(i) Describe how a stream cypher operates. What is a one-time pad?

A one-time pad is used to send the message $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} y_{7}$ which is encoded as 0101011. By mistake, it is reused to send the message $y_{0} x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ which is encoded as 0100010. Show that $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}$ is one of two possible messages, and find the two possibilities.

(ii) Describe the RSA system associated with a public key $e$, a private key $d$ and the product $N$ of two large primes.

Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack. [You are not asked to give an explicit signature system.]

Explain how to factorise $N$ when $e, d$ and $N$ are known.