A large class of checksum algorithms have the following pattern:

- Think of the bits in a file as the coefficients in a polynomial
*P*(*x*). - Divide
*P*(*x*) by a fixed polynomial*Q*(*x*) mod 2 and keep the remainder. - Report the remainder as a sequence of bits.

In practice there’s a little more to the algorithm than this, such as appending the length of the file, but the above pattern is at the heart of the algorithm.

There’s a common misconception that the polynomial *Q*(*x*) is irreducible, i.e. cannot be factored. This may or may not be the case.

## CRC-32

Perhaps the most common choice of *Q* is

*Q*(*x*) = *x*^{32} + *x*^{26} + *x*^{23} + *x*^{22} + *x*^{16} + *x*^{12} + *x*^{11} + *x*^{10} + *x*^{8} + *x*^{7} + *x*^{5} + *x*^{4} + *x*^{3} + *x*^{2} + *x* + 1

This polynomial is used in the `cksum`

utility and is part of numerous standards. It’s know as CRC-32 polynomial, though there are other polynomials occasionally used in 32-bit implementations of the CRC algorithm. And it is far from irreducible as the following Mathematica code shows. The command

Factor[x^32 + x^26 + x^23 + x^22 + x^16 + x^12 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x + 1, Modulus -> 2]

shows that *Q* can be factored as

(1 + *x*)^{5} (1 + *x* + *x*^{3} + *x*^{4} + *x*^{6}) (1 + *x* + *x*^{2} + *x*^{5} + *x*^{6})

(1 + *x* + *x*^{4} + *x*^{6} + *x*^{7}) (1 + *x* + *x*^{4} + *x*^{5} + *x*^{6} + *x*^{7} + *x*^{8})

(Mathematica displays polynomials in increasing order of terms.)

Note that the factorization is valid when done over the field with 2 elements, *GF*(2). Whether a polynomial can be factored, and what the factors are, depends on what field you do your arithmetic in. The polynomial *Q*(*x*) above *is* irreducible as a polynomial with real coefficients. It can be factored working mod 3, for example, but it factors differently mod 3 than it factors mod 2. Here’s the factorization mod 3:

(1 + 2 *x*^{2} + 2 *x*^{3} + *x*^{4} + *x*^{5}) (2 + *x* + 2 *x*^{2} + *x*^{3} + 2 *x*^{4} + *x*^{6} + *x*^{7})

(2 + *x* + *x*^{3} + 2 *x*^{7} + *x*^{8} + *x*^{9} + *x*^{10} + 2 *x*^{12} + *x*^{13} + *x*^{15} + 2 *x*^{16} + *x*^{17} + *x*^{18} + *x*^{19} + *x*^{20})

## CRC-64

The polynomial

*Q(x) = x ^{64}* +

*x*

^{4}+

*x*

^{3}+

*x*+ 1

is known as CRC-64, and is part of several standards, including ISO 3309. This polynomial *is* irreducible mod 2 as the following Mathematica code confirms.

IrreduciblePolynomialQ[x^64 + x^4 + x^3 + x + 1, Modulus -> 2]

The CRC algorithm uses this polynomial mod 2, but out of curiosity I checked whether it is irreducible in other contexts. The following code tests whether the polynomial is irreducible modulo the first 100 primes.

Table[IrreduciblePolynomialQ[x^64 + x^4 + x^3 + x + 1, Modulus -> Prime[n]], {n, 1, 100}]

It is irreducible mod *p* for *p* = 2, 233, or 383, but not for any other primes up to 541. It’s also irreducible over the real numbers.

Since *Q* is irreducible mod 2, the check sum essentially views its input *P*(*x*) as a member of the finite field *GF*(2^{64}).

Ben Eater, one of my favorite YouTube channels did an amazing breakdown of CRC including building the circuitry on breadboards.

https://youtu.be/izG7qT0EpBw?si=u3Olf3seXa4eU6MO