Pseudo-Hadamard transform

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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 it can be split into two bit strings a and b of equal lengths, each of n bits. To compute the transform, a' and b', from these we use the equations:

To reverse this, clearly:


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:

See also[edit]


  • 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[edit]