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Mathematics of cyclic redundancy checks. The cyclic redundancy check (CRC) is based on division in the ring of polynomials over the finite field GF (2) (the 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 ...
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. [ 1][ 2] Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents.
Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions.
Usually, the second sum will be multiplied by 2 32 and added to the simple checksum, effectively stacking the sums side-by-side in a 64-bit word with the simple checksum at the least significant end. This algorithm is then called the Fletcher-64 checksum. The use of the modulus 2 32 − 1 = 4,294,967,295 is also generally implied. The rationale ...
For instance, in GF(5), 4 + 3 = 7 is reduced to 2 modulo 5. Division is multiplication by the inverse modulo p, which may be computed using the extended Euclidean algorithm. A particular case is GF(2), where addition is exclusive OR (XOR) and multiplication is AND. Since the only invertible element is 1, division is the identity function.
In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context. The logical operator XOR sums 2 bits ...
up to 2 bits of triplet omitted (cases not shown in table). Though simple to implement and widely used, this triple modular redundancy is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of ...
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.