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Cyclic Shift
In combinatorial mathematics, a circular shift is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse operation. A circular shift is a special kind of cyclic permutation, which in turn is a special kind of permutation. Formally, a circular shift is a permutation σ of the ''n'' entries in the tuple such that either :\sigma(i)\equiv (i+1) modulo ''n'', for all entries ''i'' = 1, ..., ''n'' or :\sigma(i)\equiv (i-1) modulo ''n'', for all entries ''i'' = 1, ..., ''n''. The result of repeatedly applying circular shifts to a given tuple are also called the circular shifts of the tuple. For example, repeatedly applying circular shifts to the four-tuple (''a'', ''b'', ''c'', ''d'') successively gives * (''d'', ''a'', ''b'', ''c''), * (''c'', ''d'', ''a'', ''b''), * (''b'', ''c'', ''d'', ''a''), * (''a'', ''b'', ''c'', ''d'') (the original four-tup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permutation of elements. Pre- multiplying an -row matrix by a permutation matrix , forming , results in permuting the rows of , while post-multiplying an -column matrix , forming , permutes the columns of . Every permutation matrix ''P'' is orthogonal, with its inverse equal to its transpose: P^=P^\mathsf. Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative. The two permutation/matrix correspondences There are two natural one-to-one correspondences between permutations and permutation matrices, one of which works along the rows of the matrix, the other along its columns. Here is an example, starting with a permutation in two-line form at the upper left: :\begin \pi\colon\begin1&2&3&4\\3&2&4&1\e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, 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 (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String (computer Science)
In computer programming, a string is traditionally a sequence of character (computing), characters, either as a literal (computer programming), literal constant or as some kind of Variable (computer science), variable. The latter may allow its elements to be Immutable object, mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or word (computer architecture), words) that stores a sequence of elements, typically characters, using some character encoding. More general, ''string'' may also denote a sequence (or List (abstract data type), list) of data other than just characters. Depending on the programming language and precise data type used, a variable (programming), variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Code
In coding theory, block codes are a large and important family of Channel coding, error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of ''all'' block codes in a unified way. Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay code (other), Golay codes, Reed–Muller codes and Polar code (coding theory), Polar codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Code
In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction. Definition Let \mathcal be a linear code over a finite field (also called '' Galois field'') GF(q) of block length n. \mathcal is called a cyclic code if, for every Code word (communication), codeword c=(c_1,\ldots,c_n) from \mathcal, the word (c_n,c_1,\ldots,c_) in GF(q)^n obtained by a circular shift, cyclic right shift of components is again a codeword. Because one cyclic right shift is equal to n-1 cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore, the linear code \mathcal is cyclic precisely when it is invariant under all cyclic shifts. Cyclic codes have some additional structural constraint on the codes. They are based on Galois fields and because of their structural properti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotate Right
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The ends of the external ''axis of revolution'' can be ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
