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8 maj 2024 · Use modulo-2 binary division to divide binary data by the key and store remainder of division. Perform modulo-2 division again and if the remainder is 0, then there are no errors. In this article we will focus only on finding the remainder i.e. check word and the code word.
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16 lip 2017 · A CRC treats the data as a string of 1 bit coefficients of a polynomial, since the coefficients are numbers modulo 2. From a math perspective, for an n bit CRC, the data polynomial is multiplied by x^n, effectively adding n 0 bit coefficients to the data, then dividing that data + zeroes by a n+1 bit CRC polynomial, resulting in a n bit ...
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.
We must then calculate the required remainder from a modulo-2 divide and add this to the data, in order that the remainder will be zero when we perform the divide. To take a simple example, we have 32, and make it divisible by 9, we add a ‘0’ to make ‘320’, and now divide by 9, to give 35 remainder 4.
The division uses the Modulo-2 arithmetic. Modulo-2 calculation is simply realized by XOR’ing two numbers. 0 1. 1 0. In this example calculation, the message is two bytes long. In general, the message can have any length in bytes.
Arithmetic over the field of integers mod 2 is simply arithmetic on single bit binary numbers with all carries (overflows) ignored. So 1 + 1 = 0 and so does 1 - 1. In fact, addition and subtraction are equivalent in this form of arithmetic.
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.
