Whenever you're trying to attack a scheme that is [algebraically] relatively simple like this one, a sensible first step is to write out everything you know. Now, considering the information you've been given, try and substitute things into oneanother, and see where this leads you.

Let $(m,c)$ be the first 1024 bits of the plaintext-ciphertext pair.

Now, some pointers

1. Given this is a stream cipher, how is $c$ created? In particular, $c=g(m,r)$ for some function $g$. What is this function? Substituting in the data you know, what do you learn?
2. Write out $F(x)$ and $F(a \cdot x)$. If I tell you $F(x)=y$, calculate $F(a\cdot x)$
3. Now, consider everything you know. Put it together and finish the problem yourself.

Whenever you're trying to attack a scheme that is [algebraically] relatively simple like this one, a sensible first step is to write out everything you know. Now, considering the information you've been given, try and substitute things into oneanother, and see where this leads you.

Let $(m,c)$ be the first 1024 bits of the plaintext-ciphertext pair.

Now, some pointers

1. Given this is a stream cipher, how is $c$ created? In particular, $c=g(m,K)$ for some function $g$. What is this function? Substituting in the data you know, what do you learn?
2. Write out $F(x)$ and $F(a \cdot x)$. If I tell you $F(x)=y$, calculate $F(a\cdot x)$
3. Now, consider everything you know. Put it together and finish the problem yourself.