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:
- <math>a' = a + b \, \pmod{2^n}</math>
- <math>b' = a + 2b\, \pmod{2^n}</math>
To reverse this, clearly:
- <math>b = b' - a' \, \pmod{2^n}</math>
- <math>a = 2a' - b' \, \pmod{2^n}</math>
On the other hand, the transformation for SAFER+ encryption is as follows:
- <math>a' = 2a + b \, \pmod{2^n}</math>
- <math>b' = a + b\, \pmod{2^n}</math>
GeneralizationEdit
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:
- <math>H_1 = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}</math>
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:
- <math>H_n = \begin{bmatrix} 2 \times H_{n-1} & H_{n-1} \\ H_{n-1} & H_{n-1} \end{bmatrix}</math>
For example:
- <math>H_2 = \begin{bmatrix} 4 & 2 & 2 & 1 \\ 2 & 2 & 1 & 1 \\ 2 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}</math>
See alsoEdit
This is the Kronecker product of an Arnold Cat Map matrix with a Hadamard matrix.
ReferencesEdit
- James Massey, "On the Optimality of SAFER+ Diffusion", 2nd AES Conference, 1999. [1]
- Bruce Schneier, John Kelsey, Doug Whiting, David Wagner, Chris Hall, "Twofish: A 128-Bit Block Cipher", 1998. [2]
- Helger Lipmaa. On Differential Properties of Pseudo-Hadamard Transform and Related Mappings. INDOCRYPT 2002, LNCS 2551, pp 48-61, 2002.[3]