Martin Finn-Sell

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Topics in Algebra: Cryptography - Blatt 0

http://www.mat.univie.ac.at/~gagt/crypto2019

Goulnara Arzhantseva

goulnara.arzhantseva@univie.ac.at

Martin Finn-Sell

martin.finn-sell@univie.ac.at

This exercise sheet is intended to provide some warm up number theory questions that encourage us to think computationally. Terms to refresh yourself aree given in bold.

The following exercises are using Euclid’s algorithmfor computing greatest common divisors.

Question 1. Define the heighth(a)of a natural numbera ≥2to be the greatestnsuch that Euclid’s algorithm computesgcd(a, b)innsteps for some natural numberb < a. Show that h(a) =1if and only ifa=2and computeh(a)fora≤8.

Question 2. The Fibonacci numbers1, 1, 2, 3, 5, ...defined byf₁ =f₂ =1andf_n+2=f_n+1+f_n for alln ≥ 1. Show that 0 < f_n < f_n+1 for alln > 1. What happens if we apply Euclid’s algorithm to a consecutive pairf_n, f_n+1of Fibonacci numbers? Show thath(f_n+2)≥n.

The following exercises are aboutprimes numbersand elementaryprimality testing.

Question 3. Supposep > 1andpdivides(p−1)! +1. Then showpis prime.

Question 4. (Fermat) Show that if2^m+1is prime, thenm=2ⁿfor some natural numbern.

Question 5. Describe the “Sieve of Erathosthenes” and use it to calculate all the primesp ≤ 100.

The following are exercises that recall arithmetic in a finite fields and rings (Zp, wherepis prime, orZn, for any natural numbern).

Question 6. State and proveFermat’s little theorem.

Question 7. (Wilson’s Theorem) Show an integer nis prime if and only if (n −1)! ≡ −1 mod n.

Now we considerEuler’s Totient Functionφ

Question 8. Leta, b ∈ N. Adapt your solution for question 6 to prove that ifgcd(a, b) = 1 thena^φ(b)≡1 mod b.

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Question 9. Computeφ(p^e)wherepis prime ande≥1.

Question 10. Recall that an elementa 6= 0 ∈ Zⁿ is a primitive rootif multiplication by a inZ^×n has orderφ(n). Show that ifais a primitive root, thena^k−l ≡ 1 mod nif and only if k≡l mod φ(n).

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