. He was a plenary speaker at the ICM Satellite Conference on "Geometry Topology and Dynamics in Negative Curvature" held at the Raman Research Institute of the International Centre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amos Fiat
Amos Fiat (; born December 1, 1956) is an Israeli computer scientist, a professor of computer science at Tel Aviv University. He is known for his work in cryptography, online algorithms, and algorithmic game theory. Biography Fiat earned his Ph.D. in 1987 from the Weizmann Institute of Science under the supervision of Adi Shamir. After postdoctoral studies with Richard Karp and Manuel Blum at the University of California, Berkeley, he returned to Israel, taking a faculty position at Tel Aviv University. Research Many of Fiat's most highly cited publications concern cryptography, including his work with Adi Shamir on digital signatures (leading to the Fiat–Shamir heuristic for turning interactive identification protocols into signature schemes) and his work with David Chaum and Moni Naor on electronic money, used as the basis for the ecash system. With Shamir and Uriel Feige in 1988, Fiat invented the Feige–Fiat–Shamir identification scheme, a method for using public-key ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adi Shamir
Adi Shamir (; born July 6, 1952) is an Israeli cryptographer and inventor. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the Feige–Fiat–Shamir identification scheme (along with Uriel Feige and Amos Fiat), one of the inventors of differential cryptanalysis and has made numerous contributions to the fields of cryptography and computer science. Biography Adi Shamir was born in Tel Aviv. He received a Bachelor of Science (BSc) degree in mathematics from Tel Aviv University in 1973 and obtained an MSc and PhD in computer science from the Weizmann Institute in 1975 and 1977 respectively. He spent a year as a postdoctoral researcher at the University of Warwick and did research at Massachusetts Institute of Technology (MIT) from 1977 to 1980. Scientific career In 1980, he returned to Israel, joining the faculty of Mathematics and Computer Science at the Weizmann Institute. Starting from 2006, he is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meet-in-the-middle Approach
The meet-in-the-middle attack (MITM), a known-plaintext attack, is a generic space–time tradeoff cryptographic attack against encryption schemes that rely on performing multiple encryption operations in sequence. The MITM attack is the primary reason why Double DES is not used and why a Triple DES key (168-bit) can be brute-forced by an attacker with 256 space and 2112 operations. Description When trying to improve the security of a block cipher, a tempting idea is to encrypt the data several times using multiple keys. One might think this doubles or even ''n''-tuples the security of the multiple-encryption scheme, depending on the number of times the data is encrypted, because an exhaustive search on all possible combinations of keys (simple brute force) would take 2''n''·''k'' attempts if the data is encrypted with ''k''-bit keys ''n'' times. The MITM attack is a generic attack which weakens the security benefits of using multiple encryptions by storing intermediate value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
