Also, it must be kept in mind that CRCs, i.e. the remainders of polynomial division, are not the only simple way to obtain predictable error properties.
A simple alternative is to evaluate the polynomial corresponding to the data at a certain point and use the value as the error detection code. The properties about short bursts are identical.
Polynomial evaluation is advantageous at large hash sizes instead of CRC, because it matters very little at which point you choose to evaluate the polynomial, as long as it is non-null and non-unit.
With CRCs, the choice of the primitive polynomial to be used as divisor is critical and you cannot choose just a random 128-bit sequence and hope that it happens to correspond to a primitive polynomial and also to one with good error detection properties.
For example, if you want guaranteed error-burst detection for a data size where the known 32-bit or 64-bit CRCs are inadequate, you may use the Poly1305 evaluation functions available in most cryptographic libraries for data authentication.
When used in cryptography, the point chosen for the polynomial evaluation must be secret, but for error detection it may be fixed and public and you get error detection properties equivalent with an 128-bit CRC.
An optimized 128-bit CRC might be a little better, but good luck to finding the right 128-bit primitive polynomial.
A simple alternative is to evaluate the polynomial corresponding to the data at a certain point and use the value as the error detection code. The properties about short bursts are identical.
Polynomial evaluation is advantageous at large hash sizes instead of CRC, because it matters very little at which point you choose to evaluate the polynomial, as long as it is non-null and non-unit.
With CRCs, the choice of the primitive polynomial to be used as divisor is critical and you cannot choose just a random 128-bit sequence and hope that it happens to correspond to a primitive polynomial and also to one with good error detection properties.
For example, if you want guaranteed error-burst detection for a data size where the known 32-bit or 64-bit CRCs are inadequate, you may use the Poly1305 evaluation functions available in most cryptographic libraries for data authentication.
When used in cryptography, the point chosen for the polynomial evaluation must be secret, but for error detection it may be fixed and public and you get error detection properties equivalent with an 128-bit CRC.
An optimized 128-bit CRC might be a little better, but good luck to finding the right 128-bit primitive polynomial.