# Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 1 Solutions - Ex1.9

- 1 min- Return to index
- Exercise 1.1
- Exercise 1.2
- Exercise 1.3
- Exercise 1.4
- Exercise 1.5
- Exercise 1.6
- Exercise 1.7
- Exercise 1.8
- Exercise 1.9
- Exercise 1.10
- Exercise 1.11
- Exercise 1.12
- Exercise 1.13
- Exercise 1.14

## Exercise 1.9

Compute as far as possible without a calculator. Where appropriate, make use of a smart decomposition of the exponent as shown in the example in Sect. 1.4.1:

- x = 3
^{2}mod 13 - x = 7
^{2}mod 13 - x = 3
^{10}mod 13 - x = 7
^{100}mod 13 - 7
^{x}= 11, mod 13

The last problem is called a discrete logarithm and points to a hard problem which we discuss in Chap. 8. The security of many public-key schemes is based on the hardness of solving the discrete logarithm for large numbers, e.g., with more than 1000 bits.

### Solution

*This solution is verified as correct by the official Solutions for Odd-Numbered Questions manual.*

These can be performed by performing a smaller exponentiation and reducing:

1.

2.

3.

4.

5. Through trial and error, we can discover the value of :