Explain how the Euclidian algorithm is used to compute the gcd of two numbers.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. Explain how the Euclidian algorithm is used to compute the gcd of two numbers.

 

2.  Define the Euler function φ(n). Compute φ(6), φ(7), φ(9), φ(12).

 

3. Define the characteristic of a field, give an example of a field with 7, 11 elements. Does there exist a field of 12 elements?

 

4. Let F7 := Z/7Z be the field of 7 elements. Find the following expressions:

(3+6)11,(2+3)8

1. Define affine ciphers, shift ciphers, block ciphers

 

6. Encipher the following messages using the alphabet A of 29 characters and a shift cipher f(x) = x + 3.

1) The strongest survive

2) Is it over yet?
3) That is enough

4) The big day is coming! Be ready!

5) When you feel how depressingly, slowly you climb, it is well to remember, that things take time!

 

7. Using the following matrix

encrypt the following message:

                   ’See you on Monday!’

 

8. Describe the mathematical encryption function for Enigma.

 

9. Describe the algorithm behind RSA

 

10. Name and describe briefly the two compositeness tests

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