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problemes 2
March 1, 2025
Amina Aasifar El Ouahabi - 1672430
Criptografia i Seguretat
Activity Cryptographical Mathematical Background
In these set of exercises you are required to implement the specified code described in the questions.
- Make a program that for a given number n tells wether it is prime or not looking for factors in [1 ∶ 𝑛]
[1]: def is_prime(n): for num in range( 1 ,n+ 1 ): res = n% num #print(res) if res != 0 : continue else : if num == 1 or num == n: continue else : return "It's not prime :(", False
return "It's prime!! :)", True
[2]: print(is_prime( 48 ))
("It's not prime :(", False)
- Make a program that for a given number n tells wether it is prime or not looking for factors in [1 ∶ 𝑛/2].
[3]: def is_prime2(n): fi = n% 2 if fi != 0 : print("Odd number!") for num in range( 1 , n// 2 ): res = n% num #print(res) if res != 0 :
continue else : if num == 1 or num == n/ 2 : continue else : return "It's not prime :("
return "It's prime!! :)"
[4]: def is_prime2(n): if n <= 1 : return "It's not prime :(" if n == 2 : return "It's prime!! :)" if n% 2 == 0: return "It's not prime :(" for num in range( 3 , n// 2 + 1 , n// 2 ): #comprovar els números imparells if n% num == 0: return "It's not prime :(", False return "It's prime!! :)", True
#// -> entero / -> flotante
[5]: print(is_prime2( 3 ))
("It's prime!! :)", True)
- Make a program that for a given number n tells wether it is prime or not using Fermat’s theorem.
[6]: from random import randint
#k --> al número d'iteracions per verificar la primalitat
def fermat(n,k): if n <= 3 : return (str(n)+' is prime! :)', True ) for i in range(k): a = randint( 2 , n- 2 ) if (a**(n- 1 ))% n != 1: return (str(n)+' is not prime :(', False ) return (str(n)+' is prime! :)', True )
#probability = 1-(1/2)**k #return str(n)+' is prime with a probability of'+ str(probability)
[7]: print(fermat( 561 , 10 ))
60 sec completed! Results: [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109]
[10]: #tercer programa -> primer per fermrat import time
ll_prime2 = [] times2 = []
initial_time = time.time()
n = 1 while time.time() - initial_time < 60 : temps_actual = time.time() prime = fermat(n, 10 ) if prime[- 1 ] == True : #the last value ll_prime2.append(n) elapsed_time = time.time() - temps_actual times2.append(elapsed_time) time.sleep( 2 ) n+= 1
print("60 sec completed! Results:", ll_prime)
60 sec completed! Results: [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109]
- Draw a graph (x-axis number, y-axis time) using both programs to calculate the time spent for every number.
[11]: import matplotlib.pyplot as plt
#primer algoritme -> primer per n plt.bar(ll_prime, times, color='blue') plt.xlabel("Nombres primers trobats") plt.ylabel("Temps emprat (s)") plt.title("Temps de càlcul de cada número primer (primer algoritme)") plt.xticks(rotation= 45 ) plt.grid(axis='y', linestyle='--', alpha=0.7) plt.show()
[12]: import matplotlib.pyplot as plt
#primer algoritme -> primer per n/ plt.bar(ll_prime1, times1, color='blue') plt.xlabel("Nombres primers trobats") plt.ylabel("Temps emprat (s)") plt.title("Temps de càlcul de cada número primer (segon algoritme)") plt.xticks(rotation= 45 ) plt.grid(axis='y', linestyle='--', alpha=0.7) plt.show()
