Whirlpool is an interesting little hash function in the Miyaguchi-Preneel family.
In my mind, it's most interesting feature is the design of internal cipher W, where the distinction between key and message is dropped, providing a symmetric symmetric cipher. This conceivably removes the need to test for weaknesses to related-key attacks, since if you can do something in the key you could just do it in the message, and makes a lot of sense as a hash function internal.
However, why did they change the field polynomial from that used in AES?
Table 1: Differences between RIJNDAEL and W
Block size (bits) 128, 160, 192, 224, or 256 always 512
Number of rounds 10, 11, 12, 13, or 14 always 10
Key schedule dedicated a priori algorithm the round function itself
$GF(2^8)$ reduction polynomial
Origin of the S-box
mapping u → u-1 over $GF(2^8)$,
plus affine transformrecursive structure (see below)
Origin of the round constants polynomials $x^i$ over $GF(2^8)$ successive entries of the S-box
left-multiplication by the
4×4 circulant MDS matrix
cir(2, 3, 1, 1)right-multiplication by the
8×8 circulant MDS matrix
cir(1, 1, 4, 1, 8, 5, 2, 9)
The W S-box, which in the original submission is generated entirely at random (i.e. lacks any internal structure), by a recursive structure: the new 8×8 substitution box is composed of smaller 4×4 "mini-boxes" (the exponential E-box, its inverse, and the pseudo-randomly generated R-box).
More generally, why were several of the internal constants and methods changed?