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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 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 ( mod 2 n ) {\displaystyle a'=a+b\,{\pmod {2^{n}}}}
b ′ = a + 2 b ( mod 2 n ) {\displaystyle b'=a+2b\,{\pmod {2^{n}}}}

To reverse this, clearly:

b = b ′ − a ′ ( mod 2 n ) {\displaystyle b=b'-a'\,{\pmod {2^{n}}}}
a = 2 a ′ − b ′ ( mod 2 n ) {\displaystyle a=2a'-b'\,{\pmod {2^{n}}}}

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

a ′ = 2 a + b ( mod 2 n ) {\displaystyle a'=2a+b\,{\pmod {2^{n}}}}
b ′ = a + b ( mod 2 n ) {\displaystyle 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 = [ 2 1 1 1 ] {\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:

H n = [ 2 × H n − 1 H n − 1 H n − 1 H n − 1 ] {\displaystyle H_{n}={\begin{bmatrix}2\times H_{n-1}&H_{n-1}\\H_{n-1}&H_{n-1}\end{bmatrix}}}

For example:

H 2 = [ 4 2 2 1 2 2 1 1 2 1 2 1 1 1 1 1 ] {\displaystyle H_{2}={\begin{bmatrix}4&2&2&1\\2&2&1&1\\2&1&2&1\\1&1&1&1\end{bmatrix}}}

See also

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.
  • Bruce Schneier, John Kelsey, Doug Whiting, David Wagner, Chris Hall, "Twofish: A 128-Bit Block Cipher", 1998.
  • Helger Lipmaa. On Differential Properties of Pseudo-Hadamard Transform and Related Mappings. INDOCRYPT 2002, LNCS 2551, pp 48-61, 2002.
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