National Semiconductor 48-bit and 56-bit ECC polynomials

[Eric Smith]

I'm not looking forward to trying to reverse-engineer 48-bit and 56-bit ECC
polynomials. However, they usually tried to choose polynomials with
relatively few terms, to minimize the number of XOR gates needed in the
hardware.

The common "Glover" 32-bit polynomial was:
    x^32 + x^28 + x^26 + x^19 + x^17 + x^10 + x^6 + x^2 + x^0

WD's 56-bit polynomial was
    x^56 + x^52 + x^50 + x^43 + x^41 + x^34 + x^30 + x^26 + x^24 + x^8 + x^0

Assuming that National's 56-bit polynomials don't use any fewer terms than
the 32-bit, nor any more terms than the WD 56-bit, there are not quite 8
billion possibilities to brute-force. x^n and x^0 are always used, so the
size of the search space is comb(55,9) + comb(55,8) + comb(55, 7). That
would take to long to brute-force search in software on a general purpose
CPU, so I think I'd have to code it for a GPU or FPGA.

There might be some other short-cuts to reducing the search space, but I
haven't yet given it a lot of thought.