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:

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

See also

• SAFER
• Twofish

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

References

• 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]

External links

• Fast Pseudo-Hadamard Transforms