Optimize y = x*x in Galois field arithmetic

I have this C-code to do multiplications over GF(8):

int32_t GaloisMultiply (int32_t a, int32_t b) 
{
    int32_t i;
    int32_t mask = 0x100;
    int32_t y = 0;

    for(i=0;i<8;i++) 
    {
        if(b & mask) 
        {
            y ^= a;
        }
        mask >>= 1;
        y <<= 1;
    }

    if(b & 0x1) 
    {
        y ^= a;
    }

    return(y);
}

That's more or less the text-book implementation.

I wonder if I there is a clever optimization for above algorithm if I can assert that a is always b, e.g. I do squaring instead of multiplication. I'm not after a cryptographic use btw. I just want to make use of the fact that x*x in GF(8) interleaves the bits of x with zero bits one by one.

There are already quite clever methods to do the bit interleaving, but since I've found out that x*x in GF(8) does the bit interleaving thing (by accident) I can't stop trying to use it for bit-interleaving optimizations.

Any ideas?

Table-based? link

And when you are limited to x*x, it's a sparse matrix.

Here's another good paper (and a library)