Introduction to Cryptography with Coding Theory, 3rd Edition

10.7 Computer Problems

1. Let $p=53047$. Verify that $L_3(8576)=1234$.

2. Let $p=31$. Evaluate $L_3(24)$.

3. Let $p=3989$. Then 2 is a primitive root mod $p$.

   1. Show that $L_2(3925)=2000$ and $L_2(1046)=3000$.

   2. Compute $L_2(3925\cdot 1046)$. (Note: The answer should be less than 3988.)

4. Let $p=1201$.

   1. Show that ${11}^{1200/q}\cancel{\equiv }1(\text{mod }1201)\text{for}q=2\text{,}3\text{,}5$.

   2. Use method of Exercise 54 in Chapter 3 plus the result of part (a) to show that 11 is a primitive root mod 1201.

   3. Use the Pohlig-Hellman algorithm to find $L_{11}(2)$.

   4. Use the Baby Step, Giant Step method to find $L_{11}(2)$.