The pseudo-Hadamard transform is a reversible transformation of a bit string that provides cryptographic diffusion. See Hadamard transform.
The bit string must be of even length so that it can be split into two bit strings a and b of equal lengths, each of n bits. To compute the transform for Twofish algorithm, a' and b', from these we use the equations:
To reverse this, clearly:
On the other hand, the transformation for SAFER+ encryption is as follows:
The above equations can be expressed in matrix algebra, by considering a and b as two elements of a vector, and the transform itself as multiplication by a matrix of the form:
The inverse can then be derived by inverting the matrix.
However, the matrix can be generalised to higher dimensions, allowing vectors of any power-of-two size to be transformed, using the following recursive rule:
For example:
This is the Kronecker product of an Arnold Cat Map matrix with a Hadamard matrix.
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