For clarification: The Pseudo-Hadamard transformation is a reversible transformation of a bit string that provides cryptographic diffusion. Splitting a bit string (with bit length of $2n$) into two equally large bit strings $a$ and $b$ with size $n$:
\begin{align}
a' &= a + b & \pmod{2^n}\\
b' &= a + 2b& \pmod{2^n}
\end{align}
Reverse:
\begin{align}
b &=b' - a' & \pmod{2^n}\\
a &= 2a' - b' & \pmod{2^n}
\end{align}
(Source: Wikipedia: Pseudo-Hadamard transform)
What about the most significant bit (the most right bit) of $b$ in $b'$? Multiplication with $2$ and $\mod{2^n}$ cut this off. We lose any possible diffusion of this bit in $b$. That can't be good. Especially SAFER K and SAFER SK are heavily relying on the pseudo-hadamard transformation, see SAFER at Wikipedia.
My first question:
How does this affect the whole security of a cypher with this transformation? Is there a know attack on SAFER which exploits this characteristic?
My second question:
How can we fix this? Is there any public solution for this problem? Maybe we can replace the multiplication? I did try something with a left rotation of 1, but that combination doesn't work. Either this doesn't work in general, or I just don't know the "other secret key" to reverse the transformation with rotation instead of multiplication:
\begin{align} a' &= a + b & \pmod{2^n}\\ b' &= a + rotL(b, 1)& \pmod{2^n} \end{align}
Reverse: (Warning, wrong equation)
\begin{align}
b &= b' - a' & \pmod{2^n}\\
a &= rotL(a', 1) - b' & \pmod{2^n}
\end{align}
