Modulo two arithmetic is simple single-bit binary arithmetic with all carries or borrows ignored. Each digit is considered independently. This article talks about how modulo two addition is equivalent to modulo two subtraction, and can be performed using an exclusive OR operation followed by a brief on Polynomial division where remainder forms the CRC checksum.

For example, we can add two binary numbers X and Y as follows:

10101001 (X) + 00111010 (Y) = 10010011 (Z)

From this example the modulo two addition is equivalent to an exclusive OR operation. What is less obvious is that modulo two subtraction gives the same results as an addition.

From the previous example let’s add X and Z:

10101001 (X) + 10010011 (Z) = 00111010 (Y)

In our previous example we have seen how X + Y = Z therefore Y = Z – X, but the example above shows that Z+X = Y also, hence modulo two addition is equivalent to modulo two subtraction, and can be performed using an exclusive OR operation.

In integer division dividing A by B will result in a quotient Q, and a remainder R. Polynomial division is similar except that when A and B are polynomials, the remainder is a polynomial, whose degree is less than B.

The key point here is that any change to the polynomial A causes a change to the remainder R. This behavior forms the basis of the cyclic redundancy checking.

If we consider a polynomial, whose coefficients are zeros and ones (modulo two), this polynomial can be easily represented by its coefficients as binary powers of two.

In terms of cyclic redundancy calculations, the polynomial A would be the binary message string or data and polynomial B would be the generator polynomial. The remainder R would be the cyclic redundancy checksum. If the data changed or became corrupt, then a different remainder would be calculated.

Although the algorithm for cyclic redundancy calculations looks complicated, it only involves shifting and exclusive OR operations. Using modulo two arithmetic, division is just a shift operation and subtraction is an exclusive OR operation.

Cyclic redundancy calculations can therefore be efficiently implemented in hardware, using a shift register modified with XOR gates. The shift register should have the same number of bits as the degree of the generator polynomial and an XOR gate at each bit, where the generator polynomial coefficient is one.

my 2 cents...

ReplyDeleteThe polynomial represents the behaviour of the output of the last flip flop.

The exponent on the variable x represents the number of clock cycles of delay.

From polynomials to hardware:

The maximum exponent denotes the number of flops The other exponents denote the flops that tap off of feedback line from last flop

Here is an article that describes different parallel CRC algorithms for Verilog/VHDL: CRC article

ReplyDeleteThe online tool that generates parallel/serial CRC is here:CRC generation tool

Thanks

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