  In the generalized Russian cards problem, Alice, Bob and Cath draw $a$, $b$
and $c$ cards, respectively, from a deck of size $a+b+c$. Alice and Bob must
then communicate their entire hand to each other, without Cath learning the
owner of a single card she does not hold. Unlike many traditional problems in
cryptography, however, they are not allowed to encode or hide the messages they
exchange from Cath. The problem is then to find methods through which they can
achieve this. We propose a general four-step solution based on finite vector
spaces, and call it the "colouring protocol", as it involves colourings of
lines.
  Our main results show that the colouring protocol may be used to solve the
generalized Russian cards problem in cases where $a$ is a power of a prime,
$c=O(a^2)$ and $b=O(c^2)$. This improves substantially on the set of parameters
for which solutions are known to exist; in particular, it had not been shown
previously that the problem could be solved in cases where the eavesdropper has
more cards than one of the communicating players.
