If a cipher has key length shorter than plaintext, then it is not perfectly secure I am trying to verify the statement above. So far I only know that a One-Time-Pad is the only “perfectly secure” cipher. It has a key length which is exactly the same as the plaintext. I think the statement is true but I don't know how to prove it…

Can anyone give me some hints or point me to an according reference or two?
Here's a more "down to earth" example. The following cryptosystem with plaintext space $\mathcal{M} = \{a,b,c,d\}$, keyspace $\mathcal{K} = \{1,2,3,4\}$ and ciphertext space $\mathcal{C} = \{A,B,C,D\}$ has perfect secrecy:

$$\begin{array}{c|c c c c}
&amp; 1 &amp; 2 &amp; 3 &amp; 4 \\
\hline
a &amp; A &amp; B &amp; C &amp; D \\
b &amp; B &amp; C &amp; D &amp; A \\
c &amp; C &amp; D &amp; A &amp; B \\
d &amp; D &amp; A &amp; B &amp; C
\end{array}$$

if the key is chosen uniformly at random independently of the plaintext (the table should be read as saying that, for example, encryption of the plaintext $a$ with the key $3$ yields the ciphertext $C$). The perfect secrecy means that an attacker who obtains a ciphertext has no "hint" about what the plaintext may be: for example if an attacker obtains the ciphertext $C$, it could be the result of encrypting the plaintext $a$ with the key $3$, or the plaintext $b$ with the key $2$, or the plaintext $c$ with the key $1$, or the plaintext $d$ with the key $4$, and all those cases will occur with probability $1/4$. By comparison, the following cryptosystem:

$$\begin{array}{c|c c c c}
&amp; 1 &amp; 2 &amp; 3 &amp; 4 \\
\hline
a &amp; A &amp; A &amp; C &amp; D \\
b &amp; B &amp; C &amp; D &amp; A \\
c &amp; C &amp; D &amp; A &amp; B \\
d &amp; D &amp; B &amp; B &amp; C
\end{array}$$

does not have perfect secrecy because if an attacker obtains the ciphertext $A$, he knows that the plaintext is not $d$, and it is also more likely that the plaintext is $a$ (which is the case with probability $1/2$) than $b$ or $c$ (probability $1/4$ each).

On the other hand, with the following cryptosystem with keyspace $\mathcal{K}' = \{1,2,3\}$:

$$\begin{array}{c|c c c c}
&amp; 1 &amp; 2 &amp; 3 \\
\hline
a &amp; A &amp; B &amp; C \\
b &amp; B &amp; C &amp; D \\
c &amp; C &amp; D &amp; A \\
d &amp; D &amp; A &amp; B
\end{array}$$

an attacker obtaining the ciphertext $A$ knows that the plaintext is not $b$, so this system does not have perfect secrecy either, and it is easily seen that any cryptosystem with fewer keys than plaintexts will have the same problem.