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:

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

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

To reverse this, clearly:

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

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

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

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

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

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:

${\displaystyle H_{1}={\begin{bmatrix}2&1\\1&1\end{bmatrix}}}$

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:

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

For example:

${\displaystyle H_{2}={\begin{bmatrix}4&2&2&1\\2&2&1&1\\2&1&2&1\\1&1&1&1\end{bmatrix}}}$

• SAFER
• Twofish

This is the Kronecker product of an Arnold Cat Map matrix with a Hadamard matrix.

• 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]

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• Fast Pseudo-Hadamard Transforms