Understanding Cyclic Redundancy Code algorithm for beginners

Question

at section 5.5 of the PNG Specification, it discusses this concept in the PNG file format called \"CRC\" or \"Cyclic Redundancy Code\". I\'ve never heard of it before, so I\

Answer 1

I would recommend reading Ross Williams' classic "A Painless Guide to CRC Error Detection Algorithms". Therein you will find in-depth explanations and examples.

The polynomial is simply a different way to interpret a string of bits. When you have n bits in a register, they are most commonly interpreted either as just that, a list of n independent bits, or they are interpreted as an integer, where you multiply each bit by two raised to the powers 0 to n-1 and add them up. The polynomial representation is where you instead interpret each bit as the coefficient of a polynomial. Since a bit can only be a 0 or a 1, the resulting polynomials never actually show the 0 or 1. Instead the xⁿ term is either there or not. So the four bits 1011 can be interpreted to be 1 x³ + 0 x² + 1 x¹ + 1 x⁰ = x³ + x + 1. Note that I made the choice that the most significant bit was the coefficient of the x³ term. That is an arbitrary choice, where I could have chosen the other direction.

As for what x is, it is simply a placeholder for the coefficient and the power of x. You never set x to some value, nor determine anything about x. What it does is allow you to operate on those bit strings as polynomials. When doing operations on these polynomials, you treat them just like the polynomials you had in algebra class, except that the coefficients are constrained to the field GF(2), where the coefficients can only be 0 or 1. Multiplication becomes the and operation, and addition becomes the exclusive-or operation. So 1 plus 1 is 0. You get a new and different way to add, multiply, and divide strings of bits. That different way is key to many error detection and correction schemes.

It is interesting, but ultimately irrelevant, that if you set x to 2 in the polynomial representation of a string of bits (with the proper ordering choice), you get the integer interpretation of that string of bits.