What is the logical relationship between "a brute-force attack is impossible" and "a cipher is secure"? Which of $=, \neq, \Rightarrow, \Leftarrow, \Leftrightarrow, \not\Rightarrow, \not\Leftarrow, \not\Leftrightarrow$ fit between them?

Secure implies brute force is impossible, but not the other way round. So the correct signs are $\neq$, $\Leftarrow$, $\not\Rightarrow$ and $\not\Leftrightarrow$.

Read the statement as "brute force impossible" ? "cipher secure":

• $\Leftarrow$ (secure ⇒ brute force impossible): a secure cipher must at least resist exhaustive key search, otherwise it would not be secure.
• $\not\Rightarrow$ (brute force impossible ⇏ secure): a huge key space says nothing about other weaknesses. 104-bit RC4 is far beyond brute force, yet it can be broken on home equipment.
• $\not\Leftrightarrow$ and $\neq$ follow, because the implication holds in one direction only.

The lesson: a large key is a necessary condition for security, never a sufficient one. The size of the brute-force effort is no argument for an algorithm's security.