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Mathematics Of CRC
The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around. Any string of bits can be interpreted as the coefficients of a polynomial of this sort, and a message has a valid CRC if it divisible by (i.e. is a multiple of) an agreed-on ''generator polynomial''. CRCs are convenient and popular because they have good error-detection properties and such a multiple may be easily constructed from any ''message polynomial'' M(x) by appending an n-bit ''remainder polynomial'' R(x) to produce W(x) = M(x) \cdot x^n + R(x), where n is the degree of the generator polynomial. Although the separation of W(x) into the message part M(x) and the checksum part R(x) is convenient for use of CRCs, the error-detection properties do not make a distinction; errors are detected equally anywh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Redundancy Check
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short ''check value'' attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption. CRCs can be used for error correction (see bitfilters). CRCs are so called because the ''check'' (data verification) value is a ''redundancy'' (it expands the message without adding information) and the algorithm is based on ''cyclic'' codes. CRCs are popular because they are simple to implement in binary hardware, easy to analyze mathematically, and particularly good at detecting common errors caused by noise in transmission channels. Because the check value has a fixed length, the function that generates it is occasionally used as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Checksum
A checksum is a small-sized block of data derived from another block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data integrity but are not relied upon to verify data authenticity. The procedure which generates this checksum is called a checksum function or checksum algorithm. Depending on its design goals, a good checksum algorithm usually outputs a significantly different value, even for small changes made to the input. This is especially true of cryptographic hash functions, which may be used to detect many data corruption errors and verify overall data integrity; if the computed checksum for the current data input matches the stored value of a previously computed checksum, there is a very high probability the data has not been accidentally altered or corrupted. Checksum functions are related to hash functions, fingerprints, randomization functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Burst
In telecommunications, a burst error or error burst is a contiguous sequence of symbols, received over a communication channel, such that the first and last symbols are in error and there exists no contiguous subsequence of ''m'' correctly received symbols within the error burst. The integer parameter ''m'' is referred to as the ''guard band'' of the error burst. The last symbol in a burst and the first symbol in the following burst are accordingly separated by ''m'' correct symbols or more. The parameter ''m'' should be specified when describing an error burst. Channel model The Gilbert–Elliott model is a simple channel model introduced by Edgar Gilbert and E. O. Elliott that is widely used for describing burst error patterns in transmission channels and enables simulations of the digital error performance of communications links. It is based on a Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Polynomial (field Theory)
In field theory (mathematics), finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial (field theory), minimal polynomial of a primitive element (finite field), primitive element of the finite field . This means that a polynomial of degree with coefficients in is a ''primitive polynomial'' if it is monic polynomial, monic and has a root in such that \ is the entire field . This implies that is a primitive root of unity, primitive ()-root of unity in . Properties * Because all minimal polynomials are irreducible polynomial, irreducible, all primitive polynomials are also irreducible. * A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by ''x''. Over GF(2), is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by (it has 1 as a root). * An irreducible polynomial ''F''(''x'') of degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Polynomial
In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n = x^n p(x^). That is, the coefficients of are the coefficients of in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when :p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n, the conjugate reciprocal polynomial, denoted , is defined by, :p^(z) = \overline + \overlinez + \cdots + \overlinez^n = z^n\overline, where \overline denotes the complex conjugate of a_i, and is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if . The coefficients of a self-reciprocal polynomial satisfy for all . Properties Reciprocal polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nibble
In computing, a nibble, or spelled nybble to match byte, is a unit of information that is an aggregation of four- bits; half of a byte/ octet. The unit is alternatively called nyble, nybl, half-byte or tetrade. In networking or telecommunications, the unit is often called a semi-octet, quadbit, or quartet. As a nibble can represent sixteen () possible values, a nibble value is often shown as a hexadecimal digit (hex digit). A byte is two nibbles, and therefore, a value can be shown as two hex digits. Four-bit computers use nibble-sized data for storage and operations; as the word unit. Such computers were used in early microprocessors, pocket calculators and pocket computers. They continue to be used in some microcontrollers. In this context, 4-bit groups were sometimes also called characters rather than nibbles. History The term ''nibble'' originates from its representing half a byte, with ''byte'' a homophone of the English word ''bite''. In 2014, David B. Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CRC Polynomial Representation Example
CRC may refer to: Science and technology * Carboniferous Rainforest Collapse, an event at the end of the Carboniferous period * Class-responsibility-collaboration card, used as a brainstorming tool in the design of object-oriented software * Clinical research coordinator, responsible for conducting clinical trials * Colorectal cancer, the development of cancer in the colon or rectum * CRC Energy Efficiency Scheme, formerly the Carbon Reduction Commitment, a UK wide scheme designed to increase energy efficiency in large energy users * ''CRC Handbook of Chemistry and Physics'', a science reference handbook published by CRC Press * ''CRC Standard Mathematical Tables'', a mathematics reference handbook published by CRC Press * Cyclic redundancy check, a type of hash function used to produce a checksum in order to detect errors in data storage or transmission Organizations * California Rehabilitation Center, a state prison in the United States Of America * Canadian Red Cross * Capita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Register
A shift register is a type of digital circuit using a cascade of flip-flop (electronics), flip-flops where the output of one flip-flop is connected to the input of the next. They share a single clock signal, which causes the data stored in the system to shift from one location to the next. By connecting the last flip-flop back to the first, the data can cycle within the shifters for extended periods, and in this configuration they were used as computer memory, displacing delay-line memory systems in the late 1960s and early 1970s. In most cases, several parallel shift registers would be used to build a larger memory pool known as a "bit array". Data was stored into the array and read back out in parallel, often as a computer word, while each bit was stored serially in the shift registers. There is an inherent trade-off in the design of bit arrays; putting more flip-flops in a row allows a single shifter to store more bits, but requires more clock cycles to push the data through all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asynchronous Transfer Mode
Asynchronous Transfer Mode (ATM) is a telecommunications standard defined by the American National Standards Institute and International Telecommunication Union Telecommunication Standardization Sector (ITU-T, formerly CCITT) for digital transmission of multiple types of traffic. ATM was developed to meet the needs of the Broadband Integrated Services Digital Network as defined in the late 1980s, and designed to integrate telecommunication networks. It can handle both traditional high-throughput data traffic and Real-time computing, real-time, low-latency content such as telephony (voice) and video.ATM Forum, The User Network Interface (UNI), v. 3.1, , Prentice Hall PTR, 1995, page 2. ATM is a cell switching technology, providing functionality that combines features of circuit switching and packet switching networks by using asynchronous communication, asynchronous time-division multiplexing.McDysan (1999), p. 287. ATM was seen in the 1990s as a competitor to Ethernet and networ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endianness
file:Gullivers_travels.jpg, ''Gulliver's Travels'' by Jonathan Swift, the novel from which the term was coined In computing, endianness is the order in which bytes within a word (data type), word of digital data are transmitted over a data communication medium or Memory_address, addressed (by rising addresses) in computer memory, counting only byte Bit_numbering#Bit significance and indexing, significance compared to earliness. Endianness is primarily expressed as big-endian (BE) or little-endian (LE), terms introduced by Danny Cohen (computer scientist), Danny Cohen into computer science for data ordering in an Internet Experiment Note published in 1980. Also published at The adjective ''endian'' has its origin in the writings of 18th century Anglo-Irish writer Jonathan Swift. In the 1726 novel ''Gulliver's Travels'', he portrays the conflict between sects of Lilliputians divided into those breaking the shell of a boiled egg from the big end or from the little end. By analogy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carry-less Product
The carry-less product of two binary numbers is the result of carry-less multiplication of these numbers. This operation conceptually works like long multiplication except for the fact that the carry is discarded instead of applied to the more significant position. It can be used to model operations over finite fields, in particular multiplication of polynomials from GF(2) 'X'' the polynomial ring over GF(2). The operation is also known as an XOR multiplication, as carry-discarding addition is equivalent to an exclusive or. Definition Given two numbers \textstyle a=\sum_i a_i2^i and \textstyle b=\sum_i b_i2^i, with a_i,b_i\in\ denoting the bits of these numbers, the carry-less product of these two numbers is defined to be \textstyle c=\sum_i c_i2^i, with each bit c_i computed as the exclusive or of products of bits from the input numbers as follows: : c_i=\bigoplus_^i a_jb_ Example Consider ''a'' = 101000102 and ''b'' = 100101102, with all numbers given in binary. Then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
