Pseudo-Hadamard transform

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:


 * $$a' = a + b \, \pmod{2^n}$$


 * $$b' = a + 2b\, \pmod{2^n}$$

To reverse this, clearly:


 * $$b = b' - a' \, \pmod{2^n}$$


 * $$a = 2a' - b' \, \pmod{2^n}$$

On the other hand, the transformation for SAFER+ encryption is as follows:


 * $$a' = 2a + b \, \pmod{2^n}$$


 * $$b' = a + b\, \pmod{2^n}$$

Generalization
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:


 * $$H_1 = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$$

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:


 * $$H_n = \begin{bmatrix} 2 \times H_{n-1} & H_{n-1} \\ H_{n-1} & H_{n-1} \end{bmatrix}$$

For example:


 * $$H_2 = \begin{bmatrix} 4 & 2 & 2 & 1 \\ 2 & 2 & 1 & 1 \\ 2 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}$$