Introduction to Cryptography with Coding Theory, 3rd Edition

11.6 Exercises

1. Let $p$ be a prime and let $\alpha$ be an integer with $p\nmid \alpha$. Let $h(x)\equiv {\alpha }^x(\text{m}\text{o}\text{d}p)$. Explain why $h(x)$ is not a good cryptographic hash function.

2. 1. Alice claims that she knows who will win the next World Cup. She takes the name of the team, $T$, and encrypts it with a one-time pad $K$, and sends $C=T\oplus K$ to Bob. After the World Cup is finished, Alice reveals $K$, and Bob computes $T=C\oplus K$ to determine Alice’s guess. Why should Bob not believe that Alice actually guessed the correct team, even if $T=C\oplus K$ is correct?

   2. To keep Alice from changing $K$, Bob requires Alice to send not only $C=T\oplus K$ but also $H(K)$, where $H$ is a good cryptographic hash function. How does the use of the hash function convince Bob that Alice is not changing $K$?

   3. In the procedure in (b), ...